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AI、哪些還不成熟。",[26,27,28],"現況盤點：梳理日常營運中最耗時、最依賴人力的環節","痛點診斷：判斷哪些工作現在就能交給 AI，哪些建議先保留人力","可行性評估：依你的業務特性與預算，篩選適合的模型與工具","《企業 AI 導入建議報告書》（PDF）","含 1–2 個「開發成本最低、成效最明顯」的優先項目、技術棧建議、預期節省的人力時間與下一步","訪談 60 分鐘，報告書 3 個工作天內交付",{"title":33,"tagline":34,"summary":35,"points":36,"deliverable":40,"detail":41,"duration":42},"AI Readiness Audit","Look before you spend","A 60-minute remote audit of the workflows that eat the most time and people-power — and an honest answer on what AI can take over today, and what it cannot.",[37,38,39],"Audit the daily workflows that consume the most time and headcount","Triage which tasks AI can own today, and which should stay human","Shortlist the models and tools that fit your budget and constraints","AI Adoption Recommendation Report (PDF)","One or two highest-ROI entry points, a recommended tech stack, expected hours saved, and the concrete next step","60-minute call, report delivered within 3 business 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的加權和，權重是 ",[125,1925,1927,1945],{"className":1926},[128],[125,1928,1930],{"className":1929},[132],[134,1931,1932],{"xmlns":136},[138,1933,1934,1942],{},[141,1935,1936],{},[393,1937,1938,1940],{},[144,1939,975],{},[144,1941,891],{},[170,1943,1944],{"encoding":172},"q_i",[125,1946,1948],{"className":1947,"ariaHidden":178},[177],[125,1949,1951,1955],{"className":1950},[182],[125,1952],{"className":1953,"style":1954},[186],"height:0.625em;vertical-align:-0.1944em;",[125,1956,1958,1961],{"className":1957},[191],[125,1959,975],{"className":1960,"style":501},[191,192],[125,1962,1964],{"className":1963},[222],[125,1965,1967,1987],{"className":1966},[226,478],[125,1968,1970,1984],{"className":1969},[230],[125,1971,1973],{"className":1972,"style":1082},[234],[125,1974,1975,1978],{"style":1809},[125,1976],{"className":1977,"style":242},[241],[125,1979,1981],{"className":1980},[246,247,248,249],[125,1982,891],{"className":1983},[191,192,249],[125,1985,506],{"className":1986},[505],[125,1988,1990],{"className":1989},[230],[125,1991,1993],{"className":1992,"style":1104},[234],[125,1994],{}," 與 ",[125,1997,1999,2017],{"className":1998},[128],[125,2000,2002],{"className":2001},[132],[134,2003,2004],{"xmlns":136},[138,2005,2006,2014],{},[141,2007,2008],{},[393,2009,2010,2012],{},[144,2011,984],{},[144,2013,946],{},[170,2015,2016],{"encoding":172},"k_j",[125,2018,2020],{"className":2019,"ariaHidden":178},[177],[125,2021,2023,2027],{"className":2022},[182],[125,2024],{"className":2025,"style":2026},[186],"height:0.9805em;vertical-align:-0.2861em;",[125,2028,2030,2033],{"className":2029},[191],[125,2031,984],{"className":2032,"style":1410},[191,192],[125,2034,2036],{"className":2035},[222],[125,2037,2039,2060],{"className":2038},[226,478],[125,2040,2042,2057],{"className":2041},[230],[125,2043,2045],{"className":2044,"style":1082},[234],[125,2046,2048,2051],{"style":2047},"top:-2.55em;margin-left:-0.0315em;margin-right:0.05em;",[125,2049],{"className":2050,"style":242},[241],[125,2052,2054],{"className":2053},[246,247,248,249],[125,2055,946],{"className":2056,"style":1189},[191,192,249],[125,2058,506],{"className":2059},[505],[125,2061,2063],{"className":2062},[230],[125,2064,2066],{"className":2065,"style":1828},[234],[125,2067],{}," 的相似度；左邊的分母只是把權重歸一化，讓它們加起來等於 1。",[112,2070,2071,2072,2284],{},"如果把 ",[125,2073,2075,2129],{"className":2074},[128],[125,2076,2078],{"className":2077},[132],[134,2079,2080],{"xmlns":136},[138,2081,2082,2126],{},[141,2083,2084,2087,2089,2091,2093,2095,2097,2099,2101,2103,2105],{},[144,2085,2086],{},"κ",[148,2088,544],{"stretchy":543},[144,2090,975],{},[148,2092,549],{"separator":178},[144,2094,984],{},[148,2096,558],{"stretchy":543},[148,2098,561],{},[144,2100,957],{},[148,2102,540],{},[568,2104,570],{},[141,2106,2107,2109,2115,2117,2120,2124],{},[148,2108,544],{"fence":178},[152,2110,2111,2113],{},[144,2112,975],{},[144,2114,588],{"mathvariant":536},[144,2116,984],{},[144,2118,2119],{"mathvariant":536},"\u002F",[590,2121,2122],{},[144,2123,168],{},[148,2125,558],{"fence":178},[170,2127,2128],{"encoding":172},"\\kappa(q, k) = \\exp\\!\\left(q^{\\top}k \u002F \\sqrt{d}\\right)",[125,2130,2132,2168],{"className":2131,"ariaHidden":178},[177],[125,2133,2135,2138,2141,2144,2147,2150,2153,2156,2159,2162,2165],{"className":2134},[182],[125,2136],{"className":2137,"style":613},[186],[125,2139,2086],{"className":2140},[191,192],[125,2142,544],{"className":2143},[625],[125,2145,975],{"className":2146,"style":501},[191,192],[125,2148,549],{"className":2149},[632],[125,2151],{"className":2152,"style":636},[196],[125,2154,984],{"className":2155,"style":1410},[191,192],[125,2157,558],{"className":2158},[652],[125,2160],{"className":2161,"style":197},[196],[125,2163,561],{"className":2164},[201],[125,2166],{"className":2167,"style":197},[196],[125,2169,2171,2175,2178,2181,2184],{"className":2170},[182],[125,2172],{"className":2173,"style":2174},[186],"height:1.8em;vertical-align:-0.65em;",[125,2176,957],{"className":2177},[617],[125,2179],{"className":2180,"style":678},[196],[125,2182],{"className":2183,"style":636},[196],[125,2185,2187,2193,2225,2228,2231,2278],{"className":2186},[685],[125,2188,2190],{"className":2189,"style":690},[625,689],[125,2191,544],{"className":2192},[694,1244],[125,2194,2196,2199],{"className":2195},[191],[125,2197,975],{"className":2198,"style":501},[191,192],[125,2200,2202],{"className":2201},[222],[125,2203,2205],{"className":2204},[226],[125,2206,2208],{"className":2207},[230],[125,2209,2211],{"className":2210,"style":211},[234],[125,2212,2213,2216],{"style":237},[125,2214],{"className":2215,"style":242},[241],[125,2217,2219],{"className":2218},[246,247,248,249],[125,2220,2222],{"className":2221},[191,249],[125,2223,588],{"className":2224},[191,249],[125,2226,984],{"className":2227,"style":1410},[191,192],[125,2229,2119],{"className":2230},[191],[125,2232,2234],{"className":2233},[191,728],[125,2235,2237,2270],{"className":2236},[226,478],[125,2238,2240,2267],{"className":2239},[230],[125,2241,2243,2255],{"className":2242,"style":738},[234],[125,2244,2246,2249],{"className":2245,"style":743},[742],[125,2247],{"className":2248,"style":721},[241],[125,2250,2252],{"className":2251,"style":750},[191],[125,2253,168],{"className":2254},[191,192],[125,2256,2257,2260],{"style":756},[125,2258],{"className":2259,"style":721},[241],[125,2261,2263],{"className":2262,"style":764},[763],[766,2264,2265],{"xmlns":768,"width":769,"height":770,"viewBox":771,"preserveAspectRatio":772},[774,2266],{"d":776},[125,2268,506],{"className":2269},[505],[125,2271,2273],{"className":2272},[230],[125,2274,2276],{"className":2275,"style":786},[234],[125,2277],{},[125,2279,2281],{"className":2280,"style":690},[652,689],[125,2282,558],{"className":2283},[694,1244]," 當成一個核函數，上面這一坨就是",[125,2286,2288],{"className":2287},[520],[125,2289,2291,2381],{"className":2290},[128],[125,2292,2294],{"className":2293},[132],[134,2295,2296],{"xmlns":136,"display":529},[138,2297,2298,2378],{},[141,2299,2300,2306,2308,2376],{},[393,2301,2302,2304],{},[144,2303,927],{},[144,2305,891],{},[148,2307,561],{},[576,2309,2310,2348],{},[141,2311,2312,2319,2321,2323,2329,2331,2337,2339,2342],{},[2313,2314,2315,2317],"munder",{},[148,2316,941],{},[144,2318,946],{},[144,2320,2086],{},[148,2322,544],{"stretchy":543},[393,2324,2325,2327],{},[144,2326,975],{},[144,2328,891],{},[148,2330,549],{"separator":178},[393,2332,2333,2335],{},[144,2334,984],{},[144,2336,946],{},[148,2338,558],{"stretchy":543},[568,2340,2341],{}," ",[393,2343,2344,2346],{},[144,2345,399],{},[144,2347,946],{},[141,2349,2350,2356,2358,2360,2366,2368,2374],{},[2313,2351,2352,2354],{},[148,2353,941],{},[144,2355,946],{},[144,2357,2086],{},[148,2359,544],{"stretchy":543},[393,2361,2362,2364],{},[144,2363,975],{},[144,2365,891],{},[148,2367,549],{"separator":178},[393,2369,2370,2372],{},[144,2371,984],{},[144,2373,946],{},[148,2375,558],{"stretchy":543},[148,2377,549],{"separator":178},[170,2379,2380],{"encoding":172},"o_i = \\frac{\\sum_j \\kappa(q_i, k_j)\\, v_j}{\\sum_j \\kappa(q_i, k_j)} ,",[125,2382,2384,2439],{"className":2383,"ariaHidden":178},[177],[125,2385,2387,2390,2430,2433,2436],{"className":2386},[182],[125,2388],{"className":2389,"style":1063},[186],[125,2391,2393,2396],{"className":2392},[191],[125,2394,927],{"className":2395},[191,192],[125,2397,2399],{"className":2398},[222],[125,2400,2402,2422],{"className":2401},[226,478],[125,2403,2405,2419],{"className":2404},[230],[125,2406,2408],{"className":2407,"style":1082},[234],[125,2409,2410,2413],{"style":1085},[125,2411],{"className":2412,"style":242},[241],[125,2414,2416],{"className":2415},[246,247,248,249],[125,2417,891],{"className":2418},[191,192,249],[125,2420,506],{"className":2421},[505],[125,2423,2425],{"className":2424},[230],[125,2426,2428],{"className":2427,"style":1104},[234],[125,2429],{},[125,2431],{"className":2432,"style":197},[196],[125,2434,561],{"className":2435},[201],[125,2437],{"className":2438,"style":197},[196],[125,2440,2442,2446,2826],{"className":2441},[182],[125,2443],{"className":2444,"style":2445},[186],"height:2.6976em;vertical-align:-1.1218em;",[125,2447,2449,2452,2823],{"className":2448},[191],[125,2450],{"className":2451},[625,701],[125,2453,2455],{"className":2454},[576],[125,2456,2458,2814],{"className":2457},[226,478],[125,2459,2461,2811],{"className":2460},[230],[125,2462,2465,2613,2621],{"className":2463,"style":2464},[234],"height:1.5758em;",[125,2466,2468,2471],{"style":2467},"top:-2.314em;",[125,2469],{"className":2470,"style":721},[241],[125,2472,2474,2515,2518,2521,2524,2564,2567,2570,2610],{"className":2473},[191],[125,2475,2477,2480],{"className":2476},[617],[125,2478,941],{"className":2479,"style":1160},[617,1158,1159],[125,2481,2483],{"className":2482},[222],[125,2484,2486,2507],{"className":2485},[226,478],[125,2487,2489,2504],{"className":2488},[230],[125,2490,2493],{"className":2491,"style":2492},[234],"height:0.162em;",[125,2494,2495,2498],{"style":1176},[125,2496],{"className":2497,"style":242},[241],[125,2499,2501],{"className":2500},[246,247,248,249],[125,2502,946],{"className":2503,"style":1189},[191,192,249],[125,2505,506],{"className":2506},[505],[125,2508,2510],{"className":2509},[230],[125,2511,2513],{"className":2512,"style":1220},[234],[125,2514],{},[125,2516],{"className":2517,"style":636},[196],[125,2519,2086],{"className":2520},[191,192],[125,2522,544],{"className":2523},[625],[125,2525,2527,2530],{"className":2526},[191],[125,2528,975],{"className":2529,"style":501},[191,192],[125,2531,2533],{"className":2532},[222],[125,2534,2536,2556],{"className":2535},[226,478],[125,2537,2539,2553],{"className":2538},[230],[125,2540,2542],{"className":2541,"style":1082},[234],[125,2543,2544,2547],{"style":1809},[125,2545],{"className":2546,"style":242},[241],[125,2548,2550],{"className":2549},[246,247,248,249],[125,2551,891],{"className":2552},[191,192,249],[125,2554,506],{"className":2555},[505],[125,2557,2559],{"className":2558},[230],[125,2560,2562],{"className":2561,"style":1104},[234],[125,2563],{},[125,2565,549],{"className":2566},[632],[125,2568],{"className":2569,"style":636},[196],[125,2571,2573,2576],{"className":2572},[191],[125,2574,984],{"className":2575,"style":1410},[191,192],[125,2577,2579],{"className":2578},[222],[125,2580,2582,2602],{"className":2581},[226,478],[125,2583,2585,2599],{"className":2584},[230],[125,2586,2588],{"className":2587,"style":1082},[234],[125,2589,2590,2593],{"style":2047},[125,2591],{"className":2592,"style":242},[241],[125,2594,2596],{"className":2595},[246,247,248,249],[125,2597,946],{"className":2598,"style":1189},[191,192,249],[125,2600,506],{"className":2601},[505],[125,2603,2605],{"className":2604},[230],[125,2606,2608],{"className":2607,"style":1828},[234],[125,2609],{},[125,2611,558],{"className":2612},[652],[125,2614,2615,2618],{"style":791},[125,2616],{"className":2617,"style":721},[241],[125,2619],{"className":2620,"style":799},[798],[125,2622,2624,2627],{"style":2623},"top:-3.8258em;",[125,2625],{"className":2626,"style":721},[241],[125,2628,2630,2670,2673,2676,2679,2719,2722,2725,2765,2768,2771],{"className":2629},[191],[125,2631,2633,2636],{"className":2632},[617],[125,2634,941],{"className":2635,"style":1160},[617,1158,1159],[125,2637,2639],{"className":2638},[222],[125,2640,2642,2662],{"className":2641},[226,478],[125,2643,2645,2659],{"className":2644},[230],[125,2646,2648],{"className":2647,"style":2492},[234],[125,2649,2650,2653],{"style":1176},[125,2651],{"className":2652,"style":242},[241],[125,2654,2656],{"className":2655},[246,247,248,249],[125,2657,946],{"className":2658,"style":1189},[191,192,249],[125,2660,506],{"className":2661},[505],[125,2663,2665],{"className":2664},[230],[125,2666,2668],{"className":2667,"style":1220},[234],[125,2669],{},[125,2671],{"className":2672,"style":636},[196],[125,2674,2086],{"className":2675},[191,192],[125,2677,544],{"className":2678},[625],[125,2680,2682,2685],{"className":2681},[191],[125,2683,975],{"className":2684,"style":501},[191,192],[125,2686,2688],{"className":2687},[222],[125,2689,2691,2711],{"className":2690},[226,478],[125,2692,2694,2708],{"className":2693},[230],[125,2695,2697],{"className":2696,"style":1082},[234],[125,2698,2699,2702],{"style":1809},[125,2700],{"className":2701,"style":242},[241],[125,2703,2705],{"className":2704},[246,247,248,249],[125,2706,891],{"className":2707},[191,192,249],[125,2709,506],{"className":2710},[505],[125,2712,2714],{"className":2713},[230],[125,2715,2717],{"className":2716,"style":1104},[234],[125,2718],{},[125,2720,549],{"className":2721},[632],[125,2723],{"className":2724,"style":636},[196],[125,2726,2728,2731],{"className":2727},[191],[125,2729,984],{"className":2730,"style":1410},[191,192],[125,2732,2734],{"className":2733},[222],[125,2735,2737,2757],{"className":2736},[226,478],[125,2738,2740,2754],{"className":2739},[230],[125,2741,2743],{"className":2742,"style":1082},[234],[125,2744,2745,2748],{"style":2047},[125,2746],{"className":2747,"style":242},[241],[125,2749,2751],{"className":2750},[246,247,248,249],[125,2752,946],{"className":2753,"style":1189},[191,192,249],[125,2755,506],{"className":2756},[505],[125,2758,2760],{"className":2759},[230],[125,2761,2763],{"className":2762,"style":1828},[234],[125,2764],{},[125,2766,558],{"className":2767},[652],[125,2769],{"className":2770,"style":636},[196],[125,2772,2774,2777],{"className":2773},[191],[125,2775,399],{"className":2776,"style":501},[191,192],[125,2778,2780],{"className":2779},[222],[125,2781,2783,2803],{"className":2782},[226,478],[125,2784,2786,2800],{"className":2785},[230],[125,2787,2789],{"className":2788,"style":1082},[234],[125,2790,2791,2794],{"style":1809},[125,2792],{"className":2793,"style":242},[241],[125,2795,2797],{"className":2796},[246,247,248,249],[125,2798,946],{"className":2799,"style":1189},[191,192,249],[125,2801,506],{"className":2802},[505],[125,2804,2806],{"className":2805},[230],[125,2807,2809],{"className":2808,"style":1828},[234],[125,2810],{},[125,2812,506],{"className":2813},[505],[125,2815,2817],{"className":2816},[230],[125,2818,2821],{"className":2819,"style":2820},[234],"height:1.1218em;",[125,2822],{},[125,2824],{"className":2825},[652,701],[125,2827,549],{"className":2828},[632],[112,2830,2831],{},"——非參數回歸裡的 Nadaraya–Watson 估計。一模一樣。",[112,2833,2834],{},"差別只在於：這裡的「樣本」是模型自己學出來的，不是給定的資料集。",[108,2836,2837],{"id":2837},"知道這件事有什麼用",[112,2839,2840,2841,2845],{},"不是為了數學上的優雅。是因為它會告訴你",[2842,2843,2844],"strong",{},"哪裡可以砍","。",[112,2847,2848,2849,2879,2880,3030],{},"如果 ",[125,2850,2852,2866],{"className":2851},[128],[125,2853,2855],{"className":2854},[132],[134,2856,2857],{"xmlns":136},[138,2858,2859,2863],{},[141,2860,2861],{},[144,2862,2086],{},[170,2864,2865],{"encoding":172},"\\kappa",[125,2867,2869],{"className":2868,"ariaHidden":178},[177],[125,2870,2872,2876],{"className":2871},[182],[125,2873],{"className":2874,"style":2875},[186],"height:0.4306em;",[125,2877,2086],{"className":2878},[191,192]," 可以寫成有限維特徵映射的內積，也就是 ",[125,2881,2883,2931],{"className":2882},[128],[125,2884,2886],{"className":2885},[132],[134,2887,2888],{"xmlns":136},[138,2889,2890,2928],{},[141,2891,2892,2894,2896,2898,2900,2902,2904,2907,2910,2912,2914,2920,2922,2924,2926],{},[144,2893,2086],{},[148,2895,544],{"stretchy":543},[144,2897,975],{},[148,2899,549],{"separator":178},[144,2901,984],{},[148,2903,558],{"stretchy":543},[148,2905,2906],{},"≈",[144,2908,2909],{},"ϕ",[148,2911,544],{"stretchy":543},[144,2913,975],{},[152,2915,2916,2918],{},[148,2917,558],{"stretchy":543},[144,2919,588],{"mathvariant":536},[144,2921,2909],{},[148,2923,544],{"stretchy":543},[144,2925,984],{},[148,2927,558],{"stretchy":543},[170,2929,2930],{"encoding":172},"\\kappa(q,k) \\approx \\phi(q)^{\\top}\\phi(k)",[125,2932,2934,2970],{"className":2933,"ariaHidden":178},[177],[125,2935,2937,2940,2943,2946,2949,2952,2955,2958,2961,2964,2967],{"className":2936},[182],[125,2938],{"className":2939,"style":613},[186],[125,2941,2086],{"className":2942},[191,192],[125,2944,544],{"className":2945},[625],[125,2947,975],{"className":2948,"style":501},[191,192],[125,2950,549],{"className":2951},[632],[125,2953],{"className":2954,"style":636},[196],[125,2956,984],{"className":2957,"style":1410},[191,192],[125,2959,558],{"className":2960},[652],[125,2962],{"className":2963,"style":197},[196],[125,2965,2906],{"className":2966},[201],[125,2968],{"className":2969,"style":197},[196],[125,2971,2973,2977,2980,2983,2986,3018,3021,3024,3027],{"className":2972},[182],[125,2974],{"className":2975,"style":2976},[186],"height:1.0991em;vertical-align:-0.25em;",[125,2978,2909],{"className":2979},[191,192],[125,2981,544],{"className":2982},[625],[125,2984,975],{"className":2985,"style":501},[191,192],[125,2987,2989,2992],{"className":2988},[652],[125,2990,558],{"className":2991},[652],[125,2993,2995],{"className":2994},[222],[125,2996,2998],{"className":2997},[226],[125,2999,3001],{"className":3000},[230],[125,3002,3004],{"className":3003,"style":211},[234],[125,3005,3006,3009],{"style":237},[125,3007],{"className":3008,"style":242},[241],[125,3010,3012],{"className":3011},[246,247,248,249],[125,3013,3015],{"className":3014},[191,249],[125,3016,588],{"className":3017},[191,249],[125,3019,2909],{"className":3020},[191,192],[125,3022,544],{"className":3023},[625],[125,3025,984],{"className":3026,"style":1410},[191,192],[125,3028,558],{"className":3029},[652],"，那麼",[125,3032,3034],{"className":3033},[520],[125,3035,3037,3141],{"className":3036},[128],[125,3038,3040],{"className":3039},[132],[134,3041,3042],{"xmlns":136,"display":529},[138,3043,3044,3138],{},[141,3045,3046,3052,3054,3136],{},[393,3047,3048,3050],{},[144,3049,927],{},[144,3051,891],{},[148,3053,2906],{},[576,3055,3056,3100],{},[141,3057,3058,3060,3062,3068,3074,3080,3082,3084,3090,3092],{},[144,3059,2909],{},[148,3061,544],{"stretchy":543},[393,3063,3064,3066],{},[144,3065,975],{},[144,3067,891],{},[152,3069,3070,3072],{},[148,3071,558],{"stretchy":543},[144,3073,588],{"mathvariant":536},[2313,3075,3076,3078],{},[148,3077,941],{},[144,3079,946],{},[144,3081,2909],{},[148,3083,544],{"stretchy":543},[393,3085,3086,3088],{},[144,3087,984],{},[144,3089,946],{},[148,3091,558],{"stretchy":543},[971,3093,3094,3096,3098],{},[144,3095,399],{},[144,3097,946],{},[144,3099,588],{"mathvariant":536},[141,3101,3102,3104,3106,3112,3118,3124,3126,3128,3134],{},[144,3103,2909],{},[148,3105,544],{"stretchy":543},[393,3107,3108,3110],{},[144,3109,975],{},[144,3111,891],{},[152,3113,3114,3116],{},[148,3115,558],{"stretchy":543},[144,3117,588],{"mathvariant":536},[2313,3119,3120,3122],{},[148,3121,941],{},[144,3123,946],{},[144,3125,2909],{},[148,3127,544],{"stretchy":543},[393,3129,3130,3132],{},[144,3131,984],{},[144,3133,946],{},[148,3135,558],{"stretchy":543},[144,3137,600],{"mathvariant":536},[170,3139,3140],{"encoding":172},"o_i \\approx \\frac{\\phi(q_i)^{\\top} \\sum_j \\phi(k_j) v_j^{\\top}}{\\phi(q_i)^{\\top} \\sum_j \\phi(k_j)} .",[125,3142,3144,3200],{"className":3143,"ariaHidden":178},[177],[125,3145,3147,3151,3191,3194,3197],{"className":3146},[182],[125,3148],{"className":3149,"style":3150},[186],"height:0.6331em;vertical-align:-0.15em;",[125,3152,3154,3157],{"className":3153},[191],[125,3155,927],{"className":3156},[191,192],[125,3158,3160],{"className":3159},[222],[125,3161,3163,3183],{"className":3162},[226,478],[125,3164,3166,3180],{"className":3165},[230],[125,3167,3169],{"className":3168,"style":1082},[234],[125,3170,3171,3174],{"style":1085},[125,3172],{"className":3173,"style":242},[241],[125,3175,3177],{"className":3176},[246,247,248,249],[125,3178,891],{"className":3179},[191,192,249],[125,3181,506],{"className":3182},[505],[125,3184,3186],{"className":3185},[230],[125,3187,3189],{"className":3188,"style":1104},[234],[125,3190],{},[125,3192],{"className":3193,"style":197},[196],[125,3195,2906],{"className":3196},[201],[125,3198],{"className":3199,"style":197},[196],[125,3201,3203,3207,3668],{"className":3202},[182],[125,3204],{"className":3205,"style":3206},[186],"height:2.7967em;vertical-align:-1.1218em;",[125,3208,3210,3213,3665],{"className":3209},[191],[125,3211],{"className":3212},[625,701],[125,3214,3216],{"className":3215},[576],[125,3217,3219,3657],{"className":3218},[226,478],[125,3220,3222,3654],{"className":3221},[230],[125,3223,3226,3409,3417],{"className":3224,"style":3225},[234],"height:1.6749em;",[125,3227,3228,3231],{"style":2467},[125,3229],{"className":3230,"style":721},[241],[125,3232,3234,3237,3240,3280,3314,3317,3357,3360,3363,3366,3406],{"className":3233},[191],[125,3235,2909],{"className":3236},[191,192],[125,3238,544],{"className":3239},[625],[125,3241,3243,3246],{"className":3242},[191],[125,3244,975],{"className":3245,"style":501},[191,192],[125,3247,3249],{"className":3248},[222],[125,3250,3252,3272],{"className":3251},[226,478],[125,3253,3255,3269],{"className":3254},[230],[125,3256,3258],{"className":3257,"style":1082},[234],[125,3259,3260,3263],{"style":1809},[125,3261],{"className":3262,"style":242},[241],[125,3264,3266],{"className":3265},[246,247,248,249],[125,3267,891],{"className":3268},[191,192,249],[125,3270,506],{"className":3271},[505],[125,3273,3275],{"className":3274},[230],[125,3276,3278],{"className":3277,"style":1104},[234],[125,3279],{},[125,3281,3283,3286],{"className":3282},[652],[125,3284,558],{"className":3285},[652],[125,3287,3289],{"className":3288},[222],[125,3290,3292],{"className":3291},[226],[125,3293,3295],{"className":3294},[230],[125,3296,3299],{"className":3297,"style":3298},[234],"height:0.7751em;",[125,3300,3302,3305],{"style":3301},"top:-2.989em;margin-right:0.05em;",[125,3303],{"className":3304,"style":242},[241],[125,3306,3308],{"className":3307},[246,247,248,249],[125,3309,3311],{"className":3310},[191,249],[125,3312,588],{"className":3313},[191,249],[125,3315],{"className":3316,"style":636},[196],[125,3318,3320,3323],{"className":3319},[617],[125,3321,941],{"className":3322,"style":1160},[617,1158,1159],[125,3324,3326],{"className":3325},[222],[125,3327,3329,3349],{"className":3328},[226,478],[125,3330,3332,3346],{"className":3331},[230],[125,3333,3335],{"className":3334,"style":2492},[234],[125,3336,3337,3340],{"style":1176},[125,3338],{"className":3339,"style":242},[241],[125,3341,3343],{"className":3342},[246,247,248,249],[125,3344,946],{"className":3345,"style":1189},[191,192,249],[125,3347,506],{"className":3348},[505],[125,3350,3352],{"className":3351},[230],[125,3353,3355],{"className":3354,"style":1220},[234],[125,3356],{},[125,3358],{"className":3359,"style":636},[196],[125,3361,2909],{"className":3362},[191,192],[125,3364,544],{"className":3365},[625],[125,3367,3369,3372],{"className":3368},[191],[125,3370,984],{"className":3371,"style":1410},[191,192],[125,3373,3375],{"className":3374},[222],[125,3376,3378,3398],{"className":3377},[226,478],[125,3379,3381,3395],{"className":3380},[230],[125,3382,3384],{"className":3383,"style":1082},[234],[125,3385,3386,3389],{"style":2047},[125,3387],{"className":3388,"style":242},[241],[125,3390,3392],{"className":3391},[246,247,248,249],[125,3393,946],{"className":3394,"style":1189},[191,192,249],[125,3396,506],{"className":3397},[505],[125,3399,3401],{"className":3400},[230],[125,3402,3404],{"className":3403,"style":1828},[234],[125,3405],{},[125,3407,558],{"className":3408},[652],[125,3410,3411,3414],{"style":791},[125,3412],{"className":3413,"style":721},[241],[125,3415],{"className":3416,"style":799},[798],[125,3418,3419,3422],{"style":2623},[125,3420],{"className":3421,"style":721},[241],[125,3423,3425,3428,3431,3471,3503,3506,3546,3549,3552,3555,3595,3598],{"className":3424},[191],[125,3426,2909],{"className":3427},[191,192],[125,3429,544],{"className":3430},[625],[125,3432,3434,3437],{"className":3433},[191],[125,3435,975],{"className":3436,"style":501},[191,192],[125,3438,3440],{"className":3439},[222],[125,3441,3443,3463],{"className":3442},[226,478],[125,3444,3446,3460],{"className":3445},[230],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",[125,3675,3677,3700],{"className":3676},[128],[125,3678,3680],{"className":3679},[132],[134,3681,3682],{"xmlns":136},[138,3683,3684,3697],{},[141,3685,3686,3689,3691,3693,3695],{},[144,3687,3688],{},"O",[148,3690,544],{"stretchy":543},[144,3692,162],{},[144,3694,289],{},[148,3696,558],{"stretchy":543},[170,3698,3699],{"encoding":172},"O(nm)",[125,3701,3703],{"className":3702,"ariaHidden":178},[177],[125,3704,3706,3709,3713,3716,3720],{"className":3705},[182],[125,3707],{"className":3708,"style":613},[186],[125,3710,3688],{"className":3711,"style":3712},[191,192],"margin-right:0.0278em;",[125,3714,544],{"className":3715},[625],[125,3717,3719],{"className":3718},[191,192],"nm",[125,3721,558],{"className":3722},[652]," 掉到 ",[125,3725,3727,3752],{"className":3726},[128],[125,3728,3730],{"className":3729},[132],[134,3731,3732],{"xmlns":136},[138,3733,3734,3749],{},[141,3735,3736,3738,3740,3742,3745,3747],{},[144,3737,3688],{},[148,3739,544],{"stretchy":543},[144,3741,162],{},[148,3743,3744],{},"+",[144,3746,289],{},[148,3748,558],{"stretchy":543},[170,3750,3751],{"encoding":172},"O(n + m)",[125,3753,3755,3779],{"className":3754,"ariaHidden":178},[177],[125,3756,3758,3761,3764,3767,3770,3773,3776],{"className":3757},[182],[125,3759],{"className":3760,"style":613},[186],[125,3762,3688],{"className":3763,"style":3712},[191,192],[125,3765,544],{"className":3766},[625],[125,3768,162],{"className":3769},[191,192],[125,3771],{"className":3772,"style":415},[196],[125,3774,3744],{"className":3775},[259],[125,3777],{"className":3778,"style":415},[196],[125,3780,3782,3785,3788],{"className":3781},[182],[125,3783],{"className":3784,"style":613},[186],[125,3786,289],{"className":3787},[191,192],[125,3789,558],{"className":3790},[652],"：",[125,3793,3795],{"className":3794},[520],[125,3796,3798,3903],{"className":3797},[128],[125,3799,3801],{"className":3800},[132],[134,3802,3803],{"xmlns":136,"display":529},[138,3804,3805,3900],{},[3806,3807,3811,3850],"mtable",{"rowspacing":3808,"columnalign":3809,"columnspacing":3810},"0.25em","right left","0em",[3812,3813,3814,3824],"mtr",{},[3815,3816,3817],"mtd",{},[3818,3819,3821],"mstyle",{"scriptlevel":3820,"displaystyle":178},"0",[568,3822,3823],{},"標準 attention",[3815,3825,3826],{},[3818,3827,3828],{"scriptlevel":3820,"displaystyle":178},[141,3829,3830,3832,3835,3838,3840,3842,3844,3846,3848],{},[141,3831],{},[148,3833,3834],{},":",[196,3836],{"width":3837},"1em",[144,3839,3688],{},[148,3841,544],{"stretchy":543},[144,3843,162],{},[144,3845,289],{},[144,3847,168],{},[148,3849,558],{"stretchy":543},[3812,3851,3852,3859],{},[3815,3853,3854],{},[3818,3855,3856],{"scriptlevel":3820,"displaystyle":178},[568,3857,3858],{},"線性 attention",[3815,3860,3861],{},[3818,3862,3863],{"scriptlevel":3820,"displaystyle":178},[141,3864,3865,3867,3869,3871,3873,3875],{},[141,3866],{},[148,3868,3834],{},[196,3870],{"width":3837},[144,3872,3688],{},[568,3874,570],{},[141,3876,3877,3879,3881,3888,3890,3892,3898],{},[148,3878,544],{"fence":178},[144,3880,162],{},[152,3882,3883,3885],{},[144,3884,168],{},[950,3886,3887],{},"2",[148,3889,3744],{},[144,3891,289],{},[152,3893,3894,3896],{},[144,3895,168],{},[950,3897,3887],{},[148,3899,558],{"fence":178},[170,3901,3902],{"encoding":172},"\\begin{aligned}\n\\text{標準 attention} &: \\quad O(n m d) \\\\\n\\text{線性 attention} &: \\quad O\\!\\left(n d^2 + m d^2\\right)\n\\end{aligned}",[125,3904,3906],{"className":3905,"ariaHidden":178},[177],[125,3907,3909,3913],{"className":3908},[182],[125,3910],{"className":3911,"style":3912},[186],"height:2.7241em;vertical-align:-1.1121em;",[125,3914,3916],{"className":3915},[191],[125,3917,3919,3986],{"className":3918},[3806],[125,3920,3923],{"className":3921},[3922],"col-align-r",[125,3924,3926,3977],{"className":3925},[226,478],[125,3927,3929,3974],{"className":3928},[230],[125,3930,3933,3955],{"className":3931,"style":3932},[234],"height:1.6121em;",[125,3934,3936,3939],{"style":3935},"top:-3.7721em;",[125,3937],{"className":3938,"style":721},[241],[125,3940,3942],{"className":3941},[191],[125,3943,3946,3951],{"className":3944},[191,3945],"text",[125,3947,3950],{"className":3948},[191,3949],"cjk_fallback","標準",[125,3952,3954],{"className":3953},[191]," attention",[125,3956,3958,3961],{"style":3957},"top:-2.2479em;",[125,3959],{"className":3960,"style":721},[241],[125,3962,3964],{"className":3963},[191],[125,3965,3967,3971],{"className":3966},[191,3945],[125,3968,3970],{"className":3969},[191,3949],"線性",[125,3972,3954],{"className":3973},[191],[125,3975,506],{"className":3976},[505],[125,3978,3980],{"className":3979},[230],[125,3981,3984],{"className":3982,"style":3983},[234],"height:1.1121em;",[125,3985],{},[125,3987,3990],{"className":3988},[3989],"col-align-l",[125,3991,3993,4163],{"className":3992},[226,478],[125,3994,3996,4160],{"className":3995},[230],[125,3997,3999,4038],{"className":3998,"style":3932},[234],[125,4000,4001,4004],{"style":3935},[125,4002],{"className":4003,"style":721},[241],[125,4005,4007,4010,4013,4016,4020,4023,4026,4029,4032,4035],{"className":4006},[191],[125,4008],{"className":4009},[191],[125,4011],{"className":4012,"style":197},[196],[125,4014,3834],{"className":4015},[201],[125,4017],{"className":4018,"style":4019},[196],"margin-right:1em;",[125,4021],{"className":4022,"style":197},[196],[125,4024,3688],{"className":4025,"style":3712},[191,192],[125,4027,544],{"className":4028},[625],[125,4030,3719],{"className":4031},[191,192],[125,4033,168],{"className":4034},[191,192],[125,4036,558],{"className":4037},[652],[125,4039,4040,4043],{"style":3957},[125,4041],{"className":4042,"style":721},[241],[125,4044,4046,4049,4052,4055,4058,4061,4064,4067,4070],{"className":4045},[191],[125,4047],{"className":4048},[191],[125,4050],{"className":4051,"style":197},[196],[125,4053,3834],{"className":4054},[201],[125,4056],{"className":4057,"style":4019},[196],[125,4059],{"className":4060,"style":197},[196],[125,4062,3688],{"className":4063,"style":3712},[191,192],[125,4065],{"className":4066,"style":678},[196],[125,4068],{"className":4069,"style":636},[196],[125,4071,4073,4079,4082,4113,4116,4119,4122,4125,4154],{"className":4072},[685],[125,4074,4076],{"className":4075,"style":690},[625,689],[125,4077,544],{"className":4078},[694,497],[125,4080,162],{"className":4081},[191,192],[125,4083,4085,4088],{"className":4084},[191],[125,4086,168],{"className":4087},[191,192],[125,4089,4091],{"className":4090},[222],[125,4092,4094],{"className":4093},[226],[125,4095,4097],{"className":4096},[230],[125,4098,4101],{"className":4099,"style":4100},[234],"height:0.8641em;",[125,4102,4104,4107],{"style":4103},"top:-3.113em;margin-right:0.05em;",[125,4105],{"className":4106,"style":242},[241],[125,4108,4110],{"className":4109},[246,247,248,249],[125,4111,3887],{"className":4112},[191,249],[125,4114],{"className":4115,"style":415},[196],[125,4117,3744],{"className":4118},[259],[125,4120],{"className":4121,"style":415},[196],[125,4123,289],{"className":4124},[191,192],[125,4126,4128,4131],{"className":4127},[191],[125,4129,168],{"className":4130},[191,192],[125,4132,4134],{"className":4133},[222],[125,4135,4137],{"className":4136},[226],[125,4138,4140],{"className":4139},[230],[125,4141,4143],{"className":4142,"style":4100},[234],[125,4144,4145,4148],{"style":4103},[125,4146],{"className":4147,"style":242},[241],[125,4149,4151],{"className":4150},[246,247,248,249],[125,4152,3887],{"className":4153},[191,249],[125,4155,4157],{"className":4156,"style":690},[652,689],[125,4158,558],{"className":4159},[694,497],[125,4161,506],{"className":4162},[505],[125,4164,4166],{"className":4165},[230],[125,4167,4169],{"className":4168,"style":3983},[234],[125,4170],{},[112,4172,4173],{},"這就是 linear attention 那一整家族在做的事。",[108,4175,4176],{"id":4176},"代價在分母",[112,4178,4179,4180],{},"這裡有個很容易被忽略的地方：",[2842,4181,4182],{},"誤差最大的地方不是分子，是分母。",[112,4184,4185,4186,4418,4419,4608],{},"分子是「加權和」，近似得差一點，輸出只是稍微偏掉；分母是「歸一化常數」，它一錯，整個輸出會跟著縮放。序列越長，",[125,4187,4189,4235],{"className":4188},[128],[125,4190,4192],{"className":4191},[132],[134,4193,4194],{"xmlns":136},[138,4195,4196,4232],{},[141,4197,4198,4200,4202,4208,4214,4220,4222,4224,4230],{},[144,4199,2909],{},[148,4201,544],{"stretchy":543},[393,4203,4204,4206],{},[144,4205,975],{},[144,4207,891],{},[152,4209,4210,4212],{},[148,4211,558],{"stretchy":543},[144,4213,588],{"mathvariant":536},[393,4215,4216,4218],{},[148,4217,941],{},[144,4219,946],{},[144,4221,2909],{},[148,4223,544],{"stretchy":543},[393,4225,4226,4228],{},[144,4227,984],{},[144,4229,946],{},[148,4231,558],{"stretchy":543},[170,4233,4234],{"encoding":172},"\\phi(q_i)^{\\top}\\sum_j \\phi(k_j)",[125,4236,4238],{"className":4237,"ariaHidden":178},[177],[125,4239,4241,4245,4248,4251,4291,4323,4326,4366,4369,4372,4375,4415],{"className":4240},[182],[125,4242],{"className":4243,"style":4244},[186],"height:1.2849em;vertical-align:-0.4358em;",[125,4246,2909],{"className":4247},[191,192],[125,4249,544],{"className":4250},[625],[125,4252,4254,4257],{"className":4253},[191],[125,4255,975],{"className":4256,"style":501},[191,192],[125,4258,4260],{"className":4259},[222],[125,4261,4263,4283],{"className":4262},[226,478],[125,4264,4266,4280],{"className":4265},[230],[125,4267,4269],{"className":4268,"style":1082},[234],[125,4270,4271,4274],{"style":1809},[125,4272],{"className":4273,"style":242},[241],[125,4275,4277],{"className":4276},[246,247,248,249],[125,4278,891],{"className":4279},[191,192,249],[125,4281,506],{"className":4282},[505],[125,4284,4286],{"className":4285},[230],[125,4287,4289],{"className":4288,"style":1104},[234],[125,4290],{},[125,4292,4294,4297],{"className":4293},[652],[125,4295,558],{"className":4296},[652],[125,4298,4300],{"className":4299},[222],[125,4301,4303],{"className":4302},[226],[125,4304,4306],{"className":4305},[230],[125,4307,4309],{"className":4308,"style":211},[234],[125,4310,4311,4314],{"style":237},[125,4312],{"className":4313,"style":242},[241],[125,4315,4317],{"className":4316},[246,247,248,249],[125,4318,4320],{"className":4319},[191,249],[125,4321,588],{"className":4322},[191,249],[125,4324],{"className":4325,"style":636},[196],[125,4327,4329,4332],{"className":4328},[617],[125,4330,941],{"className":4331,"style":1160},[617,1158,1159],[125,4333,4335],{"className":4334},[222],[125,4336,4338,4358],{"className":4337},[226,478],[125,4339,4341,4355],{"className":4340},[230],[125,4342,4344],{"className":4343,"style":2492},[234],[125,4345,4346,4349],{"style":1176},[125,4347],{"className":4348,"style":242},[241],[125,4350,4352],{"className":4351},[246,247,248,249],[125,4353,946],{"className":4354,"style":1189},[191,192,249],[125,4356,506],{"className":4357},[505],[125,4359,4361],{"className":4360},[230],[125,4362,4364],{"className":4363,"style":1220},[234],[125,4365],{},[125,4367],{"className":4368,"style":636},[196],[125,4370,2909],{"className":4371},[191,192],[125,4373,544],{"className":4374},[625],[125,4376,4378,4381],{"className":4377},[191],[125,4379,984],{"className":4380,"style":1410},[191,192],[125,4382,4384],{"className":4383},[222],[125,4385,4387,4407],{"className":4386},[226,478],[125,4388,4390,4404],{"className":4389},[230],[125,4391,4393],{"className":4392,"style":1082},[234],[125,4394,4395,4398],{"style":2047},[125,4396],{"className":4397,"style":242},[241],[125,4399,4401],{"className":4400},[246,247,248,249],[125,4402,946],{"className":4403,"style":1189},[191,192,249],[125,4405,506],{"className":4406},[505],[125,4408,4410],{"className":4409},[230],[125,4411,4413],{"className":4412,"style":1828},[234],[125,4414],{},[125,4416,558],{"className":4417},[652]," 和真正的 ",[125,4420,4422,4460],{"className":4421},[128],[125,4423,4425],{"className":4424},[132],[134,4426,4427],{"xmlns":136},[138,4428,4429,4457],{},[141,4430,4431,4437,4439,4441,4447,4449,4455],{},[393,4432,4433,4435],{},[148,4434,941],{},[144,4436,946],{},[144,4438,2086],{},[148,4440,544],{"stretchy":543},[393,4442,4443,4445],{},[144,4444,975],{},[144,4446,891],{},[148,4448,549],{"separator":178},[393,4450,4451,4453],{},[144,4452,984],{},[144,4454,946],{},[148,4456,558],{"stretchy":543},[170,4458,4459],{"encoding":172},"\\sum_j \\kappa(q_i,k_j)",[125,4461,4463],{"className":4462,"ariaHidden":178},[177],[125,4464,4466,4470,4510,4513,4516,4519,4559,4562,4565,4605],{"className":4465},[182],[125,4467],{"className":4468,"style":4469},[186],"height:1.1858em;vertical-align:-0.4358em;",[125,4471,4473,4476],{"className":4472},[617],[125,4474,941],{"className":4475,"style":1160},[617,1158,1159],[125,4477,4479],{"className":4478},[222],[125,4480,4482,4502],{"className":4481},[226,478],[125,4483,4485,4499],{"className":4484},[230],[125,4486,4488],{"className":4487,"style":2492},[234],[125,4489,4490,4493],{"style":1176},[125,4491],{"className":4492,"style":242},[241],[125,4494,4496],{"className":4495},[246,247,248,249],[125,4497,946],{"className":4498,"style":1189},[191,192,249],[125,4500,506],{"className":4501},[505],[125,4503,4505],{"className":4504},[230],[125,4506,4508],{"className":4507,"style":1220},[234],[125,4509],{},[125,4511],{"className":4512,"style":636},[196],[125,4514,2086],{"className":4515},[191,192],[125,4517,544],{"className":4518},[625],[125,4520,4522,4525],{"className":4521},[191],[125,4523,975],{"className":4524,"style":501},[191,192],[125,4526,4528],{"className":4527},[222],[125,4529,4531,4551],{"className":4530},[226,478],[125,4532,4534,4548],{"className":4533},[230],[125,4535,4537],{"className":4536,"style":1082},[234],[125,4538,4539,4542],{"style":1809},[125,4540],{"className":4541,"style":242},[241],[125,4543,4545],{"className":4544},[246,247,248,249],[125,4546,891],{"className":4547},[191,192,249],[125,4549,506],{"className":4550},[505],[125,4552,4554],{"className":4553},[230],[125,4555,4557],{"className":4556,"style":1104},[234],[125,4558],{},[125,4560,549],{"className":4561},[632],[125,4563],{"className":4564,"style":636},[196],[125,4566,4568,4571],{"className":4567},[191],[125,4569,984],{"className":4570,"style":1410},[191,192],[125,4572,4574],{"className":4573},[222],[125,4575,4577,4597],{"className":4576},[226,478],[125,4578,4580,4594],{"className":4579},[230],[125,4581,4583],{"className":4582,"style":1082},[234],[125,4584,4585,4588],{"style":2047},[125,4586],{"className":4587,"style":242},[241],[125,4589,4591],{"className":4590},[246,247,248,249],[125,4592,946],{"className":4593,"style":1189},[191,192,249],[125,4595,506],{"className":4596},[505],[125,4598,4600],{"className":4599},[230],[125,4601,4603],{"className":4602,"style":1828},[234],[125,4604],{},[125,4606,558],{"className":4607},[652]," 差得越多，輸出就越不可信。",[112,4610,4611],{},"所以我們在評估這類方法時，看的第一個數字不是整體 perplexity，而是分母的相對誤差。",[108,4613,4614],{"id":4614},"實務上的三個檢查點",[4616,4617,4618,4631],"table",{},[4619,4620,4621],"thead",{},[4622,4623,4624,4628],"tr",{},[4625,4626,4627],"th",{},"檢查",[4625,4629,4630],{},"為什麼",[4632,4633,4634,4643,4722],"tbody",{},[4622,4635,4636,4640],{},[4637,4638,4639],"td",{},"分母的相對誤差",[4637,4641,4642],{},"比分子更致命，因為它會整體縮放輸出",[4622,4644,4645,4719],{},[4637,4646,4647,4648],{},"特徵維度 ",[125,4649,4651,4669],{"className":4650},[128],[125,4652,4654],{"className":4653},[132],[134,4655,4656],{"xmlns":136},[138,4657,4658,4666],{},[141,4659,4660],{},[393,4661,4662,4664],{},[144,4663,168],{},[144,4665,2909],{},[170,4667,4668],{"encoding":172},"d_\\phi",[125,4670,4672],{"className":4671,"ariaHidden":178},[177],[125,4673,4675,4678],{"className":4674},[182],[125,4676],{"className":4677,"style":2026},[186],[125,4679,4681,4684],{"className":4680},[191],[125,4682,168],{"className":4683},[191,192],[125,4685,4687],{"className":4686},[222],[125,4688,4690,4711],{"className":4689},[226,478],[125,4691,4693,4708],{"className":4692},[230],[125,4694,4697],{"className":4695,"style":4696},[234],"height:0.3361em;",[125,4698,4699,4702],{"style":1085},[125,4700],{"className":4701,"style":242},[241],[125,4703,4705],{"className":4704},[246,247,248,249],[125,4706,2909],{"className":4707},[191,192,249],[125,4709,506],{"className":4710},[505],[125,4712,4714],{"className":4713},[230],[125,4715,4717],{"className":4716,"style":1828},[234],[125,4718],{},[4637,4720,4721],{},"太少則近似粗糙，太多則省不下計算",[4622,4723,4724,4727],{},[4637,4725,4726],{},"序列長度分佈",[4637,4728,4729],{},"短序列幾乎沒有收益，長序列才值得換",[4731,4732,4737],"pre",{"className":4733,"code":4734,"language":4735,"meta":4736,"style":4736},"language-python shiki shiki-themes material-theme-lighter material-theme material-theme-palenight","# 概念示意：先把 K、V 的累加量算好\nkv = einsum(\"m d, m e -> d e\", phi(K), V)   # O(m d^2)\nz  = phi(K).sum(dim=0)                       # O(m d)\nout = einsum(\"n d, d e -> n e\", phi(Q), kv) \u002F einsum(\"n d, d -> n\", phi(Q), z)\n","python","",[4738,4739,4740,4747,4753,4759],"code",{"__ignoreMap":4736},[125,4741,4744],{"class":4742,"line":4743},"line",1,[125,4745,4746],{},"# 概念示意：先把 K、V 的累加量算好\n",[125,4748,4750],{"class":4742,"line":4749},2,[125,4751,4752],{},"kv = einsum(\"m d, m e -> d e\", phi(K), V)   # O(m d^2)\n",[125,4754,4756],{"class":4742,"line":4755},3,[125,4757,4758],{},"z  = phi(K).sum(dim=0)                       # O(m d)\n",[125,4760,4762],{"class":4742,"line":4761},4,[125,4763,4764],{},"out = einsum(\"n d, d e -> n e\", phi(Q), kv) \u002F einsum(\"n d, d -> n\", phi(Q), z)\n",[108,4766,4767],{"id":4767},"給下一個要調這個模型的人",[112,4769,4770],{},"如果你正在為長序列煩惱，我會建議先把 attention 當成核平滑器看一遍，再決定要不要換模型。",[112,4772,4773,4774,4777],{},"因為換成 linear attention 不是「更快的 attention」，而是",[2842,4775,4776],{},"用特徵映射換掉精確的歸一化","——你付出的是準確度，不是免費的午餐。",[112,4779,4780],{},"知道自己在付什麼，比知道自己在用什麼重要。",[4782,4783,4784],"style",{},"html .light .shiki span {color: var(--shiki-light);background: var(--shiki-light-bg);font-style: var(--shiki-light-font-style);font-weight: var(--shiki-light-font-weight);text-decoration: var(--shiki-light-text-decoration);}html.light .shiki span {color: var(--shiki-light);background: var(--shiki-light-bg);font-style: var(--shiki-light-font-style);font-weight: var(--shiki-light-font-weight);text-decoration: var(--shiki-light-text-decoration);}html .default .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .dark .shiki span {color: var(--shiki-dark);background: var(--shiki-dark-bg);font-style: var(--shiki-dark-font-style);font-weight: var(--shiki-dark-font-weight);text-decoration: var(--shiki-dark-text-decoration);}html.dark .shiki span {color: var(--shiki-dark);background: var(--shiki-dark-bg);font-style: var(--shiki-dark-font-style);font-weight: var(--shiki-dark-font-weight);text-decoration: var(--shiki-dark-text-decoration);}",{"title":4736,"searchDepth":4749,"depth":4749,"links":4786},[4787,4788,4789,4790,4791,4792],{"id":110,"depth":4749,"text":110},{"id":120,"depth":4749,"text":120},{"id":2837,"depth":4749,"text":2837},{"id":4176,"depth":4749,"text":4176},{"id":4614,"depth":4749,"text":4614},{"id":4767,"depth":4749,"text":4767},"\u002Fimages\u002Fblog\u002Fattention-kernel.svg","Attention 與核函數對應關係的示意圖","2026-10-20","第一次看到「attention 是核方法」這個說法時，我以為那只是類比。自己推過一次才發現不是——而這件事直接決定了我們該在哪裡砍計算量。",false,"md","zh-TW",{},true,"\u002Fblog\u002Fzh-tw\u002Fattention-as-kernel",{"title":102,"description":4796},"attention-as-kernel","blog\u002Fzh-TW\u002Fattention-as-kernel",[4807,4808,4809],"transformer","數學","推論","2026-10-22","ipCLaHTj0DVQiZoXlJ6DPzX7hl68udhNuMH4wa4gke8",null,[4814,10510],{"id":4815,"title":4816,"author":103,"body":4817,"cover":10498,"coverAlt":4816,"date":10499,"description":10500,"draft":4797,"extension":4798,"featured":4797,"lang":4799,"meta":10501,"navigation":4801,"path":10502,"seo":10503,"slug":10504,"stem":10505,"tags":10506,"translationKey":10504,"updated":4812,"__hash__":10509},"blog\u002Fblog\u002Fzh-TW\u002Farxiv-2512-22471v2.md","Transformer 注意力機制的貝葉斯幾何學：機制與理論深度分析報告",{"type":105,"value":4818,"toc":10465},[4819,4825,4828,4831,4834,4840,4847,4850,4853,5011,5017,5024,5097,5100,5339,5342,5372,5375,5704,5707,5921,5979,5982,6375,6383,6389,6400,6406,6409,6859,7111,7114,7396,7402,7405,7691,8146,8149,8227,8357,8363,8366,8630,8636,8639,8645,8648,8873,8879,8882,9425,9431,9434,9458,9461,9772,9778,9784,9795,9801,9804,9824,9830,9833,9906,9912,9915,9935,9941,9944,9950,9953,10111,10117,10198,10204,10207,10213,10216,10236,10242,10248,10278,10284,10291,10309,10315,10318,10408,10414,10420,10435,10438,10441,10448],[108,4820,4822],{"id":4821},"_1-執行摘要",[2842,4823,4824],{},"1. 執行摘要",[112,4826,4827],{},"在當前的人工智慧領域中，Transformer 架構已成為大型語言模型（LLMs）的基石。然而，關於其強大的上下文學習（In-Context Learning）能力背後的確切運作機制，學術界長期以來存在著激烈的爭論。核心問題在於：這些模型究竟是僅僅進行高維度的模式匹配與記憶，還是在演算法層面上真正實現了機率推理？由 Naman Agarwal、Siddhartha R. Dalal 和 Vishal Misra 於 2026 年發表的開創性研究《Transformer 注意力機制的貝葉斯幾何學》（The Bayesian Geometry of Transformer Attention）為此提供了具體的實證解答。",[112,4829,4830],{},"本報告旨在對該研究進行詳盡的深度分析與總結。研究團隊透過構建「貝葉斯風洞」（Bayesian Wind Tunnels）——一種具備解析解且無法通過記憶解決的受控實驗環境——證明了小型 Transformer 模型不僅能夠模擬貝葉斯推理，更能以機器級的精度（誤差低至10^{-3} 到 10^{-4} bits）追蹤確切的貝葉斯後驗分佈。與此形成鮮明對比的是，參數規模相當的多層感知機（MLP）在相同任務中表現出災難性的失敗，這確立了注意力機制在動態推理中的不可或缺性。",[112,4832,4833],{},"進一步的幾何診斷揭示了 Transformer 實現此一過程的三階段機制：首先在底層構建正交的假設框架（Hypothesis Frame），接著在中層透過注意力機制進行順序性的假設消除（Sequential Elimination），最後在頂層將後驗機率精確編碼於低維度的值流形（Value Manifold）上。這些發現不僅解釋了 Transformer 的成功原因，更為提示工程（Prompt Engineering）提供了理論依據——即提示應被視為消除假設空間的證據。",[108,4835,4837],{"id":4836},"_2-理論背景與研究動機",[2842,4838,4839],{},"2. 理論背景與研究動機",[4841,4842,4844],"h3",{"id":4843},"_21-大型語言模型的驗證危機",[2842,4845,4846],{},"2.1 大型語言模型的驗證危機",[112,4848,4849],{},"隨著大型語言模型（如 GPT-4, Claude 等）展現出驚人的推理能力，一個根本性的科學難題隨之浮現：我們無法驗證這些模型是否真的在「推理」。自然語言數據本質上缺乏「基本真理」（Ground Truth）的後驗機率分佈。例如，在一段文本續寫任務中，下一個詞的真實機率分佈是未知的，且往往受到多種潛在因素的影響。此外，現代 LLM 的訓練數據量極其龐大，使得研究人員難以區分模型是在進行真正的即時機率計算，還是僅僅在提取海量訓練數據中記憶的片段。",[112,4851,4852],{},"這種「驗證危機」導致了對模型能力的解釋往往停留在定性描述或擬人化的比喻上，缺乏嚴謹的定量評估。為了突破這一局限，Agarwal 等人提出了「貝葉斯風洞」的概念。這是一種合成的實驗環境，設計上必須滿足三個嚴格條件：",[4854,4855,4856,4863,5005],"ol",{},[4857,4858,4859,4862],"li",{},[2842,4860,4861],{},"解析可解性（Analytic Tractability）："," 在序列的每一步，真實的貝葉斯後驗分佈必須是已知且可精確計算的閉式解（Closed-form）。",[4857,4864,4865,4868,4869,4937,4938,5004],{},[2842,4866,4867],{},"抗記憶性（Memorization Resistance）："," 假設空間必須大到組合爆炸的程度（例如 ",[125,4870,4872,4892],{"className":4871},[128],[125,4873,4875],{"className":4874},[132],[134,4876,4877],{"xmlns":136},[138,4878,4879,4889],{},[141,4880,4881],{},[152,4882,4883,4886],{},[950,4884,4885],{},"10",[950,4887,4888],{},"18",[170,4890,4891],{"encoding":172},"10^{18}",[125,4893,4895],{"className":4894,"ariaHidden":178},[177],[125,4896,4898,4902,4905],{"className":4897},[182],[125,4899],{"className":4900,"style":4901},[186],"height:0.8141em;",[125,4903,952],{"className":4904},[191],[125,4906,4908,4911],{"className":4907},[191],[125,4909,3820],{"className":4910},[191],[125,4912,4914],{"className":4913},[222],[125,4915,4917],{"className":4916},[226],[125,4918,4920],{"className":4919},[230],[125,4921,4923],{"className":4922,"style":4901},[234],[125,4924,4925,4928],{"style":237},[125,4926],{"className":4927,"style":242},[241],[125,4929,4931],{"className":4930},[246,247,248,249],[125,4932,4934],{"className":4933},[191,249],[125,4935,4888],{"className":4936},[191,249]," 或 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種可能性），從而數學上排除模型通過背誦訓練樣本來解決任務的可能性。",[4857,5006,5007,5010],{},[2842,5008,5009],{},"推理必要性（Inference Necessity）："," 任務必須要求模型根據上下文中的新證據動態更新信念，這正是貝葉斯推理的定義。",[4841,5012,5014],{"id":5013},"_22-交叉熵與貝葉斯推理的數學對偶性",[2842,5015,5016],{},"2.2 交叉熵與貝葉斯推理的數學對偶性",[112,5018,5019,5020,5023],{},"本研究的理論基礎建立在深度學習訓練目標與機率論之間的深層聯繫上。研究團隊提出了",[2842,5021,5022],{},"定理 1：交叉熵的群體最優解","（Population Optimum of Cross-Entropy）。",[112,5025,5026,5027],{},"設想一個由潛在參數\n",[125,5028,5030,5057],{"className":5029},[128],[125,5031,5033],{"className":5032},[132],[134,5034,5035],{"xmlns":136},[138,5036,5037,5054],{},[141,5038,5039,5042,5045,5048,5050,5052],{},[144,5040,5041],{},"θ",[148,5043,5044],{},"∼",[144,5046,5047],{},"π",[148,5049,544],{"stretchy":543},[144,5051,5041],{},[148,5053,558],{"stretchy":543},[170,5055,5056],{"encoding":172},"\\theta \\sim 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al{L}(q)=\\mathbb{E}_{\\theta \\sim \\pi} \\mathbb{E}_{c, (x,y) \\sim p(\\cdot|\\theta)} [-\\log q(y|x,c)]",[125,5485,5487,5515],{"className":5486,"ariaHidden":178},[177],[125,5488,5490,5493,5497,5500,5503,5506,5509,5512],{"className":5489},[182],[125,5491],{"className":5492,"style":613},[186],[125,5494,5393],{"className":5495},[191,5496],"mathcal",[125,5498,544],{"className":5499},[625],[125,5501,975],{"className":5502,"style":501},[191,192],[125,5504,558],{"className":5505},[652],[125,5507],{"className":5508,"style":197},[196],[125,5510,561],{"className":5511},[201],[125,5513],{"className":5514,"style":197},[196],[125,5516,5518,5522,5571,5656,5659,5662,5665,5673,5676,5679,5682,5685,5688,5691,5694,5697,5700],{"className":5517},[182],[125,5519],{"className":5520,"style":5521},[186],"height:1.1052em;vertical-align:-0.3552em;",[125,5523,5525,5528],{"className":5524},[191],[125,5526,5406],{"className":5527},[191,218],[125,5529,5531],{"className":5530},[222],[125,5532,5534,5563],{"className":5533},[226,478],[125,5535,5537,5560],{"className":5536},[230],[125,5538,5540],{"className":5539,"style":4696},[234],[125,5541,5542,5545],{"style":1085},[125,5543],{"className":5544,"style":242},[241],[125,5546,5548],{"className":5547},[246,247,248,249],[125,5549,5551,5554,5557],{"className":5550},[191,249],[125,5552,5041],{"className":5553,"style":3712},[191,192,249],[125,5555,5044],{"className":5556},[201,249],[125,5558,5047],{"className":5559,"style":501},[191,192,249],[125,5561,506],{"className":5562},[505],[125,5564,5566],{"className":5565},[230],[125,5567,5569],{"className":5568,"style":1104},[234],[125,5570],{},[125,5572,5574,5577],{"className":5573},[191],[125,5575,5406],{"className":5576},[191,218],[125,5578,5580],{"className":5579},[222],[125,5581,5583,5647],{"className":5582},[226,478],[125,5584,5586,5644],{"className":5585},[230],[125,5587,5590],{"className":5588,"style":5589},[234],"height:0.3448em;",[125,5591,5593,5596],{"style":5592},"top:-2.5198em;margin-left:0em;margin-right:0.05em;",[125,5594],{"className":5595,"style":242},[241],[125,5597,5599],{"className":5598},[246,247,248,249],[125,5600,5602,5605,5608,5611,5614,5617,5620,5623,5626,5629,5632,5635,5638,5641],{"className":5601},[191,249],[125,5603,5116],{"className":5604},[191,192,249],[125,5606,549],{"className":5607},[632,249],[125,5609,544],{"className":5610},[625,249],[125,5612,5129],{"className":5613},[191,192,249],[125,5615,549],{"className":5616},[632,249],[125,5618,5138],{"className":5619,"style":501},[191,192,249],[125,5621,558],{"className":5622},[652,249],[125,5624,5044],{"className":5625},[201,249],[125,5627,112],{"className":5628},[191,192,249],[125,5630,544],{"className":5631},[625,249],[125,5633,5443],{"className":5634},[191,249],[125,5636,5446],{"className":5637},[191,249],[125,5639,5041],{"className":5640,"style":3712},[191,192,249],[125,5642,558],{"className":5643},[652,249],[125,5645,506],{"className":5646},[505],[125,5648,5650],{"className":5649},[230],[125,5651,5654],{"className":5652,"style":5653},[234],"height:0.3552em;",[125,5655],{},[125,5657,5453],{"className":5658},[625],[125,5660,5456],{"className":5661},[191],[125,5663],{"className":5664,"style":636},[196],[125,5666,5668,5669],{"className":5667},[617],"lo",[125,5670,5672],{"style":5671},"margin-right:0.0139em;","g",[125,5674],{"className":5675,"style":636},[196],[125,5677,975],{"className":5678,"style":501},[191,192],[125,5680,544],{"className":5681},[625],[125,5683,5138],{"className":5684,"style":501},[191,192],[125,5686,5446],{"className":5687},[191],[125,5689,5129],{"className":5690},[191,192],[125,5692,549],{"className":5693},[632],[125,5695],{"className":5696,"style":636},[196],[125,5698,5116],{"className":5699},[191,192],[125,5701,5703],{"className":5702},[652],")]",[112,5705,5706],{},"該定理證明，此目標函數的唯一全局最小值即為貝葉斯後驗預測分佈（Bayesian Posterior Predictive Distribution）：",[125,5708,5710],{"className":5709},[520],[125,5711,5713,5783],{"className":5712},[128],[125,5714,5716],{"className":5715},[132],[134,5717,5718],{"xmlns":136,"display":529},[138,5719,5720,5780],{},[141,5721,5722,5729,5731,5733,5735,5737,5739,5741,5743,5745,5748,5750,5752,5754,5756,5758,5760,5762,5764,5766,5768,5770,5772,5774,5776,5778],{},[152,5723,5724,5726],{},[144,5725,975],{},[148,5727,5728],{},"∗",[148,5730,544],{"stretchy":543},[144,5732,5138],{},[144,5734,5446],{"mathvariant":536},[144,5736,5129],{},[148,5738,549],{"separator":178},[144,5740,5116],{},[148,5742,558],{"stretchy":543},[148,5744,561],{},[148,5746,5747],{},"∫",[144,5749,112],{},[148,5751,544],{"stretchy":543},[144,5753,5138],{},[144,5755,5446],{"mathvariant":536},[144,5757,5129],{},[148,5759,549],{"separator":178},[144,5761,5041],{},[148,5763,558],{"stretchy":543},[144,5765,112],{},[148,5767,544],{"stretchy":543},[144,5769,5041],{},[144,5771,5446],{"mathvariant":536},[144,5773,5116],{},[148,5775,558],{"stretchy":543},[144,5777,168],{},[144,5779,5041],{},[170,5781,5782],{"encoding":172},"q^*(y|x,c)=\\int p(y|x,\\theta) p(\\theta|c) d\\theta",[125,5784,5786,5855],{"className":5785,"ariaHidden":178},[177],[125,5787,5789,5792,5822,5825,5828,5831,5834,5837,5840,5843,5846,5849,5852],{"className":5788},[182],[125,5790],{"className":5791,"style":613},[186],[125,5793,5795,5798],{"className":5794},[191],[125,5796,975],{"className":5797,"style":501},[191,192],[125,5799,5801],{"className":5800},[222],[125,5802,5804],{"className":5803},[226],[125,5805,5807],{"className":5806},[230],[125,5808,5811],{"className":5809,"style":5810},[234],"height:0.7387em;",[125,5812,5813,5816],{"style":4103},[125,5814],{"className":5815,"style":242},[241],[125,5817,5819],{"className":5818},[246,247,248,249],[125,5820,5728],{"className":5821},[259,249],[125,5823,544],{"className":5824},[625],[125,5826,5138],{"className":5827,"style":501},[191,192],[125,5829,5446],{"className":5830},[191],[125,5832,5129],{"className":5833},[191,192],[125,5835,549],{"className":5836},[632],[125,5838],{"className":5839,"style":636},[196],[125,5841,5116],{"className":5842},[191,192],[125,5844,558],{"className":5845},[652],[125,5847],{"className":5848,"style":197},[196],[125,5850,561],{"className":5851},[201],[125,5853],{"className":5854,"style":197},[196],[125,5856,5858,5862,5867,5870,5873,5876,5879,5882,5885,5888,5891,5894,5897,5900,5903,5906,5909,5912,5915,5918],{"className":5857},[182],[125,5859],{"className":5860,"style":5861},[186],"height:2.2222em;vertical-align:-0.8622em;",[125,5863,5747],{"className":5864,"style":5866},[617,1158,5865],"large-op","margin-right:0.4445em;position:relative;top:-0.0011em;",[125,5868],{"className":5869,"style":636},[196],[125,5871,112],{"className":5872},[191,192],[125,5874,544],{"className":5875},[625],[125,5877,5138],{"className":5878,"style":501},[191,192],[125,5880,5446],{"className":5881},[191],[125,5883,5129],{"className":5884},[191,192],[125,5886,549],{"className":5887},[632],[125,5889],{"className":5890,"style":636},[196],[125,5892,5041],{"className":5893,"style":3712},[191,192],[125,5895,558],{"className":5896},[652],[125,5898,112],{"className":5899},[191,192],[125,5901,544],{"className":5902},[625],[125,5904,5041],{"className":5905,"style":3712},[191,192],[125,5907,5446],{"className":5908},[191],[125,5910,5116],{"className":5911},[191,192],[125,5913,558],{"className":5914},[652],[125,5916,168],{"className":5917},[191,192],[125,5919,5041],{"className":5920,"style":3712},[191,192],[112,5922,5923,5924,5978],{},"其中 ",[125,5925,5927,5951],{"className":5926},[128],[125,5928,5930],{"className":5929},[132],[134,5931,5932],{"xmlns":136},[138,5933,5934,5948],{},[141,5935,5936,5938,5940,5942,5944,5946],{},[144,5937,112],{},[148,5939,544],{"stretchy":543},[144,5941,5041],{},[144,5943,5446],{"mathvariant":536},[144,5945,5116],{},[148,5947,558],{"stretchy":543},[170,5949,5950],{"encoding":172},"p(\\theta|c)",[125,5952,5954],{"className":5953,"ariaHidden":178},[177],[125,5955,5957,5960,5963,5966,5969,5972,5975],{"className":5956},[182],[125,5958],{"className":5959,"style":613},[186],[125,5961,112],{"className":5962},[191,192],[125,5964,544],{"className":5965},[625],[125,5967,5041],{"className":5968,"style":3712},[191,192],[125,5970,5446],{"className":5971},[191],[125,5973,5116],{"className":5974},[191,192],[125,5976,558],{"className":5977},[652],">",[112,5980,5981],{},"是根據貝葉斯法則計算的潛在參數後驗分佈：",[112,5983,5984],{},[125,5985,5987,6075],{"className":5986},[128],[125,5988,5990],{"className":5989},[132],[134,5991,5992],{"xmlns":136},[138,5993,5994,6072],{},[141,5995,5996,5998,6000,6002,6004,6006,6008,6011,6013,6015,6017,6019,6048,6050,6052,6058,6060,6066,6068,6070],{},[144,5997,112],{},[148,5999,544],{"stretchy":543},[144,6001,5041],{},[144,6003,5446],{"mathvariant":536},[144,6005,5116],{},[148,6007,558],{"stretchy":543},[148,6009,6010],{},"∝",[144,6012,5047],{},[148,6014,544],{"stretchy":543},[144,6016,5041],{},[148,6018,558],{"stretchy":543},[393,6020,6021,6024],{},[148,6022,6023],{},"∏",[141,6025,6026,6028,6034,6036,6042,6044,6046],{},[148,6027,544],{"stretchy":543},[393,6029,6030,6032],{},[144,6031,5129],{},[144,6033,891],{},[148,6035,549],{"separator":178},[393,6037,6038,6040],{},[144,6039,5138],{},[144,6041,891],{},[148,6043,558],{"stretchy":543},[148,6045,150],{},[144,6047,5116],{},[144,6049,112],{},[148,6051,544],{"stretchy":543},[393,6053,6054,6056],{},[144,6055,5138],{},[144,6057,891],{},[144,6059,5446],{"mathvariant":536},[393,6061,6062,6064],{},[144,6063,5129],{},[144,6065,891],{},[148,6067,549],{"separator":178},[144,6069,5041],{},[148,6071,558],{"stretchy":543},[170,6073,6074],{"encoding":172},"p(\\theta|c) \\propto \\pi(\\theta) \\prod_{(x_i, y_i) \\in c} p(y_i|x_i, \\theta)",[125,6076,6078,6111],{"className":6077,"ariaHidden":178},[177],[125,6079,6081,6084,6087,6090,6093,6096,6099,6102,6105,6108],{"className":6080},[182],[125,6082],{"className":6083,"style":613},[186],[125,6085,112],{"className":6086},[191,192],[125,6088,544],{"className":6089},[625],[125,6091,5041],{"className":6092,"style":3712},[191,192],[125,6094,5446],{"className":6095},[191],[125,6097,5116],{"className":6098},[191,192],[125,6100,558],{"className":6101},[652],[125,6103],{"className":6104,"style":197},[196],[125,6106,6010],{"className":6107},[201],[125,6109],{"className":6110,"style":197},[196],[125,6112,6114,6118,6121,6124,6127,6130,6133,6271,6274,6277,6280,6320,6323,6363,6366,6369,6372],{"className":6113},[182],[125,6115],{"className":6116,"style":6117},[186],"height:1.2247em;vertical-align:-0.4747em;",[125,6119,5047],{"className":6120,"style":501},[191,192],[125,6122,544],{"className":6123},[625],[125,6125,5041],{"className":6126,"style":3712},[191,192],[125,6128,558],{"className":6129},[652],[125,6131],{"className":6132,"style":636},[196],[125,6134,6136,6139],{"className":6135},[617],[125,6137,6023],{"className":6138,"style":1160},[617,1158,1159],[125,6140,6142],{"className":6141},[222],[125,6143,6145,6262],{"className":6144},[226,478],[125,6146,6148,6259],{"className":6147},[230],[125,6149,6152],{"className":6150,"style":6151},[234],"height:0.2253em;",[125,6153,6154,6157],{"style":1176},[125,6155],{"className":6156,"style":242},[241],[125,6158,6160],{"className":6159},[246,247,248,249],[125,6161,6163,6166,6206,6209,6250,6253,6256],{"className":6162},[191,249],[125,6164,544],{"className":6165},[625,249],[125,6167,6169,6172],{"className":6168},[191,249],[125,6170,5129],{"className":6171},[191,192,249],[125,6173,6175],{"className":6174},[222],[125,6176,6178,6198],{"className":6177},[226,478],[125,6179,6181,6195],{"className":6180},[230],[125,6182,6184],{"className":6183,"style":1423},[234],[125,6185,6186,6189],{"style":488},[125,6187],{"className":6188,"style":492},[241],[125,6190,6192],{"className":6191},[246,496,497,249],[125,6193,891],{"className":6194},[191,192,249],[125,6196,506],{"className":6197},[505],[125,6199,6201],{"className":6200},[230],[125,6202,6204],{"className":6203,"style":513},[234],[125,6205],{},[125,6207,549],{"className":6208},[632,249],[125,6210,6212,6215],{"className":6211},[191,249],[125,6213,5138],{"className":6214,"style":501},[191,192,249],[125,6216,6218],{"className":6217},[222],[125,6219,6221,6242],{"className":6220},[226,478],[125,6222,6224,6239],{"className":6223},[230],[125,6225,6227],{"className":6226,"style":1423},[234],[125,6228,6230,6233],{"style":6229},"top:-2.357em;margin-left:-0.0359em;margin-right:0.0714em;",[125,6231],{"className":6232,"style":492},[241],[125,6234,6236],{"className":6235},[246,496,497,249],[125,6237,891],{"className":6238},[191,192,249],[125,6240,506],{"className":6241},[505],[125,6243,6245],{"className":6244},[230],[125,6246,6248],{"className":6247,"style":513},[234],[125,6249],{},[125,6251,558],{"className":6252},[652,249],[125,6254,150],{"className":6255},[201,249],[125,6257,5116],{"className":6258},[191,192,249],[125,6260,506],{"className":6261},[505],[125,6263,6265],{"className":6264},[230],[125,6266,6269],{"className":6267,"style":6268},[234],"height:0.4747em;",[125,6270],{},[125,6272],{"className":6273,"style":636},[196],[125,6275,112],{"className":6276},[191,192],[125,6278,544],{"className":6279},[625],[125,6281,6283,6286],{"className":6282},[191],[125,6284,5138],{"className":6285,"style":501},[191,192],[125,6287,6289],{"className":6288},[222],[125,6290,6292,6312],{"className":6291},[226,478],[125,6293,6295,6309],{"className":6294},[230],[125,6296,6298],{"className":6297,"style":1082},[234],[125,6299,6300,6303],{"style":1809},[125,6301],{"className":6302,"style":242},[241],[125,6304,6306],{"className":6305},[246,247,248,249],[125,6307,891],{"className":6308},[191,192,249],[125,6310,506],{"className":6311},[505],[125,6313,6315],{"className":6314},[230],[125,6316,6318],{"className":6317,"style":1104},[234],[125,6319],{},[125,6321,5446],{"className":6322},[191],[125,6324,6326,6329],{"className":6325},[191],[125,6327,5129],{"className":6328},[191,192],[125,6330,6332],{"className":6331},[222],[125,6333,6335,6355],{"className":6334},[226,478],[125,6336,6338,6352],{"className":6337},[230],[125,6339,6341],{"className":6340,"style":1082},[234],[125,6342,6343,6346],{"style":1085},[125,6344],{"className":6345,"style":242},[241],[125,6347,6349],{"className":6348},[246,247,248,249],[125,6350,891],{"className":6351},[191,192,249],[125,6353,506],{"className":6354},[505],[125,6356,6358],{"className":6357},[230],[125,6359,6361],{"className":6360,"style":1104},[234],[125,6362],{},[125,6364,549],{"className":6365},[632],[125,6367],{"className":6368,"style":636},[196],[125,6370,5041],{"className":6371,"style":3712},[191,192],[125,6373,558],{"className":6374},[652],[112,6376,6377,6378,6382],{},"這一理論結果具有深遠的意義：它表明在無限數據和無限容量的極限下，通過交叉熵訓練的模型",[6379,6380,6381],"em",{},"應該","收斂為一個完美的貝葉斯推理機。然而，這僅僅定義了「目標函數」的性質。關鍵的實證問題在於：在有限的參數、有限的數據和特定的架構（Transformer vs. MLP）下，哪種模型能夠真正實現這一數學理想？這正是本研究試圖回答的核心問題。",[108,6384,6386],{"id":6385},"_3-實驗設計構建貝葉斯風洞",[2842,6387,6388],{},"3. 實驗設計：構建貝葉斯風洞",[112,6390,6391,6392,6395,6396,6399],{},"為了全面評估模型的推理能力，研究團隊設計了兩個難度遞增的「風洞」任務：",[2842,6393,6394],{},"雙射學習","（Bijection Learning）與",[2842,6397,6398],{},"隱馬可夫模型狀態追蹤","（HMM State Tracking）。這兩個任務分別測試了離散假設消除和遞迴信念更新兩種根本不同的推理模式。",[4841,6401,6403],{"id":6402},"_31-風洞一雙射學習bijection-learning",[2842,6404,6405],{},"3.1 風洞一：雙射學習（Bijection Learning）",[112,6407,6408],{},"此任務模擬了一個嚴格的邏輯消除過程，類似於數獨或邏輯謎題。",[6410,6411,6412,6799,6805],"ul",{},[4857,6413,6414,6417,6418,6447,6448,6603,6604,6657,6658,6728,6729,2845],{},[2842,6415,6416],{},"任務定義："," 潛在參數 ",[125,6419,6421,6435],{"className":6420},[128],[125,6422,6424],{"className":6423},[132],[134,6425,6426],{"xmlns":136},[138,6427,6428,6432],{},[141,6429,6430],{},[144,6431,5041],{},[170,6433,6434],{"encoding":172},"\\theta",[125,6436,6438],{"className":6437,"ariaHidden":178},[177],[125,6439,6441,6444],{"className":6440},[182],[125,6442],{"className":6443,"style":5066},[186],[125,6445,5041],{"className":6446,"style":3712},[191,192]," 是一個雙射函數（一對一映射） ",[125,6449,6451,6501],{"className":6450},[128],[125,6452,6454],{"className":6453},[132],[134,6455,6456],{"xmlns":136},[138,6457,6458,6498],{},[141,6459,6460,6462,6464,6467,6469,6471,6474,6476,6478,6481,6484,6486,6488,6490,6492,6494,6496],{},[144,6461,5047],{},[148,6463,3834],{},[148,6465,6466],{"stretchy":543},"{",[950,6468,952],{},[148,6470,549],{"separator":178},[148,6472,6473],{},"…",[148,6475,549],{"separator":178},[144,6477,379],{},[148,6479,6480],{"stretchy":543},"}",[148,6482,6483],{},"→",[148,6485,6466],{"stretchy":543},[950,6487,952],{},[148,6489,549],{"separator":178},[148,6491,6473],{},[148,6493,549],{"separator":178},[144,6495,379],{},[148,6497,6480],{"stretchy":543},[170,6499,6500],{"encoding":172},"\\pi: \\{1, \\dots, V\\} \\to \\{1, \\dots, V\\}",[125,6502,6504,6522,6567],{"className":6503,"ariaHidden":178},[177],[125,6505,6507,6510,6513,6516,6519],{"className":6506},[182],[125,6508],{"className":6509,"style":2875},[186],[125,6511,5047],{"className":6512,"style":501},[191,192],[125,6514],{"className":6515,"style":197},[196],[125,6517,3834],{"className":6518},[201],[125,6520],{"className":6521,"style":197},[196],[125,6523,6525,6528,6531,6534,6537,6540,6543,6546,6549,6552,6555,6558,6561,6564],{"className":6524},[182],[125,6526],{"className":6527,"style":613},[186],[125,6529,6466],{"className":6530},[625],[125,6532,952],{"className":6533},[191],[125,6535,549],{"className":6536},[632],[125,6538],{"className":6539,"style":636},[196],[125,6541,6473],{"className":6542},[685],[125,6544],{"className":6545,"style":636},[196],[125,6547,549],{"className":6548},[632],[125,6550],{"className":6551,"style":636},[196],[125,6553,379],{"className":6554,"style":415},[191,192],[125,6556,6480],{"className":6557},[652],[125,6559],{"className":6560,"style":197},[196],[125,6562,6483],{"className":6563},[201],[125,6565],{"className":6566,"style":197},[196],[125,6568,6570,6573,6576,6579,6582,6585,6588,6591,6594,6597,6600],{"className":6569},[182],[125,6571],{"className":6572,"style":613},[186],[125,6574,6466],{"className":6575},[625],[125,6577,952],{"className":6578},[191],[125,6580,549],{"className":6581},[632],[125,6583],{"className":6584,"style":636},[196],[125,6586,6473],{"className":6587},[685],[125,6589],{"className":6590,"style":636},[196],[125,6592,549],{"className":6593},[632],[125,6595],{"className":6596,"style":636},[196],[125,6598,379],{"className":6599,"style":415},[191,192],[125,6601,6480],{"className":6602},[652],"。模型在觀察到 ",[125,6605,6607,6625],{"className":6606},[128],[125,6608,6610],{"className":6609},[132],[134,6611,6612],{"xmlns":136},[138,6613,6614,6622],{},[141,6615,6616,6618,6620],{},[144,6617,984],{},[148,6619,5456],{},[950,6621,952],{},[170,6623,6624],{"encoding":172},"k-1",[125,6626,6628,6647],{"className":6627,"ariaHidden":178},[177],[125,6629,6631,6635,6638,6641,6644],{"className":6630},[182],[125,6632],{"className":6633,"style":6634},[186],"height:0.7778em;vertical-align:-0.0833em;",[125,6636,984],{"className":6637,"style":1410},[191,192],[125,6639],{"className":6640,"style":415},[196],[125,6642,5456],{"className":6643},[259],[125,6645],{"className":6646,"style":415},[196],[125,6648,6650,6654],{"className":6649},[182],[125,6651],{"className":6652,"style":6653},[186],"height:0.6444em;",[125,6655,952],{"className":6656},[191]," 個輸入-輸出對後，必須預測下一個新輸入 ",[125,6659,6661,6679],{"className":6660},[128],[125,6662,6664],{"className":6663},[132],[134,6665,6666],{"xmlns":136},[138,6667,6668,6676],{},[141,6669,6670],{},[393,6671,6672,6674],{},[144,6673,5129],{},[144,6675,984],{},[170,6677,6678],{"encoding":172},"x_k",[125,6680,6682],{"className":6681,"ariaHidden":178},[177],[125,6683,6685,6688],{"className":6684},[182],[125,6686],{"className":6687,"style":1063},[186],[125,6689,6691,6694],{"className":6690},[191],[125,6692,5129],{"className":6693},[191,192],[125,6695,6697],{"className":6696},[222],[125,6698,6700,6720],{"className":6699},[226,478],[125,6701,6703,6717],{"className":6702},[230],[125,6704,6706],{"className":6705,"style":4696},[234],[125,6707,6708,6711],{"style":1085},[125,6709],{"className":6710,"style":242},[241],[125,6712,6714],{"className":6713},[246,247,248,249],[125,6715,984],{"className":6716,"style":1410},[191,192,249],[125,6718,506],{"className":6719},[505],[125,6721,6723],{"className":6722},[230],[125,6724,6726],{"className":6725,"style":1104},[234],[125,6727],{}," 的對應輸出 ",[125,6730,6732,6750],{"className":6731},[128],[125,6733,6735],{"className":6734},[132],[134,6736,6737],{"xmlns":136},[138,6738,6739,6747],{},[141,6740,6741],{},[393,6742,6743,6745],{},[144,6744,5138],{},[144,6746,984],{},[170,6748,6749],{"encoding":172},"y_k",[125,6751,6753],{"className":6752,"ariaHidden":178},[177],[125,6754,6756,6759],{"className":6755},[182],[125,6757],{"className":6758,"style":1954},[186],[125,6760,6762,6765],{"className":6761},[191],[125,6763,5138],{"className":6764,"style":501},[191,192],[125,6766,6768],{"className":6767},[222],[125,6769,6771,6791],{"className":6770},[226,478],[125,6772,6774,6788],{"className":6773},[230],[125,6775,6777],{"className":6776,"style":4696},[234],[125,6778,6779,6782],{"style":1809},[125,6780],{"className":6781,"style":242},[241],[125,6783,6785],{"className":6784},[246,247,248,249],[125,6786,984],{"className":6787,"style":1410},[191,192,249],[125,6789,506],{"className":6790},[505],[125,6792,6794],{"className":6793},[230],[125,6795,6797],{"className":6796,"style":1104},[234],[125,6798],{},[4857,6800,6801,6804],{},[2842,6802,6803],{},"貝葉斯邏輯："," 由於映射是雙射的，每一個觀察到的輸出都排除了該輸出對應其他輸入的可能性。對於一個全新的輸入，其可能的輸出必須均勻分佈在尚未出現過的輸出集合中。",[4857,6806,6807,6808,6858],{},"解析後驗： 若已觀察到 ",[125,6809,6811,6828],{"className":6810},[128],[125,6812,6814],{"className":6813},[132],[134,6815,6816],{"xmlns":136},[138,6817,6818,6826],{},[141,6819,6820,6822,6824],{},[144,6821,984],{},[148,6823,5456],{},[950,6825,952],{},[170,6827,6624],{"encoding":172},[125,6829,6831,6849],{"className":6830,"ariaHidden":178},[177],[125,6832,6834,6837,6840,6843,6846],{"className":6833},[182],[125,6835],{"className":6836,"style":6634},[186],[125,6838,984],{"className":6839,"style":1410},[191,192],[125,6841],{"className":6842,"style":415},[196],[125,6844,5456],{"className":6845},[259],[125,6847],{"className":6848,"style":415},[196],[125,6850,6852,6855],{"className":6851},[182],[125,6853],{"className":6854,"style":6653},[186],[125,6856,952],{"className":6857},[191]," 個不同的輸出，則下一個輸出的真實機率分佈為均勻分佈，其熵（Entropy）為：",[125,6860,6862],{"className":6861},[520],[125,6863,6865,6930],{"className":6864},[128],[125,6866,6868],{"className":6867},[132],[134,6869,6870],{"xmlns":136,"display":529},[138,6871,6872,6927],{},[141,6873,6874,6895,6897,6899,6901,6903,6913,6915,6917,6919,6921,6923,6925],{},[393,6875,6876,6879],{},[144,6877,6878],{},"H",[141,6880,6881,6884,6887,6889,6892],{},[144,6882,6883],{},"B",[144,6885,6886],{},"a",[144,6888,5138],{},[144,6890,6891],{},"e",[144,6893,6894],{},"s",[148,6896,544],{"stretchy":543},[144,6898,984],{},[148,6900,558],{"stretchy":543},[148,6902,561],{},[393,6904,6905,6911],{},[141,6906,6907,6909],{},[144,6908,5459],{},[148,6910,540],{},[950,6912,3887],{},[148,6914,544],{"stretchy":543},[144,6916,379],{},[148,6918,5456],{},[144,6920,984],{},[148,6922,3744],{},[950,6924,952],{},[148,6926,558],{"stretchy":543},[170,6928,6929],{"encoding":172},"H_{Bayes}(k)= \\log_2(V - k + 1)",[125,6931,6933,7015,7081,7099],{"className":6932,"ariaHidden":178},[177],[125,6934,6936,6940,6997,7000,7003,7006,7009,7012],{"className":6935},[182],[125,6937],{"className":6938,"style":6939},[186],"height:1.0361em;vertical-align:-0.2861em;",[125,6941,6943,6947],{"className":6942},[191],[125,6944,6878],{"className":6945,"style":6946},[191,192],"margin-right:0.0813em;",[125,6948,6950],{"className":6949},[222],[125,6951,6953,6989],{"className":6952},[226,478],[125,6954,6956,6986],{"className":6955},[230],[125,6957,6960],{"className":6958,"style":6959},[234],"height:0.3283em;",[125,6961,6963,6966],{"style":6962},"top:-2.55em;margin-left:-0.0813em;margin-right:0.05em;",[125,6964],{"className":6965,"style":242},[241],[125,6967,6969],{"className":6968},[246,247,248,249],[125,6970,6972,6976,6979,6982],{"className":6971},[191,249],[125,6973,6883],{"className":6974,"style":6975},[191,192,249],"margin-right:0.0502em;",[125,6977,6886],{"className":6978},[191,192,249],[125,6980,5138],{"className":6981,"style":501},[191,192,249],[125,6983,6985],{"className":6984},[191,192,249],"es",[125,6987,506],{"className":6988},[505],[125,6990,6992],{"className":6991},[230],[125,6993,6995],{"className":6994,"style":1828},[234],[125,6996],{},[125,6998,544],{"className":6999},[625],[125,7001,984],{"className":7002,"style":1410},[191,192],[125,7004,558],{"className":7005},[652],[125,7007],{"className":7008,"style":197},[196],[125,7010,561],{"className":7011},[201],[125,7013],{"className":7014,"style":197},[196],[125,7016,7018,7021,7066,7069,7072,7075,7078],{"className":7017},[182],[125,7019],{"className":7020,"style":613},[186],[125,7022,7024,7029],{"className":7023},[617],[125,7025,5668,7027],{"className":7026},[617],[125,7028,5672],{"style":5671},[125,7030,7032],{"className":7031},[222],[125,7033,7035,7057],{"className":7034},[226,478],[125,7036,7038,7054],{"className":7037},[230],[125,7039,7042],{"className":7040,"style":7041},[234],"height:0.207em;",[125,7043,7045,7048],{"style":7044},"top:-2.4559em;margin-right:0.05em;",[125,7046],{"className":7047,"style":242},[241],[125,7049,7051],{"className":7050},[246,247,248,249],[125,7052,3887],{"className":7053},[191,249],[125,7055,506],{"className":7056},[505],[125,7058,7060],{"className":7059},[230],[125,7061,7064],{"className":7062,"style":7063},[234],"height:0.2441em;",[125,7065],{},[125,7067,544],{"className":7068},[625],[125,7070,379],{"className":7071,"style":415},[191,192],[125,7073],{"className":7074,"style":415},[196],[125,7076,5456],{"className":7077},[259],[125,7079],{"className":7080,"style":415},[196],[125,7082,7084,7087,7090,7093,7096],{"className":7083},[182],[125,7085],{"className":7086,"style":6634},[186],[125,7088,984],{"className":7089,"style":1410},[191,192],[125,7091],{"className":7092,"style":415},[196],[125,7094,3744],{"className":7095},[259],[125,7097],{"className":7098,"style":415},[196],[125,7100,7102,7105,7108],{"className":7101},[182],[125,7103],{"className":7104,"style":613},[186],[125,7106,952],{"className":7107},[191],[125,7109,558],{"className":7110},[652],[112,7112,7113],{},"隨著序列推進，可能的假設數量減少，熵呈現階梯狀下降。",[6410,7115,7116],{},[4857,7117,7118,7121,7122,7175,7176,7331,7332,7395],{},[2842,7119,7120],{},"複雜度控制","： 設定詞彙表大小 ",[125,7123,7125,7144],{"className":7124},[128],[125,7126,7128],{"className":7127},[132],[134,7129,7130],{"xmlns":136},[138,7131,7132,7141],{},[141,7133,7134,7136,7138],{},[144,7135,379],{},[148,7137,561],{},[950,7139,7140],{},"20",[170,7142,7143],{"encoding":172},"V=20",[125,7145,7147,7166],{"className":7146,"ariaHidden":178},[177],[125,7148,7150,7154,7157,7160,7163],{"className":7149},[182],[125,7151],{"className":7152,"style":7153},[186],"height:0.6833em;",[125,7155,379],{"className":7156,"style":415},[191,192],[125,7158],{"className":7159,"style":197},[196],[125,7161,561],{"className":7162},[201],[125,7164],{"className":7165,"style":197},[196],[125,7167,7169,7172],{"className":7168},[182],[125,7170],{"className":7171,"style":6653},[186],[125,7173,7140],{"className":7174},[191],"，可能的雙射函數總數為 ",[125,7177,7179,7233],{"className":7178},[128],[125,7180,7182],{"className":7181},[132],[134,7183,7184],{"xmlns":136},[138,7185,7186,7230],{},[141,7187,7188,7190,7192,7195,7197,7199,7201,7204,7206,7208,7211,7213,7216,7218,7220,7222,7224],{},[950,7189,7140],{},[568,7191,570],{},[196,7193],{"linebreak":7194},"newline",[144,7196,6886],{},[144,7198,112],{},[144,7200,112],{},[144,7202,7203],{},"r",[144,7205,927],{},[144,7207,5129],{},[950,7209,7210],{},"2.4",[196,7212],{"linebreak":7194},[144,7214,7215],{},"t",[144,7217,891],{},[144,7219,289],{},[144,7221,6891],{},[144,7223,6894],{},[152,7225,7226,7228],{},[950,7227,4885],{},[950,7229,4888],{},[170,7231,7232],{"encoding":172},"20\\! \\\\approx 2.4 \\\\times 10^{18}",[125,7234,7236,7248,7251,7277,7280],{"className":7235,"ariaHidden":178},[177],[125,7237,7239,7242,7245],{"className":7238},[182],[125,7240],{"className":7241,"style":6653},[186],[125,7243,7140],{"className":7244},[191],[125,7246],{"className":7247,"style":678},[196],[125,7249],{"className":7250},[196,7194],[125,7252,7254,7258,7261,7265,7268,7271,7274],{"className":7253},[182],[125,7255],{"className":7256,"style":7257},[186],"height:0.8389em;vertical-align:-0.1944em;",[125,7259,6886],{"className":7260},[191,192],[125,7262,7264],{"className":7263},[191,192],"pp",[125,7266,7203],{"className":7267,"style":3712},[191,192],[125,7269,927],{"className":7270},[191,192],[125,7272,5129],{"className":7273},[191,192],[125,7275,7210],{"className":7276},[191],[125,7278],{"className":7279},[196,7194],[125,7281,7283,7286,7289,7293,7296,7299],{"className":7282},[182],[125,7284],{"className":7285,"style":4901},[186],[125,7287,7215],{"className":7288},[191,192],[125,7290,7292],{"className":7291},[191,192],"im",[125,7294,6985],{"className":7295},[191,192],[125,7297,952],{"className":7298},[191],[125,7300,7302,7305],{"className":7301},[191],[125,7303,3820],{"className":7304},[191],[125,7306,7308],{"className":7307},[222],[125,7309,7311],{"className":7310},[226],[125,7312,7314],{"className":7313},[230],[125,7315,7317],{"className":7316,"style":4901},[234],[125,7318,7319,7322],{"style":237},[125,7320],{"className":7321,"style":242},[241],[125,7323,7325],{"className":7324},[246,247,248,249],[125,7326,7328],{"className":7327},[191,249],[125,7329,4888],{"className":7330},[191,249],"。由於訓練樣本僅有 ",[125,7333,7335,7354],{"className":7334},[128],[125,7336,7338],{"className":7337},[132],[134,7339,7340],{"xmlns":136},[138,7341,7342,7351],{},[141,7343,7344],{},[152,7345,7346,7348],{},[950,7347,4885],{},[950,7349,7350],{},"5",[170,7352,7353],{"encoding":172},"10^5",[125,7355,7357],{"className":7356,"ariaHidden":178},[177],[125,7358,7360,7363,7366],{"className":7359},[182],[125,7361],{"className":7362,"style":4901},[186],[125,7364,952],{"className":7365},[191],[125,7367,7369,7372],{"className":7368},[191],[125,7370,3820],{"className":7371},[191],[125,7373,7375],{"className":7374},[222],[125,7376,7378],{"className":7377},[226],[125,7379,7381],{"className":7380},[230],[125,7382,7384],{"className":7383,"style":4901},[234],[125,7385,7386,7389],{"style":237},[125,7387],{"className":7388,"style":242},[241],[125,7390,7392],{"className":7391},[246,247,248,249],[125,7393,7350],{"className":7394},[191,249]," 筆，模型在訓練中幾乎不可能見過測試集中的雙射函數。因此，模型無法「背誦」答案，必須學會「排除法」這一演算法。",[4841,7397,7399],{"id":7398},"_32-風洞二隱馬可夫模型狀態追蹤hmm-state-tracking",[2842,7400,7401],{},"3.2 風洞二：隱馬可夫模型狀態追蹤（HMM State Tracking）",[112,7403,7404],{},"此任務測試模型處理隨機性、噪聲以及遞迴狀態更新的能力，要求模型在內部激活中實現「前向演算法」（Forward Algorithm）。",[6410,7406,7407,7688],{},[4857,7408,7409,7411,7412,7415,7416,7446,7447,7476,7477,7615,7616,7687],{},[2842,7410,6416],{}," 每個序列由一個",[6379,7413,7414],{},"全新","的隱馬可夫模型生成。序列開頭包含一個「標頭」（Header），編碼了該特定 HMM 的轉移矩陣（",[125,7417,7419,7433],{"className":7418},[128],[125,7420,7422],{"className":7421},[132],[134,7423,7424],{"xmlns":136},[138,7425,7426,7431],{},[141,7427,7428],{},[144,7429,7430],{},"T",[170,7432,7430],{"encoding":172},[125,7434,7436],{"className":7435,"ariaHidden":178},[177],[125,7437,7439,7442],{"className":7438},[182],[125,7440],{"className":7441,"style":7153},[186],[125,7443,7430],{"className":7444,"style":7445},[191,192],"margin-right:0.1389em;","）和發射矩陣（",[125,7448,7450,7463],{"className":7449},[128],[125,7451,7453],{"className":7452},[132],[134,7454,7455],{"xmlns":136},[138,7456,7457,7461],{},[141,7458,7459],{},[144,7460,5406],{},[170,7462,5406],{"encoding":172},[125,7464,7466],{"className":7465,"ariaHidden":178},[177],[125,7467,7469,7472],{"className":7468},[182],[125,7470],{"className":7471,"style":7153},[186],[125,7473,5406],{"className":7474,"style":7475},[191,192],"margin-right:0.0576em;","）。隨後，模型觀察一連串的觀測值 ",[125,7478,7480,7510],{"className":7479},[128],[125,7481,7483],{"className":7482},[132],[134,7484,7485],{"xmlns":136},[138,7486,7487,7507],{},[141,7488,7489,7495,7497,7503,7505],{},[393,7490,7491,7493],{},[144,7492,927],{},[950,7494,952],{},[148,7496,549],{"separator":178},[393,7498,7499,7501],{},[144,7500,927],{},[950,7502,3887],{},[148,7504,549],{"separator":178},[148,7506,6473],{},[170,7508,7509],{"encoding":172},"o_1, o_2, \\dots",[125,7511,7513],{"className":7512,"ariaHidden":178},[177],[125,7514,7516,7519,7560,7563,7566,7606,7609,7612],{"className":7515},[182],[125,7517],{"className":7518,"style":1954},[186],[125,7520,7522,7525],{"className":7521},[191],[125,7523,927],{"className":7524},[191,192],[125,7526,7528],{"className":7527},[222],[125,7529,7531,7552],{"className":7530},[226,478],[125,7532,7534,7549],{"className":7533},[230],[125,7535,7538],{"className":7536,"style":7537},[234],"height:0.3011em;",[125,7539,7540,7543],{"style":1085},[125,7541],{"className":7542,"style":242},[241],[125,7544,7546],{"className":7545},[246,247,248,249],[125,7547,952],{"className":7548},[191,249],[125,7550,506],{"className":7551},[505],[125,7553,7555],{"className":7554},[230],[125,7556,7558],{"className":7557,"style":1104},[234],[125,7559],{},[125,7561,549],{"className":7562},[632],[125,7564],{"className":7565,"style":636},[196],[125,7567,7569,7572],{"className":7568},[191],[125,7570,927],{"className":7571},[191,192],[125,7573,7575],{"className":7574},[222],[125,7576,7578,7598],{"className":7577},[226,478],[125,7579,7581,7595],{"className":7580},[230],[125,7582,7584],{"className":7583,"style":7537},[234],[125,7585,7586,7589],{"style":1085},[125,7587],{"className":7588,"style":242},[241],[125,7590,7592],{"className":7591},[246,247,248,249],[125,7593,3887],{"className":7594},[191,249],[125,7596,506],{"className":7597},[505],[125,7599,7601],{"className":7600},[230],[125,7602,7604],{"className":7603,"style":1104},[234],[125,7605],{},[125,7607,549],{"className":7608},[632],[125,7610],{"className":7611,"style":636},[196],[125,7613,6473],{"className":7614},[685],"，並需預測當前隱藏狀態 ",[125,7617,7619,7637],{"className":7618},[128],[125,7620,7622],{"className":7621},[132],[134,7623,7624],{"xmlns":136},[138,7625,7626,7634],{},[141,7627,7628],{},[393,7629,7630,7632],{},[144,7631,6894],{},[144,7633,7215],{},[170,7635,7636],{"encoding":172},"s_t",[125,7638,7640],{"className":7639,"ariaHidden":178},[177],[125,7641,7643,7646],{"className":7642},[182],[125,7644],{"className":7645,"style":1063},[186],[125,7647,7649,7652],{"className":7648},[191],[125,7650,6894],{"className":7651},[191,192],[125,7653,7655],{"className":7654},[222],[125,7656,7658,7679],{"className":7657},[226,478],[125,7659,7661,7676],{"className":7660},[230],[125,7662,7665],{"className":7663,"style":7664},[234],"height:0.2806em;",[125,7666,7667,7670],{"style":1085},[125,7668],{"className":7669,"style":242},[241],[125,7671,7673],{"className":7672},[246,247,248,249],[125,7674,7215],{"className":7675},[191,192,249],[125,7677,506],{"className":7678},[505],[125,7680,7682],{"className":7681},[230],[125,7683,7685],{"className":7684,"style":1104},[234],[125,7686],{}," 的後驗機率分佈。",[4857,7689,7690],{},"貝葉斯邏輯： 真實的貝葉斯後驗由遞迴的前向演算法給出：",[125,7692,7694],{"className":7693},[520],[125,7695,7697,7789],{"className":7696},[128],[125,7698,7700],{"className":7699},[132],[134,7701,7702],{"xmlns":136,"display":529},[138,7703,7704,7786],{},[141,7705,7706,7713,7715,7717,7719,7721,7723,7725,7731,7733,7735,7737,7748,7750,7752,7754,7756,7762,7764,7776,7778,7784],{},[393,7707,7708,7711],{},[144,7709,7710],{},"α",[144,7712,7215],{},[148,7714,544],{"stretchy":543},[144,7716,6894],{},[148,7718,558],{"stretchy":543},[148,7720,6010],{},[144,7722,5406],{},[148,7724,544],{"stretchy":543},[393,7726,7727,7729],{},[144,7728,927],{},[144,7730,7215],{},[144,7732,5446],{"mathvariant":536},[144,7734,6894],{},[148,7736,558],{"stretchy":543},[2313,7738,7739,7741],{},[148,7740,941],{},[152,7742,7743,7745],{},[144,7744,6894],{},[148,7746,7747],{"mathvariant":536,"lspace":3810,"rspace":3810},"′",[144,7749,7430],{},[148,7751,544],{"stretchy":543},[144,7753,6894],{},[144,7755,5446],{"mathvariant":536},[152,7757,7758,7760],{},[144,7759,6894],{},[148,7761,7747],{"mathvariant":536,"lspace":3810,"rspace":3810},[148,7763,558],{"stretchy":543},[393,7765,7766,7768],{},[144,7767,7710],{},[141,7769,7770,7772,7774],{},[144,7771,7215],{},[148,7773,5456],{},[950,7775,952],{},[148,7777,544],{"stretchy":543},[152,7779,7780,7782],{},[144,7781,6894],{},[148,7783,7747],{"mathvariant":536,"lspace":3810,"rspace":3810},[148,7785,558],{"stretchy":543},[170,7787,7788],{"encoding":172},"\\alpha_t(s) \\propto E(o_t|s) \\sum_{s'} T(s|s') \\alpha_{t-1}(s')",[125,7790,7792,7858],{"className":7791,"ariaHidden":178},[177],[125,7793,7795,7798,7840,7843,7846,7849,7852,7855],{"className":7794},[182],[125,7796],{"className":7797,"style":613},[186],[125,7799,7801,7805],{"className":7800},[191],[125,7802,7710],{"className":7803,"style":7804},[191,192],"margin-right:0.0037em;",[125,7806,7808],{"className":7807},[222],[125,7809,7811,7832],{"className":7810},[226,478],[125,7812,7814,7829],{"className":7813},[230],[125,7815,7817],{"className":7816,"style":7664},[234],[125,7818,7820,7823],{"style":7819},"top:-2.55em;margin-left:-0.0037em;margin-right:0.05em;",[125,7821],{"className":7822,"style":242},[241],[125,7824,7826],{"className":7825},[246,247,248,249],[125,7827,7215],{"className":7828},[191,192,249],[125,7830,506],{"className":7831},[505],[125,7833,7835],{"className":7834},[230],[125,7836,7838],{"className":7837,"style":1104},[234],[125,7839],{},[125,7841,544],{"className":7842},[625],[125,7844,6894],{"className":7845},[191,192],[125,7847,558],{"className":7848},[652],[125,7850],{"className":7851,"style":197},[196],[125,7853,6010],{"className":7854},[201],[125,7856],{"className":7857,"style":197},[196],[125,7859,7861,7865,7868,7871,7911,7914,7917,7920,7923,8007,8010,8013,8016,8019,8022,8055,8058,8108,8111,8143],{"className":7860},[182],[125,7862],{"className":7863,"style":7864},[186],"height:2.344em;vertical-align:-1.294em;",[125,7866,5406],{"className":7867,"style":7475},[191,192],[125,7869,544],{"className":7870},[625],[125,7872,7874,7877],{"className":7873},[191],[125,7875,927],{"className":7876},[191,192],[125,7878,7880],{"className":7879},[222],[125,7881,7883,7903],{"className":7882},[226,478],[125,7884,7886,7900],{"className":7885},[230],[125,7887,7889],{"className":7888,"style":7664},[234],[125,7890,7891,7894],{"style":1085},[125,7892],{"className":7893,"style":242},[241],[125,7895,7897],{"className":7896},[246,247,248,249],[125,7898,7215],{"className":7899},[191,192,249],[125,7901,506],{"className":7902},[505],[125,7904,7906],{"className":7905},[230],[125,7907,7909],{"className":7908,"style":1104},[234],[125,7910],{},[125,7912,5446],{"className":7913},[191],[125,7915,6894],{"className":7916},[191,192],[125,7918,558],{"className":7919},[652],[125,7921],{"className":7922,"style":636},[196],[125,7924,7927],{"className":7925},[617,7926],"op-limits",[125,7928,7930,7998],{"className":7929},[226,478],[125,7931,7933,7995],{"className":7932},[230],[125,7934,7937,7984],{"className":7935,"style":7936},[234],"height:1.05em;",[125,7938,7940,7944],{"style":7939},"top:-1.856em;margin-left:0em;",[125,7941],{"className":7942,"style":7943},[241],"height:3.05em;",[125,7945,7947],{"className":7946},[246,247,248,249],[125,7948,7950],{"className":7949},[191,249],[125,7951,7953,7956],{"className":7952},[191,249],[125,7954,6894],{"className":7955},[191,192,249],[125,7957,7959],{"className":7958},[222],[125,7960,7962],{"className":7961},[226],[125,7963,7965],{"className":7964},[230],[125,7966,7969],{"className":7967,"style":7968},[234],"height:0.6828em;",[125,7970,7972,7975],{"style":7971},"top:-2.786em;margin-right:0.0714em;",[125,7973],{"className":7974,"style":492},[241],[125,7976,7978],{"className":7977},[246,496,497,249],[125,7979,7981],{"className":7980},[191,249],[125,7982,7747],{"className":7983},[191,249],[125,7985,7987,7990],{"style":7986},"top:-3.05em;",[125,7988],{"className":7989,"style":7943},[241],[125,7991,7992],{},[125,7993,941],{"className":7994},[617,1158,5865],[125,7996,506],{"className":7997},[505],[125,7999,8001],{"className":8000},[230],[125,8002,8005],{"className":8003,"style":8004},[234],"height:1.294em;",[125,8006],{},[125,8008],{"className":8009,"style":636},[196],[125,8011,7430],{"className":8012,"style":7445},[191,192],[125,8014,544],{"className":8015},[625],[125,8017,6894],{"className":8018},[191,192],[125,8020,5446],{"className":8021},[191],[125,8023,8025,8028],{"className":8024},[191],[125,8026,6894],{"className":8027},[191,192],[125,8029,8031],{"className":8030},[222],[125,8032,8034],{"className":8033},[226],[125,8035,8037],{"className":8036},[230],[125,8038,8041],{"className":8039,"style":8040},[234],"height:0.8019em;",[125,8042,8043,8046],{"style":4103},[125,8044],{"className":8045,"style":242},[241],[125,8047,8049],{"className":8048},[246,247,248,249],[125,8050,8052],{"className":8051},[191,249],[125,8053,7747],{"className":8054},[191,249],[125,8056,558],{"className":8057},[652],[125,8059,8061,8064],{"className":8060},[191],[125,8062,7710],{"className":8063,"style":7804},[191,192],[125,8065,8067],{"className":8066},[222],[125,8068,8070,8099],{"className":8069},[226,478],[125,8071,8073,8096],{"className":8072},[230],[125,8074,8076],{"className":8075,"style":7537},[234],[125,8077,8078,8081],{"style":7819},[125,8079],{"className":8080,"style":242},[241],[125,8082,8084],{"className":8083},[246,247,248,249],[125,8085,8087,8090,8093],{"className":8086},[191,249],[125,8088,7215],{"className":8089},[191,192,249],[125,8091,5456],{"className":8092},[259,249],[125,8094,952],{"className":8095},[191,249],[125,8097,506],{"className":8098},[505],[125,8100,8102],{"className":8101},[230],[125,8103,8106],{"className":8104,"style":8105},[234],"height:0.2083em;",[125,8107],{},[125,8109,544],{"className":8110},[625],[125,8112,8114,8117],{"className":8113},[191],[125,8115,6894],{"className":8116},[191,192],[125,8118,8120],{"className":8119},[222],[125,8121,8123],{"className":8122},[226],[125,8124,8126],{"className":8125},[230],[125,8127,8129],{"className":8128,"style":8040},[234],[125,8130,8131,8134],{"style":4103},[125,8132],{"className":8133,"style":242},[241],[125,8135,8137],{"className":8136},[246,247,248,249],[125,8138,8140],{"className":8139},[191,249],[125,8141,7747],{"className":8142},[191,249],[125,8144,558],{"className":8145},[652],[112,8147,8148],{},"這要求模型執行複雜的操作：",[4854,8150,8151,8215,8221],{},[4857,8152,8153,8156,8157,8185,8186,8214],{},[2842,8154,8155],{},"解析上下文："," 從標頭中讀取並存儲 ",[125,8158,8160,8173],{"className":8159},[128],[125,8161,8163],{"className":8162},[132],[134,8164,8165],{"xmlns":136},[138,8166,8167,8171],{},[141,8168,8169],{},[144,8170,7430],{},[170,8172,7430],{"encoding":172},[125,8174,8176],{"className":8175,"ariaHidden":178},[177],[125,8177,8179,8182],{"className":8178},[182],[125,8180],{"className":8181,"style":7153},[186],[125,8183,7430],{"className":8184,"style":7445},[191,192]," 和 ",[125,8187,8189,8202],{"className":8188},[128],[125,8190,8192],{"className":8191},[132],[134,8193,8194],{"xmlns":136},[138,8195,8196,8200],{},[141,8197,8198],{},[144,8199,5406],{},[170,8201,5406],{"encoding":172},[125,8203,8205],{"className":8204,"ariaHidden":178},[177],[125,8206,8208,8211],{"className":8207},[182],[125,8209],{"className":8210,"style":7153},[186],[125,8212,5406],{"className":8213,"style":7475},[191,192]," 矩陣。",[4857,8216,8217,8220],{},[2842,8218,8219],{},"狀態維護："," 在殘差流（Residual Stream）中維護一個信念狀態（Belief State）。",[4857,8222,8223,8226],{},[2842,8224,8225],{},"遞迴更新："," 每一步根據新的觀測值和轉移邏輯更新信念。",[6410,8228,8229],{},[4857,8230,8231,8234,8235,8185,8263,8291,8292,8356],{},[2842,8232,8233],{},"複雜度控制："," ",[125,8236,8238,8251],{"className":8237},[128],[125,8239,8241],{"className":8240},[132],[134,8242,8243],{"xmlns":136},[138,8244,8245,8249],{},[141,8246,8247],{},[144,8248,7430],{},[170,8250,7430],{"encoding":172},[125,8252,8254],{"className":8253,"ariaHidden":178},[177],[125,8255,8257,8260],{"className":8256},[182],[125,8258],{"className":8259,"style":7153},[186],[125,8261,7430],{"className":8262,"style":7445},[191,192],[125,8264,8266,8279],{"className":8265},[128],[125,8267,8269],{"className":8268},[132],[134,8270,8271],{"xmlns":136},[138,8272,8273,8277],{},[141,8274,8275],{},[144,8276,5406],{},[170,8278,5406],{"encoding":172},[125,8280,8282],{"className":8281,"ariaHidden":178},[177],[125,8283,8285,8288],{"className":8284},[182],[125,8286],{"className":8287,"style":7153},[186],[125,8289,5406],{"className":8290,"style":7475},[191,192]," 的參數從 Dirichlet 分佈中採樣。可能的 HMM 空間在連續參數下是無限的，即使離散化後也超過 ",[125,8293,8295,8312],{"className":8294},[128],[125,8296,8298],{"className":8297},[132],[134,8299,8300],{"xmlns":136},[138,8301,8302,8310],{},[141,8303,8304],{},[152,8305,8306,8308],{},[950,8307,4885],{},[950,8309,4956],{},[170,8311,4959],{"encoding":172},[125,8313,8315],{"className":8314,"ariaHidden":178},[177],[125,8316,8318,8321,8324],{"className":8317},[182],[125,8319],{"className":8320,"style":4901},[186],[125,8322,952],{"className":8323},[191],[125,8325,8327,8330],{"className":8326},[191],[125,8328,3820],{"className":8329},[191],[125,8331,8333],{"className":8332},[222],[125,8334,8336],{"className":8335},[226],[125,8337,8339],{"className":8338},[230],[125,8340,8342],{"className":8341,"style":4901},[234],[125,8343,8344,8347],{"style":237},[125,8345],{"className":8346,"style":242},[241],[125,8348,8350],{"className":8349},[246,247,248,249],[125,8351,8353],{"className":8352},[191,249],[125,8354,4956],{"className":8355},[191,249],"。這迫使模型必須成為一個「元學習者」（Meta-Learner），即學會執行前向演算法，而非記憶特定的 HMM。",[4841,8358,8360],{"id":8359},"_33-模型架構與基準對照",[2842,8361,8362],{},"3.3 模型架構與基準對照",[112,8364,8365],{},"為了隔離 Transformer 架構（特別是注意力機制）的貢獻，研究使用了嚴格控制的對照組。",[6410,8367,8368,8624],{},[4857,8369,8370,8373,8374],{},[2842,8371,8372],{},"Transformer 模型："," 採用小型但標準的架構（約 267-268 萬參數）。\n",[6410,8375,8376,8500,8621],{},[4857,8377,8378,8379,2845],{},"雙射任務：6 層，6 個注意力頭，模型維度 ",[125,8380,8382,8416],{"className":8381},[128],[125,8383,8385],{"className":8384},[132],[134,8386,8387],{"xmlns":136},[138,8388,8389,8413],{},[141,8390,8391,8408,8410],{},[393,8392,8393,8395],{},[144,8394,168],{},[141,8396,8397,8399,8401,8403,8405],{},[144,8398,289],{},[144,8400,927],{},[144,8402,168],{},[144,8404,6891],{},[144,8406,8407],{},"l",[148,8409,561],{},[950,8411,8412],{},"192",[170,8414,8415],{"encoding":172},"d_{model}=192",[125,8417,8419,8491],{"className":8418,"ariaHidden":178},[177],[125,8420,8422,8426,8482,8485,8488],{"className":8421},[182],[125,8423],{"className":8424,"style":8425},[186],"height:0.8444em;vertical-align:-0.15em;",[125,8427,8429,8432],{"className":8428},[191],[125,8430,168],{"className":8431},[191,192],[125,8433,8435],{"className":8434},[222],[125,8436,8438,8474],{"className":8437},[226,478],[125,8439,8441,8471],{"className":8440},[230],[125,8442,8444],{"className":8443,"style":4696},[234],[125,8445,8446,8449],{"style":1085},[125,8447],{"className":8448,"style":242},[241],[125,8450,8452],{"className":8451},[246,247,248,249],[125,8453,8455,8458,8461,8464,8467],{"className":8454},[191,249],[125,8456,289],{"className":8457},[191,192,249],[125,8459,927],{"className":8460},[191,192,249],[125,8462,168],{"className":8463},[191,192,249],[125,8465,6891],{"className":8466},[191,192,249],[125,8468,8407],{"className":8469,"style":8470},[191,192,249],"margin-right:0.0197em;",[125,8472,506],{"className":8473},[505],[125,8475,8477],{"className":8476},[230],[125,8478,8480],{"className":8479,"style":1104},[234],[125,8481],{},[125,8483],{"className":8484,"style":197},[196],[125,8486,561],{"className":8487},[201],[125,8489],{"className":8490,"style":197},[196],[125,8492,8494,8497],{"className":8493},[182],[125,8495],{"className":8496,"style":6653},[186],[125,8498,8412],{"className":8499},[191],[4857,8501,8502,8503,2845],{},"HMM 任務：9 層，8 個注意力頭，模型維度 ",[125,8504,8506,8539],{"className":8505},[128],[125,8507,8509],{"className":8508},[132],[134,8510,8511],{"xmlns":136},[138,8512,8513,8536],{},[141,8514,8515,8531,8533],{},[393,8516,8517,8519],{},[144,8518,168],{},[141,8520,8521,8523,8525,8527,8529],{},[144,8522,289],{},[144,8524,927],{},[144,8526,168],{},[144,8528,6891],{},[144,8530,8407],{},[148,8532,561],{},[950,8534,8535],{},"256",[170,8537,8538],{"encoding":172},"d_{model}=256",[125,8540,8542,8612],{"className":8541,"ariaHidden":178},[177],[125,8543,8545,8548,8603,8606,8609],{"className":8544},[182],[125,8546],{"className":8547,"style":8425},[186],[125,8549,8551,8554],{"className":8550},[191],[125,8552,168],{"className":8553},[191,192],[125,8555,8557],{"className":8556},[222],[125,8558,8560,8595],{"className":8559},[226,478],[125,8561,8563,8592],{"className":8562},[230],[125,8564,8566],{"className":8565,"style":4696},[234],[125,8567,8568,8571],{"style":1085},[125,8569],{"className":8570,"style":242},[241],[125,8572,8574],{"className":8573},[246,247,248,249],[125,8575,8577,8580,8583,8586,8589],{"className":8576},[191,249],[125,8578,289],{"className":8579},[191,192,249],[125,8581,927],{"className":8582},[191,192,249],[125,8584,168],{"className":8585},[191,192,249],[125,8587,6891],{"className":8588},[191,192,249],[125,8590,8407],{"className":8591,"style":8470},[191,192,249],[125,8593,506],{"className":8594},[505],[125,8596,8598],{"className":8597},[230],[125,8599,8601],{"className":8600,"style":1104},[234],[125,8602],{},[125,8604],{"className":8605,"style":197},[196],[125,8607,561],{"className":8608},[201],[125,8610],{"className":8611,"style":197},[196],[125,8613,8615,8618],{"className":8614},[182],[125,8616],{"className":8617,"style":6653},[186],[125,8619,8535],{"className":8620},[191],[4857,8622,8623],{},"配置：包含學習的位置嵌入、Pre-norm 殘差塊和標準的多頭自注意力機制。",[4857,8625,8626,8629],{},[2842,8627,8628],{},"MLP 基準模型："," 為了測試注意力機制的必要性，研究訓練了容量匹配（Capacity-matched）的多層感知機（MLP）。這些 MLP 擁有與 Transformer 幾乎相同的參數數量（誤差在 1% 以內），採用 18-20 層的深層結構，並配備了殘差連接和層歸一化（LayerNorm）。這確保了任何性能差異都源於架構本身（是否存在注意力機制），而非參數規模或訓練技巧的差異。",[108,8631,8633],{"id":8632},"_4-實證結果行為層面的深度分析",[2842,8634,8635],{},"4. 實證結果：行為層面的深度分析",[112,8637,8638],{},"實驗結果提供了決定性的證據，表明 Transformer 在演算法層面上與貝葉斯推理具有驚人的一致性，而傳統 MLP 則完全無法勝任此類任務。",[4841,8640,8642],{"id":8641},"_41-雙射任務中的精確熵追蹤",[2842,8643,8644],{},"4.1 雙射任務中的精確熵追蹤",[112,8646,8647],{},"在雙射學習任務中，Transformer 展現了近乎完美的假設消除能力。",[6410,8649,8650,8764,8867],{},[4857,8651,8652,8655,8656,8763],{},[2842,8653,8654],{},"階梯狀熵曲線："," 隨著模型接收到更多的輸入-輸出對，剩餘的可能性空間縮小，預測的不確定性（熵）應呈現離散的階梯狀下降。實驗數據顯示，Transformer 的預測熵曲線與理論上的 ",[125,8657,8659,8693],{"className":8658},[128],[125,8660,8662],{"className":8661},[132],[134,8663,8664],{"xmlns":136},[138,8665,8666,8690],{},[141,8667,8668,8684,8686,8688],{},[393,8669,8670,8672],{},[144,8671,6878],{},[141,8673,8674,8676,8678,8680,8682],{},[144,8675,6883],{},[144,8677,6886],{},[144,8679,5138],{},[144,8681,6891],{},[144,8683,6894],{},[148,8685,544],{"stretchy":543},[144,8687,984],{},[148,8689,558],{"stretchy":543},[170,8691,8692],{"encoding":172},"H_{Bayes}(k)",[125,8694,8696],{"className":8695,"ariaHidden":178},[177],[125,8697,8699,8702,8754,8757,8760],{"className":8698},[182],[125,8700],{"className":8701,"style":6939},[186],[125,8703,8705,8708],{"className":8704},[191],[125,8706,6878],{"className":8707,"style":6946},[191,192],[125,8709,8711],{"className":8710},[222],[125,8712,8714,8746],{"className":8713},[226,478],[125,8715,8717,8743],{"className":8716},[230],[125,8718,8720],{"className":8719,"style":6959},[234],[125,8721,8722,8725],{"style":6962},[125,8723],{"className":8724,"style":242},[241],[125,8726,8728],{"className":8727},[246,247,248,249],[125,8729,8731,8734,8737,8740],{"className":8730},[191,249],[125,8732,6883],{"className":8733,"style":6975},[191,192,249],[125,8735,6886],{"className":8736},[191,192,249],[125,8738,5138],{"className":8739,"style":501},[191,192,249],[125,8741,6985],{"className":8742},[191,192,249],[125,8744,506],{"className":8745},[505],[125,8747,8749],{"className":8748},[230],[125,8750,8752],{"className":8751,"style":1828},[234],[125,8753],{},[125,8755,544],{"className":8756},[625],[125,8758,984],{"className":8759,"style":1410},[191,192],[125,8761,558],{"className":8762},[652]," 曲線幾乎完全重疊（參見原論文 Figure 1）。",[4857,8765,8766,8769,8770,8866],{},[2842,8767,8768],{},"微小的誤差："," 定量分析顯示，Transformer 的平均絕對熵誤差（MAE）僅為 ",[125,8771,8773,8800],{"className":8772},[128],[125,8774,8776],{"className":8775},[132],[134,8777,8778],{"xmlns":136},[138,8779,8780,8797],{},[141,8781,8782,8785,8787],{},[950,8783,8784],{},"3",[148,8786,165],{},[152,8788,8789,8791],{},[950,8790,4885],{},[141,8792,8793,8795],{},[148,8794,5456],{},[950,8796,8784],{},[170,8798,8799],{"encoding":172},"3 \\times 10^{-3}",[125,8801,8803,8822],{"className":8802,"ariaHidden":178},[177],[125,8804,8806,8810,8813,8816,8819],{"className":8805},[182],[125,8807],{"className":8808,"style":8809},[186],"height:0.7278em;vertical-align:-0.0833em;",[125,8811,8784],{"className":8812},[191],[125,8814],{"className":8815,"style":415},[196],[125,8817,165],{"className":8818},[259],[125,8820],{"className":8821,"style":415},[196],[125,8823,8825,8828,8831],{"className":8824},[182],[125,8826],{"className":8827,"style":4901},[186],[125,8829,952],{"className":8830},[191],[125,8832,8834,8837],{"className":8833},[191],[125,8835,3820],{"className":8836},[191],[125,8838,8840],{"className":8839},[222],[125,8841,8843],{"className":8842},[226],[125,8844,8846],{"className":8845},[230],[125,8847,8849],{"className":8848,"style":4901},[234],[125,8850,8851,8854],{"style":237},[125,8852],{"className":8853,"style":242},[241],[125,8855,8857],{"className":8856},[246,247,248,249],[125,8858,8860,8863],{"className":8859},[191,249],[125,8861,5456],{"className":8862},[191,249],[125,8864,8784],{"className":8865},[191,249]," bits。這一誤差級別極低，幾乎等同於單精度浮點數運算引入的數值噪聲，意味著模型已經達到了機器精度的極限。",[4857,8868,8869,8872],{},[2842,8870,8871],{},"序列級別的動態性："," 更重要的是，模型並非僅僅擬合了平均趨勢。在單個序列的層面上（原論文 Figure 2），當模型遇到一個已知的輸入重複出現時，其預測熵會瞬間坍縮至零；而當遇到新輸入時，則準確回到消除曲線。這種「鋸齒狀」的動態行為證明了模型正在進行逐個序列的即時推理，而非簡單的統計平均。",[4841,8874,8876],{"id":8875},"_42-hmm-任務中的遞迴演算法學習",[2842,8877,8878],{},"4.2 HMM 任務中的遞迴演算法學習",[112,8880,8881],{},"HMM 任務的結果進一步證實了模型處理隨機性和遞迴計算的能力。",[6410,8883,8884,9038,9367],{},[4857,8885,8886,8889,8890,8941,8942,9037],{},[2842,8887,8888],{},"前向演算法的模擬："," 在訓練長度（",[125,8891,8893,8911],{"className":8892},[128],[125,8894,8896],{"className":8895},[132],[134,8897,8898],{"xmlns":136},[138,8899,8900,8908],{},[141,8901,8902,8904,8906],{},[144,8903,278],{},[148,8905,561],{},[950,8907,7140],{},[170,8909,8910],{"encoding":172},"K=20",[125,8912,8914,8932],{"className":8913,"ariaHidden":178},[177],[125,8915,8917,8920,8923,8926,8929],{"className":8916},[182],[125,8918],{"className":8919,"style":7153},[186],[125,8921,278],{"className":8922,"style":310},[191,192],[125,8924],{"className":8925,"style":197},[196],[125,8927,561],{"className":8928},[201],[125,8930],{"className":8931,"style":197},[196],[125,8933,8935,8938],{"className":8934},[182],[125,8936],{"className":8937,"style":6653},[186],[125,8939,7140],{"className":8940},[191],"）內，Transformer 的熵誤差低至 ",[125,8943,8945,8972],{"className":8944},[128],[125,8946,8948],{"className":8947},[132],[134,8949,8950],{"xmlns":136},[138,8951,8952,8969],{},[141,8953,8954,8957,8959],{},[950,8955,8956],{},"7.5",[148,8958,165],{},[152,8960,8961,8963],{},[950,8962,4885],{},[141,8964,8965,8967],{},[148,8966,5456],{},[950,8968,7350],{},[170,8970,8971],{"encoding":172},"7.5 \\times 10^{-5}",[125,8973,8975,8993],{"className":8974,"ariaHidden":178},[177],[125,8976,8978,8981,8984,8987,8990],{"className":8977},[182],[125,8979],{"className":8980,"style":8809},[186],[125,8982,8956],{"className":8983},[191],[125,8985],{"className":8986,"style":415},[196],[125,8988,165],{"className":8989},[259],[125,8991],{"className":8992,"style":415},[196],[125,8994,8996,8999,9002],{"className":8995},[182],[125,8997],{"className":8998,"style":4901},[186],[125,9000,952],{"className":9001},[191],[125,9003,9005,9008],{"className":9004},[191],[125,9006,3820],{"className":9007},[191],[125,9009,9011],{"className":9010},[222],[125,9012,9014],{"className":9013},[226],[125,9015,9017],{"className":9016},[230],[125,9018,9020],{"className":9019,"style":4901},[234],[125,9021,9022,9025],{"style":237},[125,9023],{"className":9024,"style":242},[241],[125,9026,9028],{"className":9027},[246,247,248,249],[125,9029,9031,9034],{"className":9030},[191,249],[125,9032,5456],{"className":9033},[191,249],[125,9035,7350],{"className":9036},[191,249]," bits。這意味著模型內部實現的計算過程與數學上的前向演算法在功能上是等價的。",[4857,9039,9040,9043,9044,8185,9096,9148,9149,9201,9202,9252,9253,9315,9316,9366],{},[2842,9041,9042],{},"長度泛化（Length Generalization）："," 這是區分「推理」與「記憶」的黃金標準。研究人員將測試序列長度延伸至 ",[125,9045,9047,9066],{"className":9046},[128],[125,9048,9050],{"className":9049},[132],[134,9051,9052],{"xmlns":136},[138,9053,9054,9063],{},[141,9055,9056,9058,9060],{},[144,9057,278],{},[148,9059,561],{},[950,9061,9062],{},"30",[170,9064,9065],{"encoding":172},"K=30",[125,9067,9069,9087],{"className":9068,"ariaHidden":178},[177],[125,9070,9072,9075,9078,9081,9084],{"className":9071},[182],[125,9073],{"className":9074,"style":7153},[186],[125,9076,278],{"className":9077,"style":310},[191,192],[125,9079],{"className":9080,"style":197},[196],[125,9082,561],{"className":9083},[201],[125,9085],{"className":9086,"style":197},[196],[125,9088,9090,9093],{"className":9089},[182],[125,9091],{"className":9092,"style":6653},[186],[125,9094,9062],{"className":9095},[191],[125,9097,9099,9118],{"className":9098},[128],[125,9100,9102],{"className":9101},[132],[134,9103,9104],{"xmlns":136},[138,9105,9106,9115],{},[141,9107,9108,9110,9112],{},[144,9109,278],{},[148,9111,561],{},[950,9113,9114],{},"50",[170,9116,9117],{"encoding":172},"K=50",[125,9119,9121,9139],{"className":9120,"ariaHidden":178},[177],[125,9122,9124,9127,9130,9133,9136],{"className":9123},[182],[125,9125],{"className":9126,"style":7153},[186],[125,9128,278],{"className":9129,"style":310},[191,192],[125,9131],{"className":9132,"style":197},[196],[125,9134,561],{"className":9135},[201],[125,9137],{"className":9138,"style":197},[196],[125,9140,9142,9145],{"className":9141},[182],[125,9143],{"className":9144,"style":6653},[186],[125,9146,9114],{"className":9147},[191],"（即訓練長度的 1.5 倍和 2.5 倍）。結果顯示，Transformer 在超過訓練邊界（",[125,9150,9152,9170],{"className":9151},[128],[125,9153,9155],{"className":9154},[132],[134,9156,9157],{"xmlns":136},[138,9158,9159,9167],{},[141,9160,9161,9163,9165],{},[144,9162,7215],{},[148,9164,561],{},[950,9166,7140],{},[170,9168,9169],{"encoding":172},"t=20",[125,9171,9173,9192],{"className":9172,"ariaHidden":178},[177],[125,9174,9176,9180,9183,9186,9189],{"className":9175},[182],[125,9177],{"className":9178,"style":9179},[186],"height:0.6151em;",[125,9181,7215],{"className":9182},[191,192],[125,9184],{"className":9185,"style":197},[196],[125,9187,561],{"className":9188},[201],[125,9190],{"className":9191,"style":197},[196],[125,9193,9195,9198],{"className":9194},[182],[125,9196],{"className":9197,"style":6653},[186],[125,9199,7140],{"className":9200},[191],"）後，誤差僅呈現平滑且緩慢的增長（",[125,9203,9205,9222],{"className":9204},[128],[125,9206,9208],{"className":9207},[132],[134,9209,9210],{"xmlns":136},[138,9211,9212,9220],{},[141,9213,9214,9216,9218],{},[144,9215,278],{},[148,9217,561],{},[950,9219,9114],{},[170,9221,9117],{"encoding":172},[125,9223,9225,9243],{"className":9224,"ariaHidden":178},[177],[125,9226,9228,9231,9234,9237,9240],{"className":9227},[182],[125,9229],{"className":9230,"style":7153},[186],[125,9232,278],{"className":9233,"style":310},[191,192],[125,9235],{"className":9236,"style":197},[196],[125,9238,561],{"className":9239},[201],[125,9241],{"className":9242,"style":197},[196],[125,9244,9246,9249],{"className":9245},[182],[125,9247],{"className":9248,"style":6653},[186],[125,9250,9114],{"className":9251},[191]," 時 MAE ",[125,9254,9256,9285],{"className":9255},[128],[125,9257,9259],{"className":9258},[132],[134,9260,9261],{"xmlns":136},[138,9262,9263,9282],{},[141,9264,9265,9267,9269,9271,9273,9275,9277,9279],{},[196,9266],{"linebreak":7194},[144,9268,6886],{},[144,9270,112],{},[144,9272,112],{},[144,9274,7203],{},[144,9276,927],{},[144,9278,5129],{},[950,9280,9281],{},"0.02",[170,9283,9284],{"encoding":172},"\\\\approx 0.02",[125,9286,9288,9291],{"className":9287,"ariaHidden":178},[177],[125,9289],{"className":9290},[196,7194],[125,9292,9294,9297,9300,9303,9306,9309,9312],{"className":9293},[182],[125,9295],{"className":9296,"style":7257},[186],[125,9298,6886],{"className":9299},[191,192],[125,9301,7264],{"className":9302},[191,192],[125,9304,7203],{"className":9305,"style":3712},[191,192],[125,9307,927],{"className":9308},[191,192],[125,9310,5129],{"className":9311},[191,192],[125,9313,9281],{"className":9314},[191]," bits），且在 ",[125,9317,9319,9336],{"className":9318},[128],[125,9320,9322],{"className":9321},[132],[134,9323,9324],{"xmlns":136},[138,9325,9326,9334],{},[141,9327,9328,9330,9332],{},[144,9329,7215],{},[148,9331,561],{},[950,9333,7140],{},[170,9335,9169],{"encoding":172},[125,9337,9339,9357],{"className":9338,"ariaHidden":178},[177],[125,9340,9342,9345,9348,9351,9354],{"className":9341},[182],[125,9343],{"className":9344,"style":9179},[186],[125,9346,7215],{"className":9347},[191,192],[125,9349],{"className":9350,"style":197},[196],[125,9352,561],{"className":9353},[201],[125,9355],{"className":9356,"style":197},[196],[125,9358,9360,9363],{"className":9359},[182],[125,9361],{"className":9362,"style":6653},[186],[125,9364,7140],{"className":9365},[191]," 處沒有出現任何突變或斷裂。這證明模型學到的是一個與位置無關的通用遞迴算子（Recursive Operator），而非死記硬背了長度為 20 的特定模式。",[4857,9368,9369,9372,9373,9424],{},[2842,9370,9371],{},"語義不變性（Semantic Invariance）："," 研究人員進行了一項巧妙的測試：隨機置換 HMM 隱藏狀態的標籤（例如將「狀態 1」與「狀態 3」互換），並重新生成數據進行測試。結果顯示，模型的校準誤差在置換前後幾乎完全一致（分佈在 ",[125,9374,9376,9394],{"className":9375},[128],[125,9377,9379],{"className":9378},[132],[134,9380,9381],{"xmlns":136},[138,9382,9383,9391],{},[141,9384,9385,9387,9389],{},[144,9386,5138],{},[148,9388,561],{},[144,9390,5129],{},[170,9392,9393],{"encoding":172},"y=x",[125,9395,9397,9415],{"className":9396,"ariaHidden":178},[177],[125,9398,9400,9403,9406,9409,9412],{"className":9399},[182],[125,9401],{"className":9402,"style":1954},[186],[125,9404,5138],{"className":9405,"style":501},[191,192],[125,9407],{"className":9408,"style":197},[196],[125,9410,561],{"className":9411},[201],[125,9413],{"className":9414,"style":197},[196],[125,9416,9418,9421],{"className":9417},[182],[125,9419],{"className":9420,"style":2875},[186],[125,9422,5129],{"className":9423},[191,192]," 對角線上，如 Figure 8 所示）。這表明模型理解的是 HMM 的拓撲結構和轉移邏輯，而非依賴於特定的狀態 ID。",[4841,9426,9428],{"id":9427},"_43-mlp-的架構性崩潰",[2842,9429,9430],{},"4.3 MLP 的架構性崩潰",[112,9432,9433],{},"與 Transformer 的卓越表現形成強烈對比的是，容量匹配的 MLP 在兩個風洞任務中均遭遇了災難性的失敗。",[6410,9435,9436,9442,9448],{},[4857,9437,9438,9441],{},[2842,9439,9440],{},"無法進行假設消除："," 在雙射任務中，MLP 的 MAE 高達約 1.85 bits，比 Transformer 差了近三個數量級。其熵曲線幾乎保持平坦，顯示它只能學習到輸出的邊際分佈，完全無法根據上下文進行動態的假設剔除。",[4857,9443,9444,9447],{},[2842,9445,9446],{},"無法進行狀態追蹤："," 在 HMM 任務中，MLP 的誤差穩定在 0.4 bits 左右，且不隨序列長度變化。這表明它無法維護或更新信念狀態，只能進行靜態的猜測。",[4857,9449,9450,9453,9454,9457],{},[2842,9451,9452],{},"根本原因分析："," 這一失敗並非源於參數不足或訓練不當，而是源於架構缺陷。MLP 缺乏",[2842,9455,9456],{},"內容定址（Content-Addressable）的路由機制","。在貝葉斯推理中，模型需要根據當前的證據（Query）去檢索相關的假設或參數（Key）。Transformer 的注意力機制天然支持這種操作，而 MLP 只能依賴固定的權重矩陣，無法動態地「查找」上下文中的信息。",[112,9459,9460],{},"為了直觀展示兩種架構的巨大差異，下表總結了關鍵的誤差數據：",[4616,9462,9463,9483],{},[4619,9464,9465],{},[4622,9466,9467,9471,9474,9477,9480],{},[4625,9468,9470],{"align":9469},"left","任務 (Task)",[4625,9472,9473],{"align":9469},"測試條件 (Condition)",[4625,9475,9476],{"align":9469},"Transformer MAE (bits)",[4625,9478,9479],{"align":9469},"MLP MAE (bits)",[4625,9481,9482],{"align":9469},"性能差距 (Degradation Factor)",[4632,9484,9485,9557,9629,9701],{},[4622,9486,9487,9491,9544,9549,9552],{},[4637,9488,9489],{"align":9469},[2842,9490,6394],{},[4637,9492,9493,9494,558],{"align":9469},"訓練長度(",[125,9495,9497,9514],{"className":9496},[128],[125,9498,9500],{"className":9499},[132],[134,9501,9502],{"xmlns":136},[138,9503,9504,9512],{},[141,9505,9506,9508,9510],{},[144,9507,278],{},[148,9509,561],{},[950,9511,7140],{},[170,9513,8910],{"encoding":172},[125,9515,9517,9535],{"className":9516,"ariaHidden":178},[177],[125,9518,9520,9523,9526,9529,9532],{"className":9519},[182],[125,9521],{"className":9522,"style":7153},[186],[125,9524,278],{"className":9525,"style":310},[191,192],[125,9527],{"className":9528,"style":197},[196],[125,9530,561],{"className":9531},[201],[125,9533],{"className":9534,"style":197},[196],[125,9536,9538,9541],{"className":9537},[182],[125,9539],{"className":9540,"style":6653},[186],[125,9542,7140],{"className":9543},[191],[4637,9545,9546],{"align":9469},[2842,9547,9548],{},"0.0030",[4637,9550,9551],{"align":9469},"1.8540",[4637,9553,9554],{"align":9469},[2842,9555,9556],{},"~618倍",[4622,9558,9559,9564,9616,9621,9624],{},[4637,9560,9561],{"align":9469},[2842,9562,9563],{},"HMM 狀態追蹤",[4637,9565,9493,9566,558],{"align":9469},[125,9567,9569,9586],{"className":9568},[128],[125,9570,9572],{"className":9571},[132],[134,9573,9574],{"xmlns":136},[138,9575,9576,9584],{},[141,9577,9578,9580,9582],{},[144,9579,278],{},[148,9581,561],{},[950,9583,7140],{},[170,9585,8910],{"encoding":172},[125,9587,9589,9607],{"className":9588,"ariaHidden":178},[177],[125,9590,9592,9595,9598,9601,9604],{"className":9591},[182],[125,9593],{"className":9594,"style":7153},[186],[125,9596,278],{"className":9597,"style":310},[191,192],[125,9599],{"className":9600,"style":197},[196],[125,9602,561],{"className":9603},[201],[125,9605],{"className":9606,"style":197},[196],[125,9608,9610,9613],{"className":9609},[182],[125,9611],{"className":9612,"style":6653},[186],[125,9614,7140],{"className":9615},[191],[4637,9617,9618],{"align":9469},[2842,9619,9620],{},"0.000075",[4637,9622,9623],{"align":9469},"0.4090",[4637,9625,9626],{"align":9469},[2842,9627,9628],{},"~5,467倍",[4622,9630,9631,9635,9688,9693,9696],{},[4637,9632,9633],{"align":9469},[2842,9634,9563],{},[4637,9636,9637,9638,558],{"align":9469},"泛化長度(",[125,9639,9641,9658],{"className":9640},[128],[125,9642,9644],{"className":9643},[132],[134,9645,9646],{"xmlns":136},[138,9647,9648,9656],{},[141,9649,9650,9652,9654],{},[144,9651,278],{},[148,9653,561],{},[950,9655,9062],{},[170,9657,9065],{"encoding":172},[125,9659,9661,9679],{"className":9660,"ariaHidden":178},[177],[125,9662,9664,9667,9670,9673,9676],{"className":9663},[182],[125,9665],{"className":9666,"style":7153},[186],[125,9668,278],{"className":9669,"style":310},[191,192],[125,9671],{"className":9672,"style":197},[196],[125,9674,561],{"className":9675},[201],[125,9677],{"className":9678,"style":197},[196],[125,9680,9682,9685],{"className":9681},[182],[125,9683],{"className":9684,"style":6653},[186],[125,9686,9062],{"className":9687},[191],[4637,9689,9690],{"align":9469},[2842,9691,9692],{},"0.0125",[4637,9694,9695],{"align":9469},"0.4000",[4637,9697,9698],{"align":9469},[2842,9699,9700],{},"32倍",[4622,9702,9703,9707,9759,9764,9767],{},[4637,9704,9705],{"align":9469},[2842,9706,9563],{},[4637,9708,9637,9709,558],{"align":9469},[125,9710,9712,9729],{"className":9711},[128],[125,9713,9715],{"className":9714},[132],[134,9716,9717],{"xmlns":136},[138,9718,9719,9727],{},[141,9720,9721,9723,9725],{},[144,9722,278],{},[148,9724,561],{},[950,9726,9114],{},[170,9728,9117],{"encoding":172},[125,9730,9732,9750],{"className":9731,"ariaHidden":178},[177],[125,9733,9735,9738,9741,9744,9747],{"className":9734},[182],[125,9736],{"className":9737,"style":7153},[186],[125,9739,278],{"className":9740,"style":310},[191,192],[125,9742],{"className":9743,"style":197},[196],[125,9745,561],{"className":9746},[201],[125,9748],{"className":9749,"style":197},[196],[125,9751,9753,9756],{"className":9752},[182],[125,9754],{"className":9755,"style":6653},[186],[125,9757,9114],{"className":9758},[191],[4637,9760,9761],{"align":9469},[2842,9762,9763],{},"0.0288",[4637,9765,9766],{"align":9469},"0.4020",[4637,9768,9769],{"align":9469},[2842,9770,9771],{},"14倍",[112,9773,9774,9775,2845],{},"這組數據強有力地證明：在需要複雜推理的任務中，注意力機制並非錦上添花，而是",[2842,9776,9777],{},"幾何上的必要條件",[108,9779,9781],{"id":9780},"_5-機制解釋注意力機制的貝葉斯幾何學",[2842,9782,9783],{},"5. 機制解釋：注意力機制的貝葉斯幾何學",[112,9785,9786,9787,9790,9791,9794],{},"既然證實了 Transformer ",[6379,9788,9789],{},"能夠","執行貝葉斯推理，接下來的關鍵問題是：它是",[6379,9792,9793],{},"如何","做到的？Agarwal 等人利用幾何診斷工具，分析了模型內部的向量表徵，揭示了一個跨越層級的三階段運作機制。這一機制將貝葉斯推理的抽象數學運算轉化為具體的幾何操作。",[4841,9796,9798],{"id":9797},"_51-第一階段基礎綁定與假設框架layer-0",[2842,9799,9800],{},"5.1 第一階段：基礎綁定與假設框架（Layer 0）",[112,9802,9803],{},"推理的起點在於構建一個能夠表示所有可能假設的幾何空間，這發生在網絡的最底層。",[6410,9805,9806,9812,9818],{},[4857,9807,9808,9811],{},[2842,9809,9810],{},"正交鍵基底（Orthogonal Key Basis）："," 在 Layer 0，模型學會將輸入 Token 的鍵向量（Keys）組織成一個近似正交的基底。幾何分析顯示，不同輸入 Token 對應的 Key 向量之間的餘弦相似度接近於零（參見 Figure 14）。這意味著每個輸入都被映射到高維空間中的一個獨立軸或子空間上，形成了一個坐標系。",[4857,9813,9814,9817],{},[2842,9815,9816],{},"假設頭（Hypothesis Head）的關鍵性："," 消融研究（Ablation Study）發現，Layer 0 中存在一個特定的注意力頭，其重要性遠超其他頭。移除這個「假設頭」會導致整體誤差激增。這個頭的功能是將原始輸入「綁定」到殘差流中的特定幾何位置，從而定義了後續推理的「假設空間」（Hypothesis Space）。如果沒有這個框架，後續層級就無法對假設進行操作。",[4857,9819,9820,9823],{},[2842,9821,9822],{},"早期穩定性："," 訓練動態顯示，這個幾何框架在訓練早期就已形成並保持穩定。模型首先學會「如何表示問題」，然後才學會「如何精確計算」。",[4841,9825,9827],{"id":9826},"_52-第二階段順序消除與信息路由middle-layers",[2842,9828,9829],{},"5.2 第二階段：順序消除與信息路由（Middle Layers）",[112,9831,9832],{},"一旦假設框架建立，中間層級開始執行核心的貝葉斯更新——即根據證據剔除不可能的假設。",[6410,9834,9835,9841,9847],{},[4857,9836,9837,9840],{},[2842,9838,9839],{},"注意力作為路由（Attention as Routing）："," 注意力機制在此充當了動態路由器的角色。當前位置的查詢向量（Query）會與上下文中的鍵向量（Key）進行比對。",[4857,9842,9843,9846],{},[2842,9844,9845],{},"漸進式銳化（Progressive Sharpening）："," 在淺層，注意力分佈可能較為分散。但隨著層數加深，Query 與 Key 之間的對齊變得越來越尖銳（Sharpened）。在雙射任務的 Layer 5 中，Query 向量幾乎只與那些與當前證據一致的「可行假設」的 Key 產生高相似度（參見 Figure 15）。",[4857,9848,9849,9852,9853,9905],{},[2842,9850,9851],{},"貝葉斯「與」運算（Bayesian AND-ing）："," 這種幾何上的聚焦效應對應於貝葉斯更新中的乘法性質。如果一個假設在第 ",[125,9854,9856,9874],{"className":9855},[128],[125,9857,9859],{"className":9858},[132],[134,9860,9861],{"xmlns":136},[138,9862,9863,9871],{},[141,9864,9865,9867,9869],{},[144,9866,7215],{},[148,9868,5456],{},[950,9870,952],{},[170,9872,9873],{"encoding":172},"t-1",[125,9875,9877,9896],{"className":9876,"ariaHidden":178},[177],[125,9878,9880,9884,9887,9890,9893],{"className":9879},[182],[125,9881],{"className":9882,"style":9883},[186],"height:0.6984em;vertical-align:-0.0833em;",[125,9885,7215],{"className":9886},[191,192],[125,9888],{"className":9889,"style":415},[196],[125,9891,5456],{"className":9892},[259],[125,9894],{"className":9895,"style":415},[196],[125,9897,9899,9902],{"className":9898},[182],[125,9900],{"className":9901,"style":6653},[186],[125,9903,952],{"className":9904},[191]," 步被證據排除，後續層級的注意力機制就會停止從該假設路由信息。前饋網絡（FFN）則負責處理這些路由過來的信息，執行數值上的後驗計算，並更新殘差流中的信念狀態。這一過程展示了 Transformer 如何利用深度（Depth）來實現序列化的邏輯消除。",[4841,9907,9909],{"id":9908},"_53-第三階段流形展開與精度精煉late-layers",[2842,9910,9911],{},"5.3 第三階段：流形展開與精度精煉（Late Layers）",[112,9913,9914],{},"最後的階段涉及將粗略的信念狀態轉化為精確的機率輸出。",[6410,9916,9917,9923,9929],{},[4857,9918,9919,9922],{},[2842,9920,9921],{},"值流形（Value Manifold）："," 研究人員對模型末層的「值向量」（Value Vectors）進行主成分分析（PCA），發現了一個驚人的幾何結構。這些向量並非隨機分佈，而是排列在一個低維流形上（在雙射任務中是一條一維曲線）。",[4857,9924,9925,9928],{},[2842,9926,9927],{},"熵參數化："," 這條曲線被後驗熵完美地參數化。也就是說，模型將其「不確定性」直接編碼為流形上的位置。高熵（不確定）的狀態聚集在曲線的一端，低熵（確定）的狀態則位於另一端（參見 Figure 16）。",[4857,9930,9931,9934],{},[2842,9932,9933],{},"框架-精度分離（Frame-Precision Dissociation）："," 一個重要的發現是，「路由」（注意力圖）的穩定早於「流形」（數值精度）的完善。模型先學會了「去哪裡找信息」（幾何路由），然後在訓練後期才慢慢學會「如何精確表達機率」（流形展開）。這種分離解釋了為什麼模型在訓練後期能顯著降低誤差——它是在一個固定的幾何框架內微調數值精度。",[108,9936,9938],{"id":9937},"_6-深入數據分析與消融研究",[2842,9939,9940],{},"6. 深入數據分析與消融研究",[112,9942,9943],{},"為了驗證上述機制的必要性與穩健性，報告提供了詳細的消融實驗數據。這些數據揭示了 Transformer 內部各組件的功能特異性。",[4841,9945,9947],{"id":9946},"_61-層級與組件的消融影響",[2842,9948,9949],{},"6.1 層級與組件的消融影響",[112,9951,9952],{},"研究人員逐一移除了模型的不同部分，觀察誤差（MAE）的變化。",[4616,9954,9955,9971],{},[4619,9956,9957],{},[4622,9958,9959,9962,9965,9968],{},[4625,9960,9961],{"align":9469},"消融類型 (Ablation Type)",[4625,9963,9964],{"align":9469},"被移除組件 (Component Removed)",[4625,9966,9967],{"align":9469},"對 MAE 的影響",[4625,9969,9970],{"align":9469},"洞察 (Insight)",[4632,9972,9973,10017,10035,10050],{},[4622,9974,9975,9980,10011,10014],{},[4637,9976,9977],{"align":9469},[2842,9978,9979],{},"逐層消融",[4637,9981,9982,9983,558],{"align":9469},"任意單一層 (Layer ",[125,9984,9986,9999],{"className":9985},[128],[125,9987,9989],{"className":9988},[132],[134,9990,9991],{"xmlns":136},[138,9992,9993,9997],{},[141,9994,9995],{},[144,9996,891],{},[170,9998,891],{"encoding":172},[125,10000,10002],{"className":10001,"ariaHidden":178},[177],[125,10003,10005,10008],{"className":10004},[182],[125,10006],{"className":10007,"style":903},[186],[125,10009,891],{"className":10010},[191,192],[4637,10012,10013],{"align":9469},"誤差增加 >10倍",[4637,10015,10016],{"align":9469},"證明計算是深度組合的（Compositional），沒有哪一層是冗餘的。每一層都執行了一個必要的消除步驟。",[4622,10018,10019,10024,10027,10032],{},[4637,10020,10021],{"align":9469},[2842,10022,10023],{},"逐頭消融",[4637,10025,10026],{"align":9469},"Layer 0 「假設頭」",[4637,10028,10029],{"align":9469},[2842,10030,10031],{},"系統性崩潰",[4637,10033,10034],{"align":9469},"此頭負責構建幾何基底。移除它等於摧毀了推理的坐標系，導致後續所有計算失效。",[4622,10036,10037,10041,10044,10047],{},[4637,10038,10039],{"align":9469},[2842,10040,10023],{},[4637,10042,10043],{"align":9469},"後層注意力頭",[4637,10045,10046],{"align":9469},"中等影響",[4637,10048,10049],{"align":9469},"後層頭部主要負責精煉值流形，具有一定的並行冗餘性。",[4622,10051,10052,10057,10060,10104],{},[4637,10053,10054],{"align":9469},[2842,10055,10056],{},"架構變體",[4637,10058,10059],{"align":9469},"頂層兩層的注意力",[4637,10061,10062,10063,10103],{"align":9469},"訓練長度內影響小，但在 ",[125,10064,10066,10085],{"className":10065},[128],[125,10067,10069],{"className":10068},[132],[134,10070,10071],{"xmlns":136},[138,10072,10073,10082],{},[141,10074,10075,10077,10080],{},[144,10076,278],{},[568,10078,10079],{}," ",[950,10081,7140],{},[170,10083,10084],{"encoding":172},"K\\>20",[125,10086,10088],{"className":10087,"ariaHidden":178},[177],[125,10089,10091,10094,10097,10100],{"className":10090},[182],[125,10092],{"className":10093,"style":7153},[186],[125,10095,278],{"className":10096,"style":310},[191,192],[125,10098],{"className":10099,"style":415},[196],[125,10101,7140],{"className":10102},[191]," 時崩潰",[4637,10105,10106,10107,10110],{"align":9469},"頂層注意力對於",[2842,10108,10109],{},"長度泛化","至關重要。它負責在長序列中維持穩定的信念更新，防止誤差累積。",[4841,10112,10114],{"id":10113},"_62-多種子穩健性分析",[2842,10115,10116],{},"6.2 多種子穩健性分析",[112,10118,10119,10120,10197],{},"為了確保這些發現不是隨機初始化的產物，研究在 5 個不同的隨機種子下重複了 HMM 實驗。結果顯示（參見原論文 Figure 11），不同種子之間的誤差曲線在 ",[125,10121,10123,10149],{"className":10122},[128],[125,10124,10126],{"className":10125},[132],[134,10127,10128],{"xmlns":136},[138,10129,10130,10146],{},[141,10131,10132,10134,10136,10138,10140,10142,10144],{},[144,10133,278],{},[148,10135,561],{},[950,10137,7140],{},[148,10139,549],{"separator":178},[950,10141,9062],{},[148,10143,549],{"separator":178},[950,10145,9114],{},[170,10147,10148],{"encoding":172},"K=20, 30, 50",[125,10150,10152,10170],{"className":10151,"ariaHidden":178},[177],[125,10153,10155,10158,10161,10164,10167],{"className":10154},[182],[125,10156],{"className":10157,"style":7153},[186],[125,10159,278],{"className":10160,"style":310},[191,192],[125,10162],{"className":10163,"style":197},[196],[125,10165,561],{"className":10166},[201],[125,10168],{"className":10169,"style":197},[196],[125,10171,10173,10176,10179,10182,10185,10188,10191,10194],{"className":10172},[182],[125,10174],{"className":10175,"style":7257},[186],[125,10177,7140],{"className":10178},[191],[125,10180,549],{"className":10181},[632],[125,10183],{"className":10184,"style":636},[196],[125,10186,9062],{"className":10187},[191],[125,10189,549],{"className":10190},[632],[125,10192],{"className":10193,"style":636},[196],[125,10195,9114],{"className":10196},[191]," 時幾乎完全重合。種子間的變異性遠小於 Transformer 與 MLP 之間的性能差距，這證實了 Transformer 收斂到貝葉斯後驗是一個穩健的架構特性，而非運氣。",[108,10199,10201],{"id":10200},"_7-對大型語言模型與提示工程的啟示",[2842,10202,10203],{},"7. 對大型語言模型與提示工程的啟示",[112,10205,10206],{},"雖然本研究是在小型受控環境中進行的，但其發現對於理解數千億參數的 LLM 及其應用（如提示工程）具有深遠的第二階與第三階啟示。",[4841,10208,10210],{"id":10209},"_71-啟示一幻覺的幾何解釋",[2842,10211,10212],{},"7.1 啟示一：「幻覺」的幾何解釋",[112,10214,10215],{},"本研究發現的「值流形」為 LLM 的幻覺（Hallucination）現象提供了一個嚴謹的幾何解釋。",[6410,10217,10218,10224,10230],{},[4857,10219,10220,10223],{},[2842,10221,10222],{},"機制："," 模型將其信心編碼為高維流形上的坐標。當模型處於高熵狀態（即不確定）時，其內部向量位於流形的「高不確定性」區域。",[4857,10225,10226,10229],{},[2842,10227,10228],{},"失效模式："," 然而，語言生成的最後一步是強制性的採樣（Sampling）。即使內部狀態準確地反映了「我不知道」（高熵），模型仍被迫選擇一個 Token 輸出。如果中間層的路由未能成功消除錯誤假設（即「不完全消除」，Incomplete Elimination），殘差流中的向量可能停留在流形的模糊地帶，導致模型自信地輸出錯誤答案。",[4857,10231,10232,10235],{},[2842,10233,10234],{},"結論："," 幻覺往往不是模型「發瘋」，而是模型在幾何上正確地表示了不確定性，但被解碼過程強制坍縮的結果。",[4841,10237,10239],{"id":10238},"_72-啟示二提示工程即證據供給",[2842,10240,10241],{},"7.2 啟示二：提示工程即「證據供給」",[112,10243,10244,10245,2845],{},"基於貝葉斯視角，提示工程（Prompt Engineering）的本質不再是「下指令」，而是",[2842,10246,10247],{},"注入證據以切割假設空間",[6410,10249,10250,10256,10262,10272],{},[4857,10251,10252,10255],{},[2842,10253,10254],{},"假設空間初始化："," 當輸入一個提示時，Layer 0 立即建立了一個巨大的假設空間（例如：「這是一個代碼任務？還是詩歌創作？是翻譯？」）。",[4857,10257,10258,10261],{},[2842,10259,10260],{},"證據消除："," 提示中的每一個詞、每一個範例都在充當「證據」，用於消除與之不符的子空間。例如，提供代碼範例可以迅速消除「詩歌」和「翻譯」的假設子空間，引導注意力機制聚焦於「編程」軌道。",[4857,10263,10264,10267,10268,10271],{},[2842,10265,10266],{},"邊界範例的重要性："," 正如機器人交換（Robot Exchange）分析所指出的，提供位於",[2842,10269,10270],{},"決策邊界","（Boundaries）的範例比提供典型範例更有效。邊界範例能更銳利地切割流形，幫助注意力機制區分相似的潛在假設。",[4857,10273,10274,10277],{},[2842,10275,10276],{},"領域干擾："," 如果提示中混合了多個領域（例如要求「用寫詩的風格寫代碼」），模型被迫在殘差流中同時維護多個互斥的假設軌道，這可能導致路由干擾（Interference），降低推理精度。清晰、單一領域的提示能讓幾何路由更乾淨地收斂。",[4841,10279,10281],{"id":10280},"_73-啟示三深度即時間depth-as-time",[2842,10282,10283],{},"7.3 啟示三：深度即時間（Depth as Time）",[112,10285,10286,10287,10290],{},"研究發現移除任何一層都會導致誤差激增，這強調了",[2842,10288,10289],{},"深度","在推理中的不可替代性。",[6410,10292,10293,10299],{},[4857,10294,10295,10298],{},[2842,10296,10297],{},"序列化推理："," 貝葉斯推理本質上是序列化的：你必須先根據證據 A 更新信念，才能處理證據 B。在 Transformer 中，層數（Depth）代表了計算步驟或「時間」。",[4857,10300,10301,10304,10305,10308],{},[2842,10302,10303],{},"思維鏈（Chain of Thought）："," 這解釋了為什麼「思維鏈」技術如此有效。通過讓模型生成中間步驟，我們實際上是在利用",[2842,10306,10307],{},"序列維度（Token位置）來擴展模型的虛擬深度","。這給予模型更多的幾何變換次數來執行必要的貝葉斯消除步驟，從而解決那些單次前向傳播（其物理深度有限）無法解決的複雜推理問題。",[108,10310,10312],{"id":10311},"_8-局限性與未來展望",[2842,10313,10314],{},"8. 局限性與未來展望",[112,10316,10317],{},"儘管這項研究具有里程碑意義，但作者也誠實地指出了其局限性，這些局限性為未來的研究指明了方向。",[6410,10319,10320,10377,10383],{},[4857,10321,10322,10325,10326,10376],{},[2842,10323,10324],{},"規模限制："," 實驗使用的是小型模型（約 300 萬參數）和小詞彙表（",[125,10327,10329,10346],{"className":10328},[128],[125,10330,10332],{"className":10331},[132],[134,10333,10334],{"xmlns":136},[138,10335,10336,10344],{},[141,10337,10338,10340,10342],{},[144,10339,379],{},[148,10341,561],{},[950,10343,7140],{},[170,10345,7143],{"encoding":172},[125,10347,10349,10367],{"className":10348,"ariaHidden":178},[177],[125,10350,10352,10355,10358,10361,10364],{"className":10351},[182],[125,10353],{"className":10354,"style":7153},[186],[125,10356,379],{"className":10357,"style":415},[191,192],[125,10359],{"className":10360,"style":197},[196],[125,10362,561],{"className":10363},[201],[125,10365],{"className":10366,"style":197},[196],[125,10368,10370,10373],{"className":10369},[182],[125,10371],{"className":10372,"style":6653},[186],[125,10374,7140],{"className":10375},[191],"）。雖然這足以證明原理，但真實語言模型的潛在空間維度要高得多，且假設空間的結構更為複雜（非結構化）。",[4857,10378,10379,10382],{},[2842,10380,10381],{},"任務簡化："," 風洞任務（雙射與 HMM）具有清晰的結構和邊界。真實世界的推理往往涉及模糊的定義和隱含的規則。",[4857,10384,10385,10388],{},[2842,10386,10387],{},"未來方向：",[4854,10389,10390,10396,10402],{},[4857,10391,10392,10395],{},[2842,10393,10394],{},"擴展風洞："," 開發更複雜的潛在變量模型（如卡爾曼濾波、因果圖模型）的風洞，以測試模型處理連續變量和因果推理的能力。",[4857,10397,10398,10401],{},[2842,10399,10400],{},"大模型診斷："," 將本研究開發的幾何診斷工具（正交性檢測、流形 PCA）應用於預訓練的 LLM（如 LLaMA 或 GPT），檢查在自然語言訓練下是否湧現出類似的貝葉斯幾何結構。",[4857,10403,10404,10407],{},[2842,10405,10406],{},"架構探索："," 測試其他架構（如 Mamba、RWKV 等狀態空間模型）是否也能形成類似的貝葉斯流形，這將有助於理解不同架構的歸納偏置（Inductive Bias)。",[108,10409,10411],{"id":10410},"_9-結論",[2842,10412,10413],{},"9. 結論",[112,10415,10416,10417],{},"《Transformer 注意力機制的貝葉斯幾何學》一文為我們理解現代 AI 提供了一個強大的理論與實證框架。它超越了以往模糊的「神經網絡是黑盒子」的認知，確立了一個具體的科學事實：",[2842,10418,10419],{},"Transformer 是貝葉斯推理的幾何實現引擎。",[112,10421,10422,10423,10426,10427,10430,10431,10434],{},"Agarwal、Dalal 和 Misra 的研究證明，Transformer 之所以能從海量數據中湧現出推理能力，是因為其架構中的",[2842,10424,10425],{},"殘差流","（作為信念載體）、",[2842,10428,10429],{},"注意力機制","（作為內容定址的路由器）和",[2842,10432,10433],{},"前饋網絡","（作為數值更新器）恰好對應了貝葉斯推理的三個核心步驟。它們共同作用，在模型的高維表示空間中雕刻出一個精確的幾何流形，用於整合證據並量化不確定性。",[112,10436,10437],{},"與之相對，MLP 的失敗提醒我們，架構設計至關重要。沒有正確的幾何歸納偏置（如注意力路由），單靠堆砌參數和數據無法產生真正的動態推理能力。",[112,10439,10440],{},"對於 AI 研究者與工程師而言，這份報告不僅證實了 Transformer 的優越性，更提供了一種全新的思維模型：我們不只是在訓練模型預測下一個詞，我們是在構建一個能夠容納並操作機率分佈的幾何機器。理解這一幾何學，將是我們從「模仿智能」邁向「真實推理」的關鍵一步。",[10442,10443,10445],"h4",{"id":10444},"works-cited",[2842,10446,10447],{},"Works cited",[4854,10449,10450,10458],{},[4857,10451,10452,10453],{},"How to Prompt LLM Coding Agents as the Bayesian Inference Engines they are, accessed January 15, 2026, ",[6886,10454,10455],{"href":10455,"rel":10456},"https:\u002F\u002Fwww.robotexchange.io\u002Ft\u002Fhow-to-prompt-llm-coding-agents-as-the-bayesian-inference-engines-they-are\u002F4177",[10457],"nofollow",[4857,10459,10460,10461],{},"The Bayesian Geometry of Transformer Attention - arXiv, accessed January 15, 2026, ",[6886,10462,10463],{"href":10463,"rel":10464},"https:\u002F\u002Farxiv.org\u002Fhtml\u002F2512.22471v2",[10457],{"title":4736,"searchDepth":4749,"depth":4749,"links":10466},[10467,10468,10472,10477,10482,10487,10491,10496,10497],{"id":4821,"depth":4749,"text":4824},{"id":4836,"depth":4749,"text":4839,"children":10469},[10470,10471],{"id":4843,"depth":4755,"text":4846},{"id":5013,"depth":4755,"text":5016},{"id":6385,"depth":4749,"text":6388,"children":10473},[10474,10475,10476],{"id":6402,"depth":4755,"text":6405},{"id":7398,"depth":4755,"text":7401},{"id":8359,"depth":4755,"text":8362},{"id":8632,"depth":4749,"text":8635,"children":10478},[10479,10480,10481],{"id":8641,"depth":4755,"text":8644},{"id":8875,"depth":4755,"text":8878},{"id":9427,"depth":4755,"text":9430},{"id":9780,"depth":4749,"text":9783,"children":10483},[10484,10485,10486],{"id":9797,"depth":4755,"text":9800},{"id":9826,"depth":4755,"text":9829},{"id":9908,"depth":4755,"text":9911},{"id":9937,"depth":4749,"text":9940,"children":10488},[10489,10490],{"id":9946,"depth":4755,"text":9949},{"id":10113,"depth":4755,"text":10116},{"id":10200,"depth":4749,"text":10203,"children":10492},[10493,10494,10495],{"id":10209,"depth":4755,"text":10212},{"id":10238,"depth":4755,"text":10241},{"id":10280,"depth":4755,"text":10283},{"id":10311,"depth":4749,"text":10314},{"id":10410,"depth":4749,"text":10413},"\u002Fimages\u002Fblog\u002F2512.22471v2.webp","2026-01-15","Naman Agarwal等人2026年開創性研究證實Transformer透過注意力機制精確實現貝葉斯推理，揭示其上下文學習的幾何本質並為提示工程奠定理論基礎。",{},"\u002Fblog\u002Fzh-tw\u002Farxiv-2512-22471v2",{"title":4816,"description":10500},"arxiv-2512-22471v2","blog\u002Fzh-TW\u002Farxiv-2512-22471v2",[10507,4807,10508],"paper","bayesian","G_DGUjWKK_hCDisM4vLGWf13eighqDB2SOshjvUv17w",{"id":10511,"title":10512,"author":103,"body":10513,"cover":10718,"coverAlt":10719,"date":10720,"description":10721,"draft":4797,"extension":4798,"featured":4801,"lang":4799,"meta":10722,"navigation":4801,"path":10723,"seo":10724,"slug":10725,"stem":10726,"tags":10727,"translationKey":10725,"updated":4812,"__hash__":10731},"blog\u002Fblog\u002Fzh-TW\u002Fhello-jecision.md","搬家記：把部落格從一個沒人找得到的網域搬回來",{"type":105,"value":10514,"toc":10710},[10515,10518,10521,10539,10545,10548,10551,10557,10572,10575,10578,10598,10616,10622,10625,10637,10647,10650,10664,10670,10673,10698,10701,10707],[108,10516,10517],{"id":10517},"起因是一行原始碼",[112,10519,10520],{},"前陣子要把一篇舊文章轉給同事看，順手抓了頁面原始碼，然後愣住。",[112,10522,10523,10526,10527,10530,10531,10534,10535,10538],{},[4738,10524,10525],{},"\u003Chtml>"," 沒有 ",[4738,10528,10529],{},"lang","。沒有 ",[4738,10532,10533],{},"\u003Ctitle>","。沒有 description，沒有 og，沒有 canonical，沒有一行 JSON-LD。公式的部分更妙——原始碼裡躺著的是 ",[4738,10536,10537],{},"\\operatorname{Attention}"," 這串字，要等瀏覽器把 JavaScript 跑完，人才看得到排版好的數學。",[112,10540,10541,10542],{},"也就是說：",[2842,10543,10544],{},"對搜尋引擎來說，那篇文章從來沒有存在過。",[112,10546,10547],{},"文章寫得再好，放在一個沒有標題、沒有摘要、沒有結構化資料的頁面上，它就只是一段沒有人索引的文字。",[108,10549,10550],{"id":10550},"當初為什麼會變這樣",[112,10552,10553,10554,2845],{},"不是不用心，是",[2842,10555,10556],{},"工具用得比它設計的方式更用力",[112,10558,10559,10560,10563,10564,10567,10568,10571],{},"當時為了讓公式能顯示，我們在 Markdown 裡塞了 ",[4738,10561,10562],{},"\u003Cdiv class=\"latex-block\">","，然後在 ",[4738,10565,10566],{},"onMounted"," 裡用 KaTeX 把它的 ",[4738,10569,10570],{},"textContent"," 換成公式。瀏覽器上看起來完全正常，所以也沒有人覺得有問題。",[112,10573,10574],{},"問題在於這個做法把兩件事綁在一起：內容要靠 JavaScript 才成立，而 SEO 需要的是「不跑 JavaScript 也成立」。",[108,10576,10577],{"id":10577},"搬家的三個決定",[112,10579,10580,10583,10584,1995,10587,10590,10591,1995,10594,10597],{},[2842,10581,10582],{},"一、內容搬家，不是版面搬家。"," 12 篇文章全部重新轉換，公式改寫成 ",[4738,10585,10586],{},"$...$",[4738,10588,10589],{},"$$...$$","，交給 ",[4738,10592,10593],{},"remark-math",[4738,10595,10596],{},"rehype-katex"," 在 build 時處理。現在公式是 HTML 的一部分，不需要任何前端 JavaScript。",[112,10599,10600,10603,10604,10607,10608,10611,10612,10615],{},[2842,10601,10602],{},"二、網址由資料決定，不由檔名決定。"," 舊站的檔名是 ",[4738,10605,10606],{},"2604.04921v1.md","，結果網址變成 ",[4738,10609,10610],{},"\u002Fen\u002Fblog\u002F04921v1","——數字前綴被吃掉，檔名和網址永遠對不起來。新站改成 frontmatter 裡寫 ",[4738,10613,10614],{},"slug","，檔名再亂都不影響網址。",[112,10617,10618,10621],{},[2842,10619,10620],{},"三、舊網址一條都不能斷。"," 已經被 Google 記住的東西，不該因為我們重寫而消失。28 條 301 對照全部列出來，包含尾斜線的版本。",[108,10623,10624],{"id":10624},"三個差點讓上線失敗的意外",[112,10626,10627,10630,10631,10633,10634,10636],{},[2842,10628,10629],{},"第一個：slug 被吃掉。"," 這件事是在舊站發現的——",[4738,10632,10606],{}," 變成 ",[4738,10635,10610],{},"。如果沒有先抓一次線上頁面，我們會以為搬家只是複製檔案。",[112,10638,10639,10642,10643,10646],{},[2842,10640,10641],{},"第二個：hreflang 指到不存在的頁面。"," 新站預設會幫每個語系產生 alternate 連結，但只有中文版的文章，英文連結就是一個 404。跑 ",[4738,10644,10645],{},"nuxt generate"," 的時候 prerender 直接掛掉，錯誤訊息長得像「Article not found，linked from \u002Fblog\u002Fxxx」。",[112,10648,10649],{},"解法不是關掉檢查，而是讓每一頁自己宣告「我實際有哪些語系版本」。只有真的存在的中英版本，才會出現在 hreflang 裡。",[112,10651,10652,10659,10660,10663],{},[2842,10653,10654,10655,10658],{},"第三個：",[4738,10656,10657],{},"\u002Fsitemap.xml"," 是 HTML。"," 這件事最陰險。build 成功、檔案存在、打開也有內容——但它是一個 ",[4738,10661,10662],{},"\u003Cmeta http-equiv=\"refresh\">"," 的轉址頁，不是 XML。Google 會直接判定 sitemap 無效。",[112,10665,10666,10667,10669],{},"後來在 build 後加了一步，把 sitemap index 的內容直接寫成 ",[4738,10668,10657],{},"，才真的能提交。",[108,10671,10672],{"id":10672},"現在的樣子",[6410,10674,10675,10678,10685,10692],{},[4857,10676,10677],{},"公式在伺服器端就排版完成，原始碼裡有 MathML",[4857,10679,10680,10681,10684],{},"每一頁都有 title、description、canonical、hreflang、og，文章另有 ",[4738,10682,10683],{},"BlogPosting"," 結構化資料",[4857,10686,10687,10688,10691],{},"RSS、",[4738,10689,10690],{},"llms.txt","、sitemap、301 對照都有",[4857,10693,10694,10695,10697],{},"還寫了一支 ",[4738,10696,20],{}," 腳本，build 完直接檢查 25 個頁面的 SEO 與壞連結。目前是 0 錯誤 0 警告",[108,10699,10700],{"id":10700},"一句話總結",[112,10702,10703,10704,2845],{},"搬家最花時間的不是搬，是",[2842,10705,10706],{},"確認搬過去的東西真的還在",[112,10708,10709],{},"我們的做法是：把「應該要有」的東西寫成檢查，讓它每次 build 都自己驗一遍。人會忘記，腳本不會。",{"title":4736,"searchDepth":4749,"depth":4749,"links":10711},[10712,10713,10714,10715,10716,10717],{"id":10517,"depth":4749,"text":10517},{"id":10550,"depth":4749,"text":10550},{"id":10577,"depth":4749,"text":10577},{"id":10624,"depth":4749,"text":10624},{"id":10672,"depth":4749,"text":10672},{"id":10700,"depth":4749,"text":10700},"\u002Fimages\u002Fblog\u002Fhello-jecision.svg","從臨時網域搬回品牌網域的示意圖","2026-10-23","舊站的每一篇文章，原始碼裡都沒有 title、沒有 description、沒有 canonical，公式還是純文字。這篇記錄我們把部落格搬回 jecision.com 的過程，以及三個差點讓上線失敗的意外。",{},"\u002Fblog\u002Fzh-tw\u002Fhello-jecision",{"title":10512,"description":10721},"hello-jecision","blog\u002Fzh-TW\u002Fhello-jecision",[10728,10729,10730],"nuxt","seo","重構","eMzmxLsypk5GpqCf6q2_9_s4WWIcInwgrCKPRumQDZc",1792775650216]