{"id":9864,"date":"2026-09-30T12:36:39","date_gmt":"2026-09-30T03:36:39","guid":{"rendered":"https:\/\/since2020.jp\/media\/?p=9864"},"modified":"2026-09-30T12:36:42","modified_gmt":"2026-09-30T03:36:42","slug":"hidden-markov-model","status":"publish","type":"post","link":"https:\/\/since2020.jp\/media\/hidden-markov-model\/","title":{"rendered":"\u305d\u306e\u884c\u52d5\u306e\u88cf\u306b\u4f55\u304c\u3042\u308b\uff1f \u96a0\u308c\u30de\u30eb\u30b3\u30d5\u30e2\u30c7\u30eb\uff08HMM\uff09\u89e3\u8aac"},"content":{"rendered":"\n<h2 class=\"wp-block-heading\">\u96a0\u308c\u30de\u30eb\u30b3\u30d5\u30e2\u30c7\u30eb\u3068\u306f<\/h2>\n\n\n\n<p>\u96a0\u308c\u30de\u30eb\u30b3\u30d5\u30e2\u30c7\u30eb\uff08Hidden Markov Model, HMM\uff09\u306f\u3001\u6642\u9593\u3068\u3068\u3082\u306b\u5909\u5316\u3059\u308b\u30c7\u30fc\u30bf\u3092\u78ba\u7387\u7684\u306b\u8868\u73fe\u3059\u308b\u30e2\u30c7\u30eb\u3067\u3059\u3002\u89b3\u6e2c\u53ef\u80fd\u306a\u30c7\u30fc\u30bf\u306e\u80cc\u5f8c\u306b\u3001\u76f4\u63a5\u306f\u89b3\u6e2c\u3067\u304d\u306a\u3044\u300c\u96a0\u308c\u72b6\u614b\u300d\u304c\u5b58\u5728\u3059\u308b\u3068\u4eee\u5b9a\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<p>\u6642\u7cfb\u5217\u30c7\u30fc\u30bf\u3092\u6271\u3046\u73fe\u5b9f\u306e\u554f\u984c\u3067\u306f\u3001\u89b3\u6e2c\u5024\u305d\u306e\u3082\u306e\u3088\u308a\u3001\u305d\u308c\u3092\u751f\u307f\u51fa\u3057\u3066\u3044\u308b\u5185\u90e8\u72b6\u614b\u306b\u95a2\u5fc3\u304c\u3042\u308b\u3053\u3068\u304c\u5c11\u306a\u304f\u3042\u308a\u307e\u305b\u3093\u3002HMM\u306f\u3001\u3053\u306e\u3088\u3046\u306a\u72b6\u6cc1\u3092\u6570\u7406\u7684\u306b\u6271\u3046\u305f\u3081\u306e\u67a0\u7d44\u307f\u3067\u3059\u3002<\/p>\n\n\n\n<p>HMM\u306e\u57fa\u76e4\u306b\u306f\u3001\u78ba\u7387\u904e\u7a0b\u3068\u3057\u3066\u306e\u30de\u30eb\u30b3\u30d5\u904e\u7a0b\u304c\u3042\u308a\u307e\u3059\u3002\u30de\u30eb\u30b3\u30d5\u904e\u7a0b\u3067\u306f\u3001\u300c\u73fe\u5728\u306e\u72b6\u614b\u304c\u5206\u304b\u308c\u3070\u3001\u5c06\u6765\u306e\u72b6\u614b\u306f\u904e\u53bb\u5168\u4f53\u306b\u4f9d\u5b58\u3057\u306a\u3044\u300d\u3068\u4eee\u5b9a\u3057\u307e\u3059\u3002HMM\u306f\u3001\u3053\u306e\u30de\u30eb\u30b3\u30d5\u6027\u3092\u6301\u3064\u72b6\u614b\u304c\u76f4\u63a5\u89b3\u6e2c\u3067\u304d\u306a\u3044\u5834\u5408\u3092\u6271\u3048\u308b\u3088\u3046\u306b\u62e1\u5f35\u3057\u305f\u30e2\u30c7\u30eb\u3067\u3059\u3002<\/p>\n\n\n\n<p>\u6642\u523b $t=1,2,\\ldots,T$ \u306b\u304a\u3051\u308b\u96a0\u308c\u72b6\u614b\u3092 $X_t$\u3001\u89b3\u6e2c\u5024\u3092 $Y_t$ \u3068\u3057\u307e\u3059\u3002\u89b3\u6e2c\u7cfb\u5217 $Y_1,Y_2,\\ldots,Y_T$ \u306f\u4e0e\u3048\u3089\u308c\u3066\u3044\u307e\u3059\u304c\u3001\u72b6\u614b\u7cfb\u5217 $X_1,X_2,\\ldots,X_T$ \u306f\u76f4\u63a5\u89b3\u6e2c\u3067\u304d\u306a\u3044\u78ba\u7387\u5909\u6570\u3068\u3057\u3066\u6271\u3044\u307e\u3059\u3002<\/p>\n\n\n\n<p>HMM\u304c\u9577\u5e74\u306b\u308f\u305f\u3063\u3066\u5229\u7528\u3055\u308c\u3066\u304d\u305f\u7406\u7531\u306f\u3001\u6642\u9593\u69cb\u9020\u3092\u6301\u3064\u30c7\u30fc\u30bf\u3092\u89e3\u91c8\u3057\u3084\u3059\u3044\u5f62\u3067\u30e2\u30c7\u30eb\u5316\u3067\u304d\u308b\u305f\u3081\u3067\u3059\u3002\u89b3\u6e2c\u3068\u72b6\u614b\u3092\u5206\u3051\u308b\u3053\u3068\u3067\u3001\u30c7\u30fc\u30bf\u751f\u6210\u306e\u4ed5\u7d44\u307f\u3092\u660e\u793a\u7684\u306b\u8868\u73fe\u3067\u304d\u307e\u3059\u3002\u307e\u305f\u3001\u5c24\u5ea6\u8a08\u7b97\u30fb\u72b6\u614b\u63a8\u5b9a\u30fb\u30d1\u30e9\u30e1\u30fc\u30bf\u5b66\u7fd2\u3092\u52b9\u7387\u7684\u306b\u884c\u3046\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u304c\u78ba\u7acb\u3055\u308c\u3066\u304a\u308a\u3001\u97f3\u58f0\u8a8d\u8b58\u3084\u81ea\u7136\u8a00\u8a9e\u51e6\u7406\u306a\u3069\u306e\u5206\u91ce\u3067\u6a19\u6e96\u7684\u306a\u30e2\u30c7\u30eb\u3068\u3057\u3066\u5229\u7528\u3055\u308c\u3066\u304d\u307e\u3057\u305f\u3002<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">\u96a0\u308c\u72b6\u614b\u3068\u89b3\u6e2c\u306e\u95a2\u4fc2<\/h2>\n\n\n\n<p>HMM\u3067\u306f\u3001\u6642\u9593\u3068\u3068\u3082\u306b\u5909\u5316\u3059\u308b\u72b6\u614b\u7cfb\u5217\u3068\u3001\u305d\u306e\u72b6\u614b\u304b\u3089\u751f\u6210\u3055\u308c\u308b\u89b3\u6e2c\u7cfb\u5217\u3092\u533a\u5225\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u72b6\u614b\u306f\u3001\u76f4\u524d\u306e\u72b6\u614b\u306e\u307f\u306b\u4f9d\u5b58\u3057\u3066\u9077\u79fb\u3059\u308b<\/li>\n\n\n\n<li>\u89b3\u6e2c\u5024\u306f\u3001\u305d\u306e\u6642\u523b\u306e\u72b6\u614b\u306b\u4f9d\u5b58\u3057\u3066\u78ba\u7387\u7684\u306b\u751f\u6210\u3055\u308c\u308b<\/li>\n<\/ul>\n\n\n\n<p>\u3053\u306e\u69cb\u9020\u306b\u3088\u308a\u3001\u72b6\u614b\u306e\u6642\u9593\u7684\u306a\u5909\u5316\u3068\u3001\u72b6\u614b\u304b\u3089\u89b3\u6e2c\u304c\u751f\u307e\u308c\u308b\u4ed5\u7d44\u307f\u3092\u5206\u96e2\u3057\u3066\u8868\u73fe\u3067\u304d\u307e\u3059\u3002\u89b3\u6e2c\u30c7\u30fc\u30bf\u3057\u304b\u5f97\u3089\u308c\u306a\u3044\u72b6\u6cc1\u3067\u3082\u3001\u305d\u306e\u80cc\u5f8c\u306b\u3042\u308b\u72b6\u614b\u306e\u9077\u79fb\u3092\u78ba\u7387\u7684\u306b\u63a8\u5b9a\u3067\u304d\u308b\u3053\u3068\u304c\u3001HMM\u306e\u4e2d\u6838\u7684\u306a\u30a2\u30a4\u30c7\u30a2\u3067\u3059\u3002<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">\u4f8b\uff1a\u5098\u304b\u3089\u5929\u6c17\u3092\u63a8\u5b9a\u3059\u308b<\/h3>\n\n\n\n<p>\u7a93\u306e\u306a\u3044\u90e8\u5c4b\u306b\u3044\u3066\u3001\u5916\u306e\u5929\u6c17\uff08\u96a0\u308c\u72b6\u614b\uff1a\u6674\u308c\uff0f\u96e8\uff09\u3092\u76f4\u63a5\u78ba\u8a8d\u3067\u304d\u306a\u3044\u3068\u3057\u307e\u3059\u3002\u4e00\u65b9\u3001\u6bce\u671d\u3001\u540c\u50da\u304c\u5098\u3092\u6301\u3063\u3066\u51fa\u793e\u3057\u305f\u304b\u3069\u3046\u304b\uff08\u89b3\u6e2c\u5024\uff1a\u5098\u3042\u308a\uff0f\u5098\u306a\u3057\uff09\u306f\u5206\u304b\u308a\u307e\u3059\u3002<\/p>\n\n\n\n<p>\u3053\u3053\u3067\u3001\u6b21\u306e2\u3064\u3092\u4eee\u5b9a\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>\u5929\u6c17\u306f\u524d\u65e5\u306e\u5929\u6c17\u306b\u4f9d\u5b58\u3057\u3066\u5909\u5316\u3059\u308b<\/li>\n\n\n\n<li>\u5098\u3092\u6301\u3063\u3066\u304f\u308b\u78ba\u7387\u306f\u3001\u305d\u306e\u65e5\u306e\u5929\u6c17\u306b\u4f9d\u5b58\u3059\u308b<\/li>\n<\/ol>\n\n\n\n<p>\u300c3\u65e5\u9593\u3001\u6bce\u65e5\u5098\u3092\u6301\u3063\u3066\u304d\u305f\u300d\u3068\u3044\u3046\u89b3\u6e2c\u7cfb\u5217\u306f\u3001\u6b21\u306e\u3088\u3046\u306b\u8868\u305b\u307e\u3059\u3002<\/p>\n\n\n\n<p>$$<br>Y_1=\\text{\u5098\u3042\u308a},\\quad<br>Y_2=\\text{\u5098\u3042\u308a},\\quad<br>Y_3=\\text{\u5098\u3042\u308a}<br>$$<\/p>\n\n\n\n<p>\u3053\u306e\u89b3\u6e2c\u3060\u3051\u304b\u3089\u3001\u5b9f\u969b\u306e\u5929\u6c17\u306e\u7cfb\u5217 $(X_1,X_2,X_3)$ \u304c\u3069\u306e\u3088\u3046\u306a\u72b6\u614b\u3060\u3063\u305f\u53ef\u80fd\u6027\u304c\u9ad8\u3044\u304b\u3092\u63a8\u5b9a\u3059\u308b\u3053\u3068\u304c\u3001HMM\u306e\u5178\u578b\u7684\u306a\u4f7f\u3044\u65b9\u3067\u3059\u3002<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" width=\"1024\" height=\"407\" src=\"https:\/\/since2020.jp\/media\/wp-content\/uploads\/2026\/09\/hmm_graphical_model-1024x407.png\" alt=\"\" class=\"wp-image-9865\" srcset=\"https:\/\/since2020.jp\/media\/wp-content\/uploads\/2026\/09\/hmm_graphical_model-1024x407.png 1024w, https:\/\/since2020.jp\/media\/wp-content\/uploads\/2026\/09\/hmm_graphical_model-300x119.png 300w, https:\/\/since2020.jp\/media\/wp-content\/uploads\/2026\/09\/hmm_graphical_model-768x305.png 768w, https:\/\/since2020.jp\/media\/wp-content\/uploads\/2026\/09\/hmm_graphical_model-1536x610.png 1536w, https:\/\/since2020.jp\/media\/wp-content\/uploads\/2026\/09\/hmm_graphical_model-2048x813.png 2048w\" sizes=\"(max-width: 1024px) 100vw, 1024px\" \/><\/figure>\n\n\n\n<p><em>\u56f31\uff1a\u7d2b\u304c\u96a0\u308c\u72b6\u614b\u3001\u7dd1\u304c\u89b3\u6e2c\u5024\u3002\u5404\u6642\u523b\u306e\u89b3\u6e2c\u5024\u306f\u3001\u305d\u306e\u6642\u523b\u306e\u96a0\u308c\u72b6\u614b\u306e\u307f\u306b\u4f9d\u5b58\u3057\u3066\u751f\u6210\u3055\u308c\u308b\u3002<\/em><\/p>\n\n\n\n<h2 class=\"wp-block-heading\">\u30de\u30eb\u30b3\u30d5\u6027\u3068\u72b6\u614b\u9077\u79fb<\/h2>\n\n\n\n<p>HMM\u3067\u306f\u3001\u72b6\u614b\u7cfb\u5217\u304c\u30de\u30eb\u30b3\u30d5\u6027\u3092\u6e80\u305f\u3059\u3068\u4eee\u5b9a\u3057\u307e\u3059\u3002\u3064\u307e\u308a\u3001\u73fe\u5728\u306e\u72b6\u614b\u304c\u5206\u304b\u308c\u3070\u3001\u6b21\u306e\u72b6\u614b\u306f\u305d\u308c\u4ee5\u524d\u306e\u72b6\u614b\u306b\u4f9d\u5b58\u3057\u307e\u305b\u3093\u3002<\/p>\n\n\n\n<div class=\"wp-block-math\"><math display=\"block\"><semantics><mrow><mi>P<\/mi><mspace width=\"-0.1667em\" style=\"margin-left:-0.1667em\"><\/mspace><mrow><mo fence=\"true\" form=\"prefix\">(<\/mo><msub><mi>X<\/mi><mi>t<\/mi><\/msub><mo lspace=\"0.22em\" rspace=\"0.22em\" stretchy=\"false\">|<\/mo><msub><mi>X<\/mi><mrow><mi>t<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msub><mo separator=\"true\">,<\/mo><msub><mi>X<\/mi><mrow><mi>t<\/mi><mo>\u2212<\/mo><mn>2<\/mn><\/mrow><\/msub><mo separator=\"true\">,<\/mo><mo>\u2026<\/mo><mo separator=\"true\">,<\/mo><msub><mi>X<\/mi><mn>1<\/mn><\/msub><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mo>=<\/mo><mi>P<\/mi><mspace width=\"-0.1667em\" style=\"margin-left:-0.1667em\"><\/mspace><mrow><mo fence=\"true\" form=\"prefix\">(<\/mo><msub><mi>X<\/mi><mi>t<\/mi><\/msub><mo lspace=\"0.22em\" rspace=\"0.22em\" stretchy=\"false\">|<\/mo><msub><mi>X<\/mi><mrow><mi>t<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msub><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><annotation encoding=\"application\/x-tex\">P\\!\\left(X_t\\mid X_{t-1},X_{t-2},\\ldots,X_1\\right)\n=\nP\\!\\left(X_t\\mid X_{t-1}\\right)\n<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<p>\u4efb\u610f\u306e\u72b6\u614b $i,j$ \u306b\u5bfe\u3059\u308b\u72b6\u614b\u9077\u79fb\u78ba\u7387\u3092\u3001\u6b21\u306e\u3088\u3046\u306b\u5b9a\u7fa9\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<div class=\"wp-block-math\"><math display=\"block\"><semantics><mrow><msub><mi>a<\/mi><mrow><mi>i<\/mi><mi>j<\/mi><\/mrow><\/msub><mo>=<\/mo><mi>P<\/mi><mspace width=\"-0.1667em\" style=\"margin-left:-0.1667em\"><\/mspace><mrow><mo fence=\"true\" form=\"prefix\">(<\/mo><msub><mi>X<\/mi><mi>t<\/mi><\/msub><mo>=<\/mo><mi>j<\/mi><mo lspace=\"0.22em\" rspace=\"0.22em\" stretchy=\"false\">|<\/mo><msub><mi>X<\/mi><mrow><mi>t<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msub><mo>=<\/mo><mi>i<\/mi><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><annotation encoding=\"application\/x-tex\">a_{ij}\n=\nP\\!\\left(X_t=j\\mid X_{t-1}=i\\right)<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<p>\u5404 $a_{ij}$ \u3092\u8981\u7d20\u3068\u3059\u308b\u884c\u5217 $A=(a_{ij})$ \u304c\u3001\u72b6\u614b\u9077\u79fb\u306e\u30eb\u30fc\u30eb\u3092\u8868\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<p>\u5098\u306e\u4f8b\u3067\u3001\u72b6\u614b\u306e\u9806\u5e8f\u3092\u300c\u6674\u308c\u3001\u96e8\u300d\u3068\u3059\u308b\u3068\u3001\u9077\u79fb\u884c\u5217\u306f\u6b21\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n<p>$$<br>A=<br>\\begin{pmatrix}<br>0.7 &amp; 0.3\\\\<br>0.4 &amp; 0.6<br>\\end{pmatrix}<br>$$<\/p>\n\n\n\n<p>\u884c\u306f\u73fe\u5728\u306e\u5929\u6c17\u3001\u5217\u306f\u7fcc\u65e5\u306e\u5929\u6c17\u3092\u8868\u3057\u307e\u3059\u3002\u4f8b\u3048\u3070\u3001\u6674\u308c\u306e\u7fcc\u65e5\u3082\u6674\u308c\u3067\u3042\u308b\u78ba\u7387\u306f $a_{\\mathrm{\u6674\u308c},\\mathrm{\u6674\u308c}}=0.7$\u3001\u96e8\u306b\u5909\u308f\u308b\u78ba\u7387\u306f $a_{\\mathrm{\u6674\u308c},\\mathrm{\u96e8}}=0.3$ \u3067\u3059\u3002<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" width=\"1024\" height=\"392\" src=\"https:\/\/since2020.jp\/media\/wp-content\/uploads\/2026\/09\/hmm_state_transition-1024x392.png\" alt=\"\" class=\"wp-image-9866\" srcset=\"https:\/\/since2020.jp\/media\/wp-content\/uploads\/2026\/09\/hmm_state_transition-1024x392.png 1024w, https:\/\/since2020.jp\/media\/wp-content\/uploads\/2026\/09\/hmm_state_transition-300x115.png 300w, https:\/\/since2020.jp\/media\/wp-content\/uploads\/2026\/09\/hmm_state_transition-768x294.png 768w, https:\/\/since2020.jp\/media\/wp-content\/uploads\/2026\/09\/hmm_state_transition-1536x587.png 1536w, https:\/\/since2020.jp\/media\/wp-content\/uploads\/2026\/09\/hmm_state_transition-2048x783.png 2048w\" sizes=\"(max-width: 1024px) 100vw, 1024px\" \/><\/figure>\n\n\n\n<p><em>\u56f32\uff1a\u5186\u4e0a\u306e\u30eb\u30fc\u30d7\u306f\u81ea\u5df1\u9077\u79fb\u78ba\u7387 $a_{ii}$\u3001\u5186\u540c\u58eb\u3092\u7d50\u3076\u77e2\u5370\u306f\u72b6\u614b\u9593\u306e\u9077\u79fb\u78ba\u7387 $a_{ij}$\uff08$i\\neq j$\uff09\u3092\u8868\u3059\u3002<\/em><\/p>\n\n\n\n<h2 class=\"wp-block-heading\">\u89b3\u6e2c\u751f\u6210\u30e2\u30c7\u30eb<\/h2>\n\n\n\n<p>HMM\u3067\u306f\u3001\u5404\u6642\u523b\u306e\u89b3\u6e2c\u5024\u306f\u3001\u305d\u306e\u6642\u523b\u306e\u96a0\u308c\u72b6\u614b\u306e\u307f\u306b\u4f9d\u5b58\u3057\u3066\u751f\u6210\u3055\u308c\u308b\u3068\u4eee\u5b9a\u3057\u307e\u3059\u3002\u3053\u306e\u6761\u4ef6\u4ed8\u304d\u72ec\u7acb\u6027\u306f\u3001\u6b21\u306e\u3088\u3046\u306b\u8868\u305b\u307e\u3059\u3002<\/p>\n\n\n\n<div class=\"wp-block-math\"><math display=\"block\"><semantics><mrow><mi>P<\/mi><mspace width=\"-0.1667em\" style=\"margin-left:-0.1667em\"><\/mspace><mrow><mo fence=\"true\" form=\"prefix\">(<\/mo><msub><mi>Y<\/mi><mi>t<\/mi><\/msub><mo lspace=\"0.22em\" rspace=\"0.22em\" stretchy=\"false\">|<\/mo><msub><mi>X<\/mi><mi>t<\/mi><\/msub><mo separator=\"true\">,<\/mo><msub><mi>X<\/mi><mrow><mi>t<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msub><mo separator=\"true\">,<\/mo><mo>\u2026<\/mo><mo separator=\"true\">,<\/mo><msub><mi>X<\/mi><mn>1<\/mn><\/msub><mo separator=\"true\">,<\/mo><msub><mi>Y<\/mi><mrow><mi>t<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msub><mo separator=\"true\">,<\/mo><mo>\u2026<\/mo><mo separator=\"true\">,<\/mo><msub><mi>Y<\/mi><mn>1<\/mn><\/msub><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mo>=<\/mo><mi>P<\/mi><mspace width=\"-0.1667em\" style=\"margin-left:-0.1667em\"><\/mspace><mrow><mo fence=\"true\" form=\"prefix\">(<\/mo><msub><mi>Y<\/mi><mi>t<\/mi><\/msub><mo lspace=\"0.22em\" rspace=\"0.22em\" stretchy=\"false\">|<\/mo><msub><mi>X<\/mi><mi>t<\/mi><\/msub><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><annotation encoding=\"application\/x-tex\">P\\!\\left(Y_t\\mid X_t,X_{t-1},\\ldots,X_1,Y_{t-1},\\ldots,Y_1\\right)\n=\nP\\!\\left(Y_t\\mid X_t\\right)<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<p>\u72b6\u614b $X_t=j$ \u306e\u3068\u304d\u306b\u89b3\u6e2c $Y_t=y$ \u304c\u5f97\u3089\u308c\u308b\u78ba\u7387\u3092\u3001\u51fa\u529b\u78ba\u7387\u307e\u305f\u306f\u89b3\u6e2c\u78ba\u7387\u3068\u547c\u3073\u307e\u3059\u3002<\/p>\n\n\n\n<div class=\"wp-block-math\"><math display=\"block\"><semantics><mrow><msub><mi>b<\/mi><mi>j<\/mi><\/msub><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>y<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>=<\/mo><mi>P<\/mi><mspace width=\"-0.1667em\" style=\"margin-left:-0.1667em\"><\/mspace><mrow><mo fence=\"true\" form=\"prefix\">(<\/mo><msub><mi>Y<\/mi><mi>t<\/mi><\/msub><mo>=<\/mo><mi>y<\/mi><mo lspace=\"0.22em\" rspace=\"0.22em\" stretchy=\"false\">|<\/mo><msub><mi>X<\/mi><mi>t<\/mi><\/msub><mo>=<\/mo><mi>j<\/mi><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><annotation encoding=\"application\/x-tex\">b_j(y)\n=\nP\\!\\left(Y_t=y\\mid X_t=j\\right)<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<p>\u89b3\u6e2c\u5024\u304c\u96e2\u6563\u5024\u306e\u5834\u5408\u306f\u30ab\u30c6\u30b4\u30ea\u5206\u5e03\u3001\u9023\u7d9a\u5024\u306e\u5834\u5408\u306f\u6b63\u898f\u5206\u5e03\u306a\u3069\u304c\u3088\u304f\u7528\u3044\u3089\u308c\u307e\u3059\u3002<\/p>\n\n\n\n<p>\u5098\u306e\u4f8b\u3067\u3001\u72b6\u614b\u306e\u9806\u5e8f\u3092\u300c\u6674\u308c\u3001\u96e8\u300d\u3001\u89b3\u6e2c\u306e\u9806\u5e8f\u3092\u300c\u5098\u3042\u308a\u3001\u5098\u306a\u3057\u300d\u3068\u3059\u308b\u3068\u3001\u89b3\u6e2c\u78ba\u7387\u884c\u5217\u306f\u6b21\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n<p>$$<br>B=<br>\\begin{pmatrix}<br>0.1 &amp; 0.9\\\\<br>0.8 &amp; 0.2<br>\\end{pmatrix}<br>$$<\/p>\n\n\n\n<p>\u4f8b\u3048\u3070\u3001\u96e8\u306e\u65e5\u306b\u5098\u3092\u6301\u3063\u3066\u304f\u308b\u78ba\u7387\u306f $b_{\\mathrm{\u96e8}}(\\mathrm{\u5098\u3042\u308a})=0.8$ \u3067\u3059\u3002<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">\u521d\u671f\u72b6\u614b\u5206\u5e03\u3068\u30e2\u30c7\u30eb\u306e\u5b9a\u7fa9<\/h2>\n\n\n\n<p>\u6642\u7cfb\u5217\u306e\u958b\u59cb\u6642\u70b9\u306b\u304a\u3051\u308b\u72b6\u614b\u3082\u78ba\u7387\u7684\u306b\u5b9a\u7fa9\u3057\u307e\u3059\u3002\u521d\u671f\u72b6\u614b\u5206\u5e03\u306f\u3001\u6b21\u306e\u3088\u3046\u306b\u8868\u3055\u308c\u307e\u3059\u3002<\/p>\n\n\n\n<p>$$<br>\\pi_i=P(X_1=i)<br>$$<\/p>\n\n\n\n<p>\u4ee5\u4e0a\u304b\u3089\u3001HMM\u306f\u6b21\u306e3\u8981\u7d20\u3067\u5b9a\u7fa9\u3055\u308c\u307e\u3059\u3002<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u521d\u671f\u72b6\u614b\u5206\u5e03\uff1a$\\boldsymbol{\\pi}$<\/li>\n\n\n\n<li>\u72b6\u614b\u9077\u79fb\u78ba\u7387\uff1a$A$<\/li>\n\n\n\n<li>\u89b3\u6e2c\u751f\u6210\u78ba\u7387\uff1a$B$<\/li>\n<\/ul>\n\n\n\n<p>\u30e2\u30c7\u30eb\u5168\u4f53\u3092\u3001\u307e\u3068\u3081\u3066\u6b21\u306e\u3088\u3046\u306b\u8868\u3059\u3053\u3068\u3082\u3042\u308a\u307e\u3059\u3002<\/p>\n\n\n\n<p>$$<br>\\lambda=(\\boldsymbol{\\pi},A,B)<br>$$<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">\u89b3\u6e2c\u7cfb\u5217\u3068\u72b6\u614b\u7cfb\u5217\u306e\u540c\u6642\u5206\u5e03<\/h2>\n\n\n\n<p>HMM\u306e\u91cd\u8981\u306a\u7279\u5fb4\u306f\u3001\u72b6\u614b\u7cfb\u5217\u3068\u89b3\u6e2c\u7cfb\u5217\u306e\u540c\u6642\u5206\u5e03\u3092\u5358\u7d14\u306a\u5f62\u306b\u5206\u89e3\u3067\u304d\u308b\u3053\u3068\u3067\u3059\u3002<\/p>\n\n\n\n<p>\u72b6\u614b\u7cfb\u5217\u3092 $X_{1:T}=(X_1,\\ldots,X_T)$\u3001\u89b3\u6e2c\u7cfb\u5217\u3092 $Y_{1:T}=(Y_1,\\ldots,Y_T)$ \u3068\u3059\u308b\u3068\u3001\u540c\u6642\u78ba\u7387\u306f\u6b21\u306e\u3088\u3046\u306b\u8868\u305b\u307e\u3059\u3002<\/p>\n\n\n\n<div class=\"wp-block-math\"><math display=\"block\"><semantics><mrow><mi>P<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><msub><mi>X<\/mi><mrow><mn>1<\/mn><mo lspace=\"0.2222em\" rspace=\"0.2222em\">:<\/mo><mi>T<\/mi><\/mrow><\/msub><mo separator=\"true\">,<\/mo><msub><mi>Y<\/mi><mrow><mn>1<\/mn><mo lspace=\"0.2222em\" rspace=\"0.2222em\">:<\/mo><mi>T<\/mi><\/mrow><\/msub><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>=<\/mo><mi>P<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><msub><mi>X<\/mi><mn>1<\/mn><\/msub><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mi>P<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><msub><mi>Y<\/mi><mn>1<\/mn><\/msub><mo lspace=\"0.22em\" rspace=\"0.22em\" stretchy=\"false\">|<\/mo><msub><mi>X<\/mi><mn>1<\/mn><\/msub><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mrow><munderover><mo movablelimits=\"false\">\u220f<\/mo><mrow><mi>t<\/mi><mo>=<\/mo><mn>2<\/mn><\/mrow><mi>T<\/mi><\/munderover><\/mrow><mi>P<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><msub><mi>X<\/mi><mi>t<\/mi><\/msub><mo lspace=\"0.22em\" rspace=\"0.22em\" stretchy=\"false\">|<\/mo><msub><mi>X<\/mi><mrow><mi>t<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msub><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mi>P<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><msub><mi>Y<\/mi><mi>t<\/mi><\/msub><mo lspace=\"0.22em\" rspace=\"0.22em\" stretchy=\"false\">|<\/mo><msub><mi>X<\/mi><mi>t<\/mi><\/msub><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">P(X_{1:T},Y_{1:T})\n=\nP(X_1)P(Y_1\\mid X_1)\n\\prod_{t=2}^{T}\nP(X_t\\mid X_{t-1})P(Y_t\\mid X_t)\n<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<p>\u3053\u306e\u5206\u89e3\u69cb\u9020\u306b\u3088\u308a\u3001\u6642\u7cfb\u5217\u5168\u4f53\u306e\u78ba\u7387\u8a08\u7b97\u3001\u72b6\u614b\u63a8\u5b9a\u3001\u30d1\u30e9\u30e1\u30fc\u30bf\u5b66\u7fd2\u3092\u52b9\u7387\u7684\u306b\u5b9f\u884c\u3067\u304d\u307e\u3059\u3002\u56f31\u306f\u3001\u3053\u306e\u4f9d\u5b58\u95a2\u4fc2\u3092\u56f3\u793a\u3057\u305f\u3082\u306e\u3067\u3059\u3002<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">HMM\u3067\u89e3\u304f3\u3064\u306e\u4ee3\u8868\u7684\u306a\u554f\u984c<\/h2>\n\n\n\n<p>HMM\u3067\u306f\u3001\u4e3b\u306b\u6b21\u306e3\u7a2e\u985e\u306e\u554f\u984c\u3092\u6271\u3044\u307e\u3059\u3002<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">\u5c24\u5ea6\u8a08\u7b97<\/h3>\n\n\n\n<p>\u4e0e\u3048\u3089\u308c\u305f\u30e2\u30c7\u30eb $\\lambda$ \u306e\u3082\u3068\u3067\u3001\u89b3\u6e2c\u7cfb\u5217 $Y_{1:T}$ \u304c\u5f97\u3089\u308c\u308b\u78ba\u7387\u3092\u8a08\u7b97\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<p>$$<br>P(Y_{1:T}\\mid\\lambda)<br>$$<\/p>\n\n\n\n<p>\u3053\u308c\u306f\u3001\u30e2\u30c7\u30eb\u304c\u89b3\u6e2c\u30c7\u30fc\u30bf\u306b\u3069\u306e\u7a0b\u5ea6\u9069\u5408\u3057\u3066\u3044\u308b\u304b\u3092\u8a55\u4fa1\u3059\u308b\u305f\u3081\u306b\u91cd\u8981\u3067\u3059\u3002<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">\u72b6\u614b\u7cfb\u5217\u306e\u63a8\u5b9a<\/h3>\n\n\n\n<p>\u89b3\u6e2c\u7cfb\u5217\u304c\u4e0e\u3048\u3089\u308c\u305f\u3068\u304d\u3001\u6700\u3082\u78ba\u304b\u3089\u3057\u3044\u72b6\u614b\u7cfb\u5217\u3092\u6c42\u3081\u307e\u3059\u3002<\/p>\n\n\n\n<div class=\"wp-block-math\"><math display=\"block\"><semantics><mrow><msubsup><mi>X<\/mi><mrow><mn>1<\/mn><mo lspace=\"0.2222em\" rspace=\"0.2222em\">:<\/mo><mi>T<\/mi><\/mrow><mo lspace=\"0em\" rspace=\"0em\">\u2217<\/mo><\/msubsup><mo>=<\/mo><munder><mi>arg\u2009max<\/mi><msub><mi>X<\/mi><mrow><mn>1<\/mn><mo lspace=\"0.2222em\" rspace=\"0.2222em\">:<\/mo><mi>T<\/mi><\/mrow><\/msub><\/munder><mo>\u2061<\/mo><mspace width=\"0.1667em\"><\/mspace><mi>P<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><msub><mi>X<\/mi><mrow><mn>1<\/mn><mo lspace=\"0.2222em\" rspace=\"0.2222em\">:<\/mo><mi>T<\/mi><\/mrow><\/msub><mo lspace=\"0.22em\" rspace=\"0.22em\" stretchy=\"false\">|<\/mo><msub><mi>Y<\/mi><mrow><mn>1<\/mn><mo lspace=\"0.2222em\" rspace=\"0.2222em\">:<\/mo><mi>T<\/mi><\/mrow><\/msub><mo separator=\"true\">,<\/mo><mi>\u03bb<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">X_{1:T}^{*}\n=\n\\operatorname*{arg\\,max}_{X_{1:T}}\nP(X_{1:T}\\mid Y_{1:T},\\lambda)<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<h3 class=\"wp-block-heading\">\u30d1\u30e9\u30e1\u30fc\u30bf\u5b66\u7fd2<\/h3>\n\n\n\n<p>\u89b3\u6e2c\u30c7\u30fc\u30bf\u304b\u3089\u3001\u30e2\u30c7\u30eb\u30d1\u30e9\u30e1\u30fc\u30bf $\\boldsymbol{\\pi},A,B$ \u3092\u63a8\u5b9a\u3057\u307e\u3059\u3002\u5178\u578b\u7684\u306b\u306f\u3001\u89b3\u6e2c\u7cfb\u5217\u306e\u5c24\u5ea6\u3092\u6700\u5927\u5316\u3059\u308b\u554f\u984c\u3068\u3057\u3066\u5b9a\u5f0f\u5316\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<div class=\"wp-block-math\"><math display=\"block\"><semantics><mrow><msup><mi>\u03bb<\/mi><mo lspace=\"0em\" rspace=\"0em\">\u2217<\/mo><\/msup><mo>=<\/mo><munder><mi>arg\u2009max<\/mi><mi>\u03bb<\/mi><\/munder><mo>\u2061<\/mo><mspace width=\"0.1667em\"><\/mspace><mi>P<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><msub><mi>Y<\/mi><mrow><mn>1<\/mn><mo lspace=\"0.2222em\" rspace=\"0.2222em\">:<\/mo><mi>T<\/mi><\/mrow><\/msub><mo lspace=\"0.22em\" rspace=\"0.22em\" stretchy=\"false\">|<\/mo><mi>\u03bb<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">\\lambda^{*}\n=\n\\operatorname*{arg\\,max}_{\\lambda}\nP(Y_{1:T}\\mid\\lambda)<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<figure class=\"wp-block-table\"><table class=\"has-fixed-layout\"><thead><tr><th>\u554f\u984c<\/th><th>\u4ee3\u8868\u7684\u306a\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0<\/th><\/tr><\/thead><tbody><tr><td>\u5c24\u5ea6\u8a08\u7b97<\/td><td>\u524d\u5411\u304d\u30fb\u5f8c\u5411\u304d\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0<\/td><\/tr><tr><td>\u6700\u3082\u78ba\u304b\u3089\u3057\u3044\u72b6\u614b\u7cfb\u5217\u306e\u63a8\u5b9a<\/td><td>\u30d3\u30bf\u30d3\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0<\/td><\/tr><tr><td>\u30d1\u30e9\u30e1\u30fc\u30bf\u5b66\u7fd2\uff08\u6559\u5e2b\u306a\u3057\uff09<\/td><td>\u30d0\u30a6\u30e0\u30fb\u30a6\u30a7\u30eb\u30c1\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\uff08EM\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u306e\u4e00\u7a2e\uff09<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<h2 class=\"wp-block-heading\">\u307e\u3068\u3081<\/h2>\n\n\n\n<p>\u96a0\u308c\u30de\u30eb\u30b3\u30d5\u30e2\u30c7\u30eb\u306f\u3001\u6642\u9593\u3068\u3068\u3082\u306b\u5909\u5316\u3059\u308b\u96a0\u308c\u72b6\u614b\u3068\u89b3\u6e2c\u30c7\u30fc\u30bf\u3092\u3001\u30de\u30eb\u30b3\u30d5\u6027\u3068\u6761\u4ef6\u4ed8\u304d\u72ec\u7acb\u6027\u306e\u4eee\u5b9a\u306b\u3088\u3063\u3066\u7d50\u3073\u3064\u3051\u305f\u78ba\u7387\u30e2\u30c7\u30eb\u3067\u3059\u3002<\/p>\n\n\n\n<p>\u8981\u70b9\u306f\u6b21\u306e3\u3064\u3067\u3059\u3002<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>\u96a0\u308c\u72b6\u614b\u306f\u3001\u76f4\u524d\u306e\u72b6\u614b\u306b\u4f9d\u5b58\u3057\u3066\u9077\u79fb\u3059\u308b<\/li>\n\n\n\n<li>\u89b3\u6e2c\u5024\u306f\u3001\u305d\u306e\u6642\u523b\u306e\u96a0\u308c\u72b6\u614b\u304b\u3089\u78ba\u7387\u7684\u306b\u751f\u6210\u3055\u308c\u308b<\/li>\n\n\n\n<li>\u3053\u306e\u69cb\u9020\u3092\u5229\u7528\u3057\u3066\u3001\u5c24\u5ea6\u8a08\u7b97\u30fb\u72b6\u614b\u63a8\u5b9a\u30fb\u30d1\u30e9\u30e1\u30fc\u30bf\u5b66\u7fd2\u3092\u52b9\u7387\u7684\u306b\u884c\u3048\u308b<\/li>\n<\/ol>\n\n\n\n<p>HMM\u306f\u6bd4\u8f03\u7684\u5358\u7d14\u306a\u69cb\u9020\u3092\u6301\u3061\u306a\u304c\u3089\u3001\u6642\u7cfb\u5217\u30c7\u30fc\u30bf\u306e\u80cc\u5f8c\u306b\u3042\u308b\u72b6\u614b\u3092\u89e3\u91c8\u53ef\u80fd\u306a\u5f62\u3067\u63a8\u5b9a\u3067\u304d\u307e\u3059\u3002\u6642\u7cfb\u5217\u89e3\u6790\u3084\u78ba\u7387\u30e2\u30c7\u30eb\u3092\u5b66\u3076\u3046\u3048\u3067\u3001\u7406\u8ad6\u3068\u5fdc\u7528\u3092\u3064\u306a\u3050\u57fa\u790e\u7684\u306a\u30e2\u30c7\u30eb\u3067\u3059\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u96a0\u308c\u30de\u30eb\u30b3\u30d5\u30e2\u30c7\u30eb\u3068\u306f 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[&hellip;]<\/p>\n","protected":false},"author":77,"featured_media":8355,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"content-type":"","swell_btn_cv_data":"","footnotes":"","_wp_rev_ctl_limit":""},"categories":[1251],"tags":[39,419,35],"class_list":["post-9864","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-data-analysis","tag-39","tag-419","tag-35"],"_links":{"self":[{"href":"https:\/\/since2020.jp\/media\/wp-json\/wp\/v2\/posts\/9864","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/since2020.jp\/media\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/since2020.jp\/media\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/since2020.jp\/media\/wp-json\/wp\/v2\/users\/77"}],"replies":[{"embeddable":true,"href":"https:\/\/since2020.jp\/media\/wp-json\/wp\/v2\/comments?post=9864"}],"version-history":[{"count":5,"href":"https:\/\/since2020.jp\/media\/wp-json\/wp\/v2\/posts\/9864\/revisions"}],"predecessor-version":[{"id":9872,"href":"https:\/\/since2020.jp\/media\/wp-json\/wp\/v2\/posts\/9864\/revisions\/9872"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/since2020.jp\/media\/wp-json\/wp\/v2\/media\/8355"}],"wp:attachment":[{"href":"https:\/\/since2020.jp\/media\/wp-json\/wp\/v2\/media?parent=9864"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/since2020.jp\/media\/wp-json\/wp\/v2\/categories?post=9864"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/since2020.jp\/media\/wp-json\/wp\/v2\/tags?post=9864"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}