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LMmodel.html
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<!doctype html>
<html>
<head>
<meta charset='UTF-8'><meta name='viewport' content='width=device-width initial-scale=1'>
<title>统计语言模型</title><link href='http://fonts.googleapis.com/css?family=Open+Sans:400italic,700italic,700,400&subset=latin,latin-ext' rel='stylesheet' type='text/css'><style type='text/css'>html, body {overflow-x: initial !important;}html { font-size: 14px; background-color: rgb(255, 255, 255); color: rgb(51, 51, 51); }
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* https://github.com/GitbookIO/gitbook/blob/master/LICENSE
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<div id='write' class = 'is-mac'><p></p><h1><a name='header-c7' class='md-header-anchor '></a>LMmodel</h1><h2><a name='header-c8' class='md-header-anchor '></a>1 简介</h2><p>目标：给每个句子或者词序列赋予概率</p><p>相关应用领域：</p><ol><li><p>机器翻译</p><p>如将中文“风很大”翻译成英文，“风”对应的词为“wind”，“很”对应的词为“too”，“very”，“大”对应的词为“big”,"large","high", 这样“风很大”对应的英文就有<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-28-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="13.947ex" height="2.577ex" viewBox="0 -806.1 6004.9 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E28-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="1" 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如输入音节是“zuoye”，其对应的中文可能是“昨夜”，也可能是“作业”，通过语言模型能判断哪种可能最大，如在上下文是“风很大”的情况下，“昨夜”的可能性更大（昨夜风很大）。</p></li></ol><h2><a name='header-c27' class='md-header-anchor '></a>2 语言模型建模</h2><h3><a name='header-c28' class='md-header-anchor '></a>2.1 任务</h3><p>任务1：计算出句子或者词序列的概率</p><div contenteditable="false" class="mathjax-block md-end-block" id="mathjax-c30" cid="c30" mdtype="math_block"><span class="MathJax_Preview"></span><span class="MathJax_SVG_Display" style="text-align: center;"><span class="MathJax_SVG" id="MathJax-Element-1-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="36.355ex" height="2.577ex" viewBox="0 -806.1 15652.9 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E1-MJMATHI-50" d="M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 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P(W) = P(w_1,w_2,w_3,w_4,w_5...w_n)
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P(the | its\ water\ is\ so\ transparent\ that) \approx P(the)
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P(the | its\ water\ is\ so\ transparent\ that) \approx P(the | transparent\ that)
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P(w_i |w_1w_2...w_{i-1}) \approx P(w_i |w_{i-n}...w_{i-1})
</script></div><h4><a name='header-c61' class='md-header-anchor '></a>2.2.2 文本生成</h4><h5><a name='header-c62' class='md-header-anchor '></a>2.2.2.1 一般方法</h5><p>已知条件: <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-29-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="12.023ex" height="2.577ex" viewBox="-38.5 -806.1 5176.3 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex; margin-left: -0.089ex;"><defs><path stroke-width="1" id="E29-MJMATHI-70" d="M23 287Q24 290 25 295T30 317T40 348T55 381T75 411T101 433T134 442Q209 442 230 378L240 387Q302 442 358 442Q423 442 460 395T497 281Q497 173 421 82T249 -10Q227 -10 210 -4Q199 1 187 11T168 28L161 36Q160 35 139 -51T118 -138Q118 -144 126 -145T163 -148H188Q194 -155 194 -157T191 -175Q188 -187 185 -190T172 -194Q170 -194 161 -194T127 -193T65 -192Q-5 -192 -24 -194H-32Q-39 -187 -39 -183Q-37 -156 -26 -148H-6Q28 -147 33 -136Q36 -130 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id="MathJax-Element-33">\sum_{j=1}^ Np(w_{j}|w_{1:i-1})=\gamma</script>,取出对应的word<sub>N</sub></p><p> 重复以上过程直到取出结束符号<EOS></p><h5><a name='header-c73' class='md-header-anchor '></a>2.2.2.2 实际效果</h5><p>从Unigram Model中生成的文本</p><div contenteditable="false" class="mathjax-block md-end-block" id="mathjax-c75" cid="c75" mdtype="math_block"><span class="MathJax_Preview"></span><span class="MathJax_SVG_Display" style="text-align: center;"><span class="MathJax_SVG" id="MathJax-Element-11-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="27.449ex" height="5.262ex" viewBox="0 -1007.2 11818.5 2265.7" role="img" focusable="false" style="vertical-align: -2.923ex;"><defs><path stroke-width="1" id="E11-MJMATHI-50" d="M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 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P(w_1w_2 ...w_n ) \approx \prod_i P(w_i )
</script></div><p>fifth, an, of, futures, the, an, incorporated, a, a, the, inflation, most, dollars, quarter, in, is, mass<br/></p><p>结论：基本是乱序，无规律的文本</p><p>从Bigram Model中生成的文本 <br/></p><div contenteditable="false" class="mathjax-block md-end-block" id="mathjax-c82" cid="c82" mdtype="math_block"><span class="MathJax_Preview"></span><span class="MathJax_SVG_Display" style="text-align: center;"><span class="MathJax_SVG" id="MathJax-Element-12-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="37.453ex" height="5.262ex" viewBox="0 -1007.2 16125.5 2265.7" role="img" focusable="false" style="vertical-align: -2.923ex;"><defs><path stroke-width="1" id="E12-MJMATHI-50" d="M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 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mode=display" id="MathJax-Element-12">
P(w_i |w_1w_2...w_{i-1}) \approx \prod_i P(w_i |w_{i-1})
</script></div><p>outside, new, car, parking, lot, of, the, agreement, reached<br/></p><p>this, would, be, a, record, november</p><p>结论： 一些短句子的语法结构基本无问题</p><p>我们可以延伸至N-gram model，生成的句子会更像自然语言，但这种N-gram model 仍然是由缺陷的，因为语言由长距离依赖问题：</p><p>“The computer which I had just put into the machine room on the fifth floor crashed.” </p><p>其中crashed出现的概率应当与computer相关，但N-gram model绝对不能覆盖到。</p><h2><a name='header-c96' class='md-header-anchor '></a>3 N-gram 概率计算</h2><h3><a name='header-c97' class='md-header-anchor '></a>3.1 一元模型先验概率（MLE）</h3><div contenteditable="false" class="mathjax-block md-end-block" id="mathjax-c98" cid="c98" mdtype="math_block"><span class="MathJax_Preview"></span><span class="MathJax_SVG_Display" style="text-align: center;"><span class="MathJax_SVG" id="MathJax-Element-13-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="45.663ex" height="6.78ex" viewBox="-38.5 -1660.6 19660.6 2919" role="img" 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y="0"></use><use transform="scale(0.707)" xlink:href="#E13-MJMATHI-69" x="1013" y="-213"></use></g><use xlink:href="#E13-MJMAIN-2208" x="13372" y="0"></use><use xlink:href="#E13-MJMATHI-56" x="14317" y="0"></use><g transform="translate(16837,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" stroke="none" transform="scale(50.259) matrix(1 0 0 -1 0 0)">式</text></g><use xlink:href="#E13-MJMAIN-28" x="17563" y="0"></use><g transform="translate(17953,0)"><use xlink:href="#E13-MJMAIN-33"></use><use xlink:href="#E13-MJMAIN-2E" x="500" y="0"></use><use xlink:href="#E13-MJMAIN-31" x="779" y="0"></use></g><use xlink:href="#E13-MJMAIN-29" x="19232" y="0"></use></g></svg></span></span><script type="math/tex; mode=display" id="MathJax-Element-13">
p(w_{1:n}|\theta_{1:|V|}) = \prod_i^n p(w_i) \ \ \ \ \ \ w_i\in V \ \ \ \ \ \ \ 式 (3.1)
</script></div><p>式3.1中<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-34-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="4.162ex" height="1.76ex" viewBox="0 -504.6 1792 757.9" role="img" focusable="false" style="vertical-align: -0.588ex;"><defs><path stroke-width="1" id="E34-MJMATHI-77" d="M580 385Q580 406 599 424T641 443Q659 443 674 425T690 368Q690 339 671 253Q656 197 644 161T609 80T554 12T482 -11Q438 -11 404 5T355 48Q354 47 352 44Q311 -11 252 -11Q226 -11 202 -5T155 14T118 53T104 116Q104 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Q21 293 29 315T52 366T96 418T161 441Q204 441 227 416T250 358Q250 340 217 250T184 111Q184 65 205 46T258 26Q301 26 334 87L339 96V119Q339 122 339 128T340 136T341 143T342 152T345 165T348 182T354 206T362 238T373 281Q402 395 406 404Q419 431 449 431Q468 431 475 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130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E34-MJMATHI-77" x="0" y="0"></use><g transform="translate(716,-150)"><use transform="scale(0.707)" xlink:href="#E34-MJMAIN-31" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E34-MJMAIN-3A" x="500" y="0"></use><use transform="scale(0.707)" xlink:href="#E34-MJMATHI-6E" x="779" y="0"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-34">w_{1:n}</script>代表一个n长的语料库，其中<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-35-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.464ex" height="1.76ex" viewBox="0 -504.6 1060.8 757.9" role="img" focusable="false" style="vertical-align: -0.588ex;"><defs><path stroke-width="1" id="E35-MJMATHI-77" d="M580 385Q580 406 599 424T641 443Q659 443 674 425T690 368Q690 339 671 253Q656 197 644 161T609 80T554 12T482 -11Q438 -11 404 5T355 48Q354 47 352 44Q311 -11 252 -11Q226 -11 202 -5T155 14T118 53T104 116Q104 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Q21 293 29 315T52 366T96 418T161 441Q204 441 227 416T250 358Q250 340 217 250T184 111Q184 65 205 46T258 26Q301 26 334 87L339 96V119Q339 122 339 128T340 136T341 143T342 152T345 165T348 182T354 206T362 238T373 281Q402 395 406 404Q419 431 449 431Q468 431 475 421T483 402Q483 389 454 274T422 142Q420 131 420 107V100Q420 85 423 71T442 42T487 26Q558 26 600 148Q609 171 620 213T632 273Q632 306 619 325T593 357T580 385Z"></path><path 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tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.89ex" height="2.461ex" viewBox="0 -806.1 813.8 1059.4" role="img" focusable="false" style="vertical-align: -0.588ex;"><defs><path stroke-width="1" id="E36-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="1" id="E36-MJMATHI-69" d="M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E36-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E36-MJMATHI-69" x="663" y="-213"></use></g></svg></span><script type="math/tex" id="MathJax-Element-36">\theta_i</script>表示每一个词在语料库中出现的概率，故式3.1亦可改写为以下</p><div contenteditable="false" class="mathjax-block md-end-block" id="mathjax-c100" cid="c100" mdtype="math_block"><span class="MathJax_Preview"></span><span class="MathJax_SVG_Display" style="text-align: center;"><span class="MathJax_SVG" id="MathJax-Element-14-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" 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287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path stroke-width="1" id="E14-MJMAIN-33" d="M127 463Q100 463 85 480T69 524Q69 579 117 622T233 665Q268 665 277 664Q351 652 390 611T430 522Q430 470 396 421T302 350L299 348Q299 347 308 345T337 336T375 315Q457 262 457 175Q457 96 395 37T238 -22Q158 -22 100 21T42 130Q42 158 60 175T105 193Q133 193 151 175T169 130Q169 119 166 110T159 94T148 82T136 74T126 70T118 67L114 66Q165 21 238 21Q293 21 321 74Q338 107 338 175V195Q338 290 274 322Q259 328 213 329L171 330L168 332Q166 335 166 348Q166 366 174 366Q202 366 232 371Q266 376 294 413T322 525V533Q322 590 287 612Q265 626 240 626Q208 626 181 615T143 592T132 580H135Q138 579 143 578T153 573T165 566T175 555T183 540T186 520Q186 498 172 481T127 463Z"></path><path stroke-width="1" id="E14-MJMAIN-2E" d="M78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z"></path><path stroke-width="1" id="E14-MJMAIN-32" d="M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 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transform="translate(469,553)"><use transform="scale(0.707)" xlink:href="#E14-MJMATHI-6D" x="0" y="0"></use><use transform="scale(0.5)" xlink:href="#E14-MJMATHI-6A" x="1242" y="-213"></use></g><use transform="scale(0.707)" xlink:href="#E14-MJMATHI-6A" x="663" y="-430"></use></g><g transform="translate(15356,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" stroke="none" transform="scale(50.259) matrix(1 0 0 -1 0 0)">式</text></g><use xlink:href="#E14-MJMAIN-28" x="16083" y="0"></use><g transform="translate(16472,0)"><use xlink:href="#E14-MJMAIN-33"></use><use xlink:href="#E14-MJMAIN-2E" x="500" y="0"></use><use xlink:href="#E14-MJMAIN-32" x="779" y="0"></use></g><use xlink:href="#E14-MJMAIN-29" x="17752" y="0"></use></g></svg></span></span><script type="math/tex; mode=display" id="MathJax-Element-14">
p(w_{1:n}|\theta_{1:|V|}) = \prod_j^{|V|} \theta_j^{m_j} \hspace{3cm} 式 (3.2)
</script></div><p>式3.2中m<sub>j</sub>表示词w<sub>j</sub>在句子中出现的次数。<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-37-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="13.294ex" height="3.745ex" viewBox="0 -1107.7 5724 1612.3" role="img" focusable="false" style="vertical-align: -1.172ex;"><defs><path stroke-width="1" id="E37-MJSZ1-2211" d="M61 748Q64 750 489 750H913L954 640Q965 609 976 579T993 533T999 516H979L959 517Q936 579 886 621T777 682Q724 700 655 705T436 710H319Q183 710 183 709Q186 706 348 484T511 259Q517 250 513 244L490 216Q466 188 420 134T330 27L149 -187Q149 -188 362 -188Q388 -188 436 -188T506 -189Q679 -189 778 -162T936 -43Q946 -27 959 6H999L913 -249L489 -250Q65 -250 62 -248Q56 -246 56 -239Q56 -234 118 -161Q186 -81 245 -11L428 206Q428 207 242 462L57 717L56 728Q56 744 61 748Z"></path><path stroke-width="1" id="E37-MJMAIN-7C" d="M139 -249H137Q125 -249 119 -235V251L120 737Q130 750 139 750Q152 750 159 735V-235Q151 -249 141 -249H139Z"></path><path stroke-width="1" id="E37-MJMATHI-56" d="M52 648Q52 670 65 683H76Q118 680 181 680Q299 680 320 683H330Q336 677 336 674T334 656Q329 641 325 637H304Q282 635 274 635Q245 630 242 620Q242 618 271 369T301 118L374 235Q447 352 520 471T595 594Q599 601 599 609Q599 633 555 637Q537 637 537 648Q537 649 539 661Q542 675 545 679T558 683Q560 683 570 683T604 682T668 681Q737 681 755 683H762Q769 676 769 672Q769 655 760 640Q757 637 743 637Q730 636 719 635T698 630T682 623T670 615T660 608T652 599T645 592L452 282Q272 -9 266 -16Q263 -18 259 -21L241 -22H234Q216 -22 216 -15Q213 -9 177 305Q139 623 138 626Q133 637 76 637H59Q52 642 52 648Z"></path><path stroke-width="1" id="E37-MJMATHI-6A" d="M297 596Q297 627 318 644T361 661Q378 661 389 651T403 623Q403 595 384 576T340 557Q322 557 310 567T297 596ZM288 376Q288 405 262 405Q240 405 220 393T185 362T161 325T144 293L137 279Q135 278 121 278H107Q101 284 101 286T105 299Q126 348 164 391T252 441Q253 441 260 441T272 442Q296 441 316 432Q341 418 354 401T367 348V332L318 133Q267 -67 264 -75Q246 -125 194 -164T75 -204Q25 -204 7 -183T-12 -137Q-12 -110 7 -91T53 -71Q70 -71 82 -81T95 -112Q95 -148 63 -167Q69 -168 77 -168Q111 -168 139 -140T182 -74L193 -32Q204 11 219 72T251 197T278 308T289 365Q289 372 288 376Z"></path><path stroke-width="1" id="E37-MJMAIN-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path stroke-width="1" id="E37-MJMAIN-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path><path stroke-width="1" id="E37-MJMATHI-6D" d="M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path stroke-width="1" id="E37-MJMATHI-6E" d="M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E37-MJSZ1-2211" x="0" y="0"></use><g transform="translate(1056,521)"><use transform="scale(0.707)" xlink:href="#E37-MJMAIN-7C" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E37-MJMATHI-56" x="278" y="0"></use><use transform="scale(0.707)" xlink:href="#E37-MJMAIN-7C" x="1048" y="0"></use></g><g transform="translate(1056,-308)"><use transform="scale(0.707)" xlink:href="#E37-MJMATHI-6A" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E37-MJMAIN-3D" x="412" y="0"></use><use transform="scale(0.707)" xlink:href="#E37-MJMAIN-31" x="1191" y="0"></use></g><g transform="translate(2519,0)"><use xlink:href="#E37-MJMATHI-6D" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E37-MJMATHI-6A" x="1242" y="-213"></use></g><use xlink:href="#E37-MJMAIN-3D" x="4067" y="0"></use><use xlink:href="#E37-MJMATHI-6E" x="5123" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-37">\sum_{j=1}^{|V|}m_j=n</script></p><p>由贝叶斯条件概率公式可得：</p><div contenteditable="false" class="mathjax-block md-end-block" id="mathjax-c105" cid="c105" mdtype="math_block"><span class="MathJax_Preview"></span><span class="MathJax_SVG_Display" style="text-align: center;"><span class="MathJax_SVG" id="MathJax-Element-15-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="52.376ex" height="6.546ex" viewBox="-38.5 -1660.6 22550.9 2818.5" role="img" focusable="false" style="vertical-align: -2.689ex; margin-left: -0.089ex;"><defs><path stroke-width="1" 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xlink:href="#E15-MJMAIN-31" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E15-MJMAIN-3A" x="500" y="0"></use><use transform="scale(0.707)" xlink:href="#E15-MJMATHI-6E" x="779" y="0"></use></g></g><use xlink:href="#E15-MJMAIN-29" x="9041" y="0"></use></g><g transform="translate(3104,-717)"><use xlink:href="#E15-MJMATHI-70" x="0" y="0"></use><use xlink:href="#E15-MJMAIN-28" x="503" y="0"></use><g transform="translate(893,0)"><use xlink:href="#E15-MJMATHI-3B8" x="0" y="0"></use><g transform="translate(469,-187)"><use transform="scale(0.707)" xlink:href="#E15-MJMAIN-31" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E15-MJMAIN-3A" x="500" y="0"></use><use transform="scale(0.707)" xlink:href="#E15-MJMAIN-7C" x="779" y="0"></use><use transform="scale(0.707)" xlink:href="#E15-MJMATHI-56" x="1057" y="0"></use><use transform="scale(0.707)" xlink:href="#E15-MJMAIN-7C" x="1827" y="0"></use></g></g><use xlink:href="#E15-MJMAIN-29" x="2951" y="0"></use></g></g></g><g transform="translate(19727,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" stroke="none" transform="scale(50.259) matrix(1 0 0 -1 0 0)">式</text></g><use xlink:href="#E15-MJMAIN-28" x="20453" y="0"></use><g transform="translate(20843,0)"><use xlink:href="#E15-MJMAIN-33"></use><use xlink:href="#E15-MJMAIN-2E" x="500" y="0"></use><use xlink:href="#E15-MJMAIN-33" x="779" y="0"></use></g><use xlink:href="#E15-MJMAIN-29" x="22122" y="0"></use></g></svg></span></span><script type="math/tex; mode=display" id="MathJax-Element-15">
p(w_{1:n}|\theta_{1:|V|}) ＝ \frac {p(\theta_{1:|V|}|w_{1:n}) *p(w_{1:n})}{p(\theta_{1:|V|})} \hspace{2cm} 式 (3.3)
</script></div><p>极大似然估计是先验概率估计，故<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-38-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="8.429ex" height="2.928ex" viewBox="-38.5 -806.1 3629.3 1260.5" role="img" focusable="false" style="vertical-align: -1.055ex; margin-left: -0.089ex;"><defs><path stroke-width="1" id="E38-MJMATHI-70" d="M23 287Q24 290 25 295T30 317T40 348T55 381T75 411T101 433T134 442Q209 442 230 378L240 387Q302 442 358 442Q423 442 460 395T497 281Q497 173 421 82T249 -10Q227 -10 210 -4Q199 1 187 11T168 28L161 36Q160 35 139 -51T118 -138Q118 -144 126 -145T163 -148H188Q194 -155 194 -157T191 -175Q188 -187 185 -190T172 -194Q170 -194 161 -194T127 -193T65 -192Q-5 -192 -24 -194H-32Q-39 -187 -39 -183Q-37 -156 -26 -148H-6Q28 -147 33 -136Q36 -130 94 103T155 350Q156 355 156 364Q156 405 131 405Q109 405 94 377T71 316T59 280Q57 278 43 278H29Q23 284 23 287ZM178 102Q200 26 252 26Q282 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683H762Q769 676 769 672Q769 655 760 640Q757 637 743 637Q730 636 719 635T698 630T682 623T670 615T660 608T652 599T645 592L452 282Q272 -9 266 -16Q263 -18 259 -21L241 -22H234Q216 -22 216 -15Q213 -9 177 305Q139 623 138 626Q133 637 76 637H59Q52 642 52 648Z"></path><path stroke-width="1" id="E38-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E38-MJMATHI-70" x="0" y="0"></use><use xlink:href="#E38-MJMAIN-28" x="503" y="0"></use><g transform="translate(893,0)"><use xlink:href="#E38-MJMATHI-3B8" x="0" y="0"></use><g transform="translate(469,-187)"><use transform="scale(0.707)" xlink:href="#E38-MJMAIN-31" x="0" y="0"></use><use transform="scale(0.707)" 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-143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E39-MJMATHI-70" x="0" y="0"></use><use xlink:href="#E39-MJMAIN-28" x="503" y="0"></use><g transform="translate(893,0)"><use xlink:href="#E39-MJMATHI-77" x="0" y="0"></use><g transform="translate(716,-150)"><use transform="scale(0.707)" xlink:href="#E39-MJMAIN-31" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E39-MJMAIN-3A" x="500" y="0"></use><use transform="scale(0.707)" xlink:href="#E39-MJMATHI-6E" x="779" y="0"></use></g></g><use xlink:href="#E39-MJMAIN-29" x="2684" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-39">p(w_{1:n})</script>是固定的，问题转化成如下：</p><div contenteditable="false" class="mathjax-block md-end-block" id="mathjax-c108" cid="c108" mdtype="math_block"><span class="MathJax_Preview"></span><span class="MathJax_SVG_Display" style="text-align: center;"><span class="MathJax_SVG" id="MathJax-Element-16-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="22.692ex" height="4.328ex" viewBox="0 -806.1 9769.9 1863.6" role="img" focusable="false" style="vertical-align: -2.456ex;"><defs><path stroke-width="1" id="E16-MJMATHI-61" d="M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z"></path><path stroke-width="1" id="E16-MJMATHI-72" d="M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 442Q161 442 183 430T214 408T225 388Q227 382 228 382T236 389Q284 441 347 441H350Q398 441 422 400Q430 381 430 363Q430 333 417 315T391 292T366 288Q346 288 334 299T322 328Q322 376 378 392Q356 405 342 405Q286 405 239 331Q229 315 224 298T190 165Q156 25 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 114 189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path stroke-width="1" id="E16-MJMATHI-67" d="M311 43Q296 30 267 15T206 0Q143 0 105 45T66 160Q66 265 143 353T314 442Q361 442 401 394L404 398Q406 401 409 404T418 412T431 419T447 422Q461 422 470 413T480 394Q480 379 423 152T363 -80Q345 -134 286 -169T151 -205Q10 -205 10 -137Q10 -111 28 -91T74 -71Q89 -71 102 -80T116 -111Q116 -121 114 -130T107 -144T99 -154T92 -162L90 -164H91Q101 -167 151 -167Q189 -167 211 -155Q234 -144 254 -122T282 -75Q288 -56 298 -13Q311 35 311 43ZM384 328L380 339Q377 350 375 354T369 368T359 382T346 393T328 402T306 405Q262 405 221 352Q191 313 171 233T151 117Q151 38 213 38Q269 38 323 108L331 118L384 328Z"></path><path stroke-width="1" id="E16-MJMATHI-6D" d="M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path stroke-width="1" id="E16-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path stroke-width="1" id="E16-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="1" id="E16-MJMATHI-70" d="M23 287Q24 290 25 295T30 317T40 348T55 381T75 411T101 433T134 442Q209 442 230 378L240 387Q302 442 358 442Q423 442 460 395T497 281Q497 173 421 82T249 -10Q227 -10 210 -4Q199 1 187 11T168 28L161 36Q160 35 139 -51T118 -138Q118 -144 126 -145T163 -148H188Q194 -155 194 -157T191 -175Q188 -187 185 -190T172 -194Q170 -194 161 -194T127 -193T65 -192Q-5 -192 -24 -194H-32Q-39 -187 -39 -183Q-37 -156 -26 -148H-6Q28 -147 33 -136Q36 -130 94 103T155 350Q156 355 156 364Q156 405 131 405Q109 405 94 377T71 316T59 280Q57 278 43 278H29Q23 284 23 287ZM178 102Q200 26 252 26Q282 26 310 49T356 107Q374 141 392 215T411 325V331Q411 405 350 405Q339 405 328 402T306 393T286 380T269 365T254 350T243 336T235 326L232 322Q232 321 229 308T218 264T204 212Q178 106 178 102Z"></path><path stroke-width="1" id="E16-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="1" id="E16-MJMAIN-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path><path stroke-width="1" id="E16-MJMAIN-3A" d="M78 370Q78 394 95 412T138 430Q162 430 180 414T199 371Q199 346 182 328T139 310T96 327T78 370ZM78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z"></path><path stroke-width="1" id="E16-MJMAIN-7C" d="M139 -249H137Q125 -249 119 -235V251L120 737Q130 750 139 750Q152 750 159 735V-235Q151 -249 141 -249H139Z"></path><path stroke-width="1" id="E16-MJMATHI-56" d="M52 648Q52 670 65 683H76Q118 680 181 680Q299 680 320 683H330Q336 677 336 674T334 656Q329 641 325 637H304Q282 635 274 635Q245 630 242 620Q242 618 271 369T301 118L374 235Q447 352 520 471T595 594Q599 601 599 609Q599 633 555 637Q537 637 537 648Q537 649 539 661Q542 675 545 679T558 683Q560 683 570 683T604 682T668 681Q737 681 755 683H762Q769 676 769 672Q769 655 760 640Q757 637 743 637Q730 636 719 635T698 630T682 623T670 615T660 608T652 599T645 592L452 282Q272 -9 266 -16Q263 -18 259 -21L241 -22H234Q216 -22 216 -15Q213 -9 177 305Q139 623 138 626Q133 637 76 637H59Q52 642 52 648Z"></path><path stroke-width="1" id="E16-MJMATHI-77" d="M580 385Q580 406 599 424T641 443Q659 443 674 425T690 368Q690 339 671 253Q656 197 644 161T609 80T554 12T482 -11Q438 -11 404 5T355 48Q354 47 352 44Q311 -11 252 -11Q226 -11 202 -5T155 14T118 53T104 116Q104 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Q21 293 29 315T52 366T96 418T161 441Q204 441 227 416T250 358Q250 340 217 250T184 111Q184 65 205 46T258 26Q301 26 334 87L339 96V119Q339 122 339 128T340 136T341 143T342 152T345 165T348 182T354 206T362 238T373 281Q402 395 406 404Q419 431 449 431Q468 431 475 421T483 402Q483 389 454 274T422 142Q420 131 420 107V100Q420 85 423 71T442 42T487 26Q558 26 600 148Q609 171 620 213T632 273Q632 306 619 325T593 357T580 385Z"></path><path stroke-width="1" id="E16-MJMATHI-6E" d="M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z"></path><path stroke-width="1" id="E16-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E16-MJMATHI-61" x="0" y="0"></use><use xlink:href="#E16-MJMATHI-72" x="529" y="0"></use><use xlink:href="#E16-MJMATHI-67" x="981" y="0"></use><use xlink:href="#E16-MJMATHI-6D" x="1461" y="0"></use><use xlink:href="#E16-MJMATHI-61" x="2340" y="0"></use><use xlink:href="#E16-MJMATHI-78" x="2869" y="0"></use><use transform="scale(0.707)" xlink:href="#E16-MJMATHI-3B8" x="2199" y="-1234"></use><use xlink:href="#E16-MJMATHI-70" x="4358" y="0"></use><use xlink:href="#E16-MJMAIN-28" x="4862" y="0"></use><g transform="translate(5251,0)"><use xlink:href="#E16-MJMATHI-3B8" x="0" y="0"></use><g transform="translate(469,-187)"><use transform="scale(0.707)" xlink:href="#E16-MJMAIN-31" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E16-MJMAIN-3A" x="500" y="0"></use><use transform="scale(0.707)" xlink:href="#E16-MJMAIN-7C" x="779" y="0"></use><use transform="scale(0.707)" xlink:href="#E16-MJMATHI-56" x="1057" y="0"></use><use transform="scale(0.707)" xlink:href="#E16-MJMAIN-7C" x="1827" y="0"></use></g></g><use xlink:href="#E16-MJMAIN-7C" x="7309" y="0"></use><g transform="translate(7588,0)"><use xlink:href="#E16-MJMATHI-77" x="0" y="0"></use><g transform="translate(716,-150)"><use transform="scale(0.707)" xlink:href="#E16-MJMAIN-31" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E16-MJMAIN-3A" x="500" y="0"></use><use transform="scale(0.707)" xlink:href="#E16-MJMATHI-6E" x="779" y="0"></use></g></g><use xlink:href="#E16-MJMAIN-29" x="9380" y="0"></use></g></svg></span></span><script type="math/tex; mode=display" id="MathJax-Element-16">
\begin{equation}
\mathop{argmax}_{\theta} \ \ \ p(\theta_{1:|V|}|w_{1:n})
\end{equation}
</script></div><p>其中<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-40-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="12.112ex" height="3.745ex" viewBox="0 -1107.7 5215 1612.3" role="img" focusable="false" style="vertical-align: -1.172ex;"><defs><path stroke-width="1" id="E40-MJSZ1-2211" d="M61 748Q64 750 489 750H913L954 640Q965 609 976 579T993 533T999 516H979L959 517Q936 579 886 621T777 682Q724 700 655 705T436 710H319Q183 710 183 709Q186 706 348 484T511 259Q517 250 513 244L490 216Q466 188 420 134T330 27L149 -187Q149 -188 362 -188Q388 -188 436 -188T506 -189Q679 -189 778 -162T936 -43Q946 -27 959 6H999L913 -249L489 -250Q65 -250 62 -248Q56 -246 56 -239Q56 -234 118 -161Q186 -81 245 -11L428 206Q428 207 242 462L57 717L56 728Q56 744 61 748Z"></path><path stroke-width="1" id="E40-MJMAIN-7C" d="M139 -249H137Q125 -249 119 -235V251L120 737Q130 750 139 750Q152 750 159 735V-235Q151 -249 141 -249H139Z"></path><path stroke-width="1" id="E40-MJMATHI-56" d="M52 648Q52 670 65 683H76Q118 680 181 680Q299 680 320 683H330Q336 677 336 674T334 656Q329 641 325 637H304Q282 635 274 635Q245 630 242 620Q242 618 271 369T301 118L374 235Q447 352 520 471T595 594Q599 601 599 609Q599 633 555 637Q537 637 537 648Q537 649 539 661Q542 675 545 679T558 683Q560 683 570 683T604 682T668 681Q737 681 755 683H762Q769 676 769 672Q769 655 760 640Q757 637 743 637Q730 636 719 635T698 630T682 623T670 615T660 608T652 599T645 592L452 282Q272 -9 266 -16Q263 -18 259 -21L241 -22H234Q216 -22 216 -15Q213 -9 177 305Q139 623 138 626Q133 637 76 637H59Q52 642 52 648Z"></path><path stroke-width="1" id="E40-MJMATHI-6A" d="M297 596Q297 627 318 644T361 661Q378 661 389 651T403 623Q403 595 384 576T340 557Q322 557 310 567T297 596ZM288 376Q288 405 262 405Q240 405 220 393T185 362T161 325T144 293L137 279Q135 278 121 278H107Q101 284 101 286T105 299Q126 348 164 391T252 441Q253 441 260 441T272 442Q296 441 316 432Q341 418 354 401T367 348V332L318 133Q267 -67 264 -75Q246 -125 194 -164T75 -204Q25 -204 7 -183T-12 -137Q-12 -110 7 -91T53 -71Q70 -71 82 -81T95 -112Q95 -148 63 -167Q69 -168 77 -168Q111 -168 139 -140T182 -74L193 -32Q204 11 219 72T251 197T278 308T289 365Q289 372 288 376Z"></path><path stroke-width="1" id="E40-MJMAIN-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path stroke-width="1" id="E40-MJMAIN-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path><path stroke-width="1" id="E40-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E40-MJSZ1-2211" x="0" y="0"></use><g transform="translate(1056,521)"><use transform="scale(0.707)" xlink:href="#E40-MJMAIN-7C" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E40-MJMATHI-56" x="278" y="0"></use><use transform="scale(0.707)" xlink:href="#E40-MJMAIN-7C" x="1048" y="0"></use></g><g transform="translate(1056,-308)"><use transform="scale(0.707)" xlink:href="#E40-MJMATHI-6A" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E40-MJMAIN-3D" x="412" y="0"></use><use transform="scale(0.707)" xlink:href="#E40-MJMAIN-31" x="1191" y="0"></use></g><g transform="translate(2519,0)"><use xlink:href="#E40-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E40-MJMATHI-6A" x="663" y="-213"></use></g><use xlink:href="#E40-MJMAIN-3D" x="3658" y="0"></use><use xlink:href="#E40-MJMAIN-31" x="4714" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-40">\sum_{j=1}^{|V|}\theta_j=1</script></p><p>使用拉格朗日条件极值，得出结论：</p><div contenteditable="false" class="mathjax-block md-end-block" id="mathjax-c113" cid="c113" mdtype="math_block"><span class="MathJax_Preview"></span><span class="MathJax_SVG_Display" style="text-align: center;"><span class="MathJax_SVG" id="MathJax-Element-17-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="37.413ex" height="4.679ex" viewBox="0 -1258.5 16108.1 2014.4" role="img" focusable="false" style="vertical-align: -1.756ex;"><defs><path stroke-width="1" id="E17-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="1" id="E17-MJMATHI-6A" d="M297 596Q297 627 318 644T361 661Q378 661 389 651T403 623Q403 595 384 576T340 557Q322 557 310 567T297 596ZM288 376Q288 405 262 405Q240 405 220 393T185 362T161 325T144 293L137 279Q135 278 121 278H107Q101 284 101 286T105 299Q126 348 164 391T252 441Q253 441 260 441T272 442Q296 441 316 432Q341 418 354 401T367 348V332L318 133Q267 -67 264 -75Q246 -125 194 -164T75 -204Q25 -204 7 -183T-12 -137Q-12 -110 7 -91T53 -71Q70 -71 82 -81T95 -112Q95 -148 63 -167Q69 -168 77 -168Q111 -168 139 -140T182 -74L193 -32Q204 11 219 72T251 197T278 308T289 365Q289 372 288 376Z"></path><path stroke-width="1" id="E17-MJMAIN-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path stroke-width="1" id="E17-MJMATHI-6D" d="M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 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y="-686"></use></g></g><g transform="translate(13323,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" stroke="none" transform="scale(50.259) matrix(1 0 0 -1 0 0)">式</text></g><use xlink:href="#E17-MJMAIN-28" x="14049" y="0"></use><g transform="translate(14439,0)"><use xlink:href="#E17-MJMAIN-33"></use><use xlink:href="#E17-MJMAIN-2E" x="500" y="0"></use><use xlink:href="#E17-MJMAIN-34" x="779" y="0"></use></g><use xlink:href="#E17-MJMAIN-29" x="15718" y="0"></use></g></svg></span></span><script type="math/tex; mode=display" id="MathJax-Element-17">
\theta_{j}=\frac{m_j}{n} \hspace{5cm}式(3.4)
</script></div><h3><a name='header-c115' class='md-header-anchor '></a>3.2 一元模型后验概率（MAP）</h3><div contenteditable="false" class="mathjax-block md-end-block" id="mathjax-c117" cid="c117" mdtype="math_block"><span class="MathJax_Preview"></span><span class="MathJax_SVG_Display" style="text-align: center;"><span class="MathJax_SVG" id="MathJax-Element-18-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="40.625ex" height="6.546ex" viewBox="0 -1660.6 17491.3 2818.5" role="img" focusable="false" style="vertical-align: -2.689ex;"><defs><path stroke-width="1" id="E18-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 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\theta_{j}=\frac{m_j＋1}{n＋|V|} \hspace{5cm}式(3.5)
</script></div><h3><a name='header-c118' class='md-header-anchor '></a>3.3 二元模型概率</h3><div contenteditable="false" class="mathjax-block md-end-block" id="mathjax-c119" cid="c119" mdtype="math_block"><span class="MathJax_Preview"></span><span class="MathJax_SVG_Display" style="text-align: center;"><span class="MathJax_SVG" id="MathJax-Element-19-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="30.734ex" height="6.079ex" viewBox="0 -1560 13232.7 2617.5" role="img" focusable="false" style="vertical-align: -2.456ex;"><defs><path stroke-width="1" id="E19-MJMATHI-50" d="M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 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y="0"></use></g><g transform="translate(812,-717)"><use xlink:href="#E19-MJMATHI-63" x="0" y="0"></use><use xlink:href="#E19-MJMATHI-6F" x="433" y="0"></use><use xlink:href="#E19-MJMATHI-75" x="919" y="0"></use><use xlink:href="#E19-MJMATHI-6E" x="1491" y="0"></use><use xlink:href="#E19-MJMATHI-74" x="2092" y="0"></use><use xlink:href="#E19-MJMAIN-28" x="2453" y="0"></use><g transform="translate(2843,0)"><use xlink:href="#E19-MJMATHI-77" x="0" y="0"></use><g transform="translate(716,-150)"><use transform="scale(0.707)" xlink:href="#E19-MJMATHI-69" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E19-MJMAIN-2212" x="345" y="0"></use><use transform="scale(0.707)" xlink:href="#E19-MJMAIN-31" x="1124" y="0"></use></g></g><use xlink:href="#E19-MJMAIN-29" x="4808" y="0"></use></g></g></g></g></svg></span></span><script type="math/tex; mode=display" id="MathJax-Element-19">
P(w_i |w_{i-1}) =
\frac{count(w_{i-1},w_{i})}{count(w_{i-1})}
</script></div><p>例子1：</p><p>语料库：<s> I am Sam </s></p><p> <s> Sam I am </s></p><p> <s> I do not like green eggs and ham </s></p><p> <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-41-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="16.772ex" height="3.278ex" viewBox="0 -956.9 7221.3 1411.3" role="img" focusable="false" style="vertical-align: -1.055ex;"><defs><path stroke-width="1" id="E41-MJMATHI-50" d="M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z"></path><path stroke-width="1" id="E41-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="1" id="E41-MJMATHI-49" d="M43 1Q26 1 26 10Q26 12 29 24Q34 43 39 45Q42 46 54 46H60Q120 46 136 53Q137 53 138 54Q143 56 149 77T198 273Q210 318 216 344Q286 624 286 626Q284 630 284 631Q274 637 213 637H193Q184 643 189 662Q193 677 195 680T209 683H213Q285 681 359 681Q481 681 487 683H497Q504 676 504 672T501 655T494 639Q491 637 471 637Q440 637 407 634Q393 631 388 623Q381 609 337 432Q326 385 315 341Q245 65 245 59Q245 52 255 50T307 46H339Q345 38 345 37T342 19Q338 6 332 0H316Q279 2 179 2Q143 2 113 2T65 2T43 1Z"></path><path stroke-width="1" 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463Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E41-MJMATHI-50" x="0" y="0"></use><use xlink:href="#E41-MJMAIN-28" x="751" y="0"></use><use xlink:href="#E41-MJMATHI-49" x="1141" y="0"></use><use xlink:href="#E41-MJMAIN-7C" x="1645" y="0"></use><use xlink:href="#E41-MJMAIN-3C" x="2201" y="0"></use><use xlink:href="#E41-MJMATHI-73" x="3258" y="0"></use><use xlink:href="#E41-MJMAIN-3E" x="4005" y="0"></use><use xlink:href="#E41-MJMAIN-29" x="4783" y="0"></use><use xlink:href="#E41-MJMAIN-3D" x="5451" y="0"></use><g transform="translate(6229,0)"><g transform="translate(397,0)"><rect stroke="none" width="473" height="60" x="0" y="220"></rect><use transform="scale(0.707)" xlink:href="#E41-MJMAIN-32" x="84" y="572"></use><use transform="scale(0.707)" xlink:href="#E41-MJMAIN-33" x="84" y="-531"></use></g></g></g></svg></span><script type="math/tex" id="MathJax-Element-41">P(I|<s>)=\frac{2}{3}</script> <span 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110T159 94T148 82T136 74T126 70T118 67L114 66Q165 21 238 21Q293 21 321 74Q338 107 338 175V195Q338 290 274 322Q259 328 213 329L171 330L168 332Q166 335 166 348Q166 366 174 366Q202 366 232 371Q266 376 294 413T322 525V533Q322 590 287 612Q265 626 240 626Q208 626 181 615T143 592T132 580H135Q138 579 143 578T153 573T165 566T175 555T183 540T186 520Q186 498 172 481T127 463Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E42-MJMATHI-50" x="0" y="0"></use><use xlink:href="#E42-MJMAIN-28" x="751" y="0"></use><use xlink:href="#E42-MJMATHI-53" x="1141" y="0"></use><use xlink:href="#E42-MJMATHI-61" x="1786" y="0"></use><use xlink:href="#E42-MJMATHI-6D" x="2316" y="0"></use><use xlink:href="#E42-MJMAIN-7C" x="3194" y="0"></use><use xlink:href="#E42-MJMAIN-3C" x="3750" y="0"></use><use xlink:href="#E42-MJMATHI-73" x="4807" y="0"></use><use xlink:href="#E42-MJMAIN-3E" x="5554" y="0"></use><use xlink:href="#E42-MJMAIN-29" 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</script></div><h2><a name='header-c135' class='md-header-anchor '></a>4 N-gram语言模型评价</h2><h3><a name='header-c136' class='md-header-anchor '></a>4.1 外在评价</h3><p>将需比较的模型用于具体任务中（如机器翻译，语音识别），看在任务中的效果</p><p>缺点：非常耗费时间，并且需要理解更为复杂的任务。</p><h3><a name='header-c141' class='md-header-anchor '></a>4.2 内在评价</h3><p>困惑度（perplexity）</p><p>缺点：只在训练集和测试集相似的情况下取得好的评价，通常只适用于小规模试验</p><div contenteditable="false" class="mathjax-block md-end-block" id="mathjax-c145" cid="c145" mdtype="math_block"><span class="MathJax_Preview"></span><span class="MathJax_SVG_Display" style="text-align: center;"><span class="MathJax_SVG" id="MathJax-Element-21-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="34.53ex" height="7.83ex" viewBox="0 -2112.9 14867.1 3371.4" role="img" focusable="false" style="vertical-align: -2.923ex;"><defs><path stroke-width="1" id="E21-MJMATHI-50" d="M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 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xlink:href="#E21-MJMATHI-77" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E21-MJMAIN-31" x="1013" y="-213"></use></g><use xlink:href="#E21-MJMAIN-2E" x="3650" y="0"></use><use xlink:href="#E21-MJMAIN-2E" x="4095" y="0"></use><use xlink:href="#E21-MJMAIN-2E" x="4541" y="0"></use><g transform="translate(4986,0)"><use xlink:href="#E21-MJMATHI-77" x="0" y="0"></use><g transform="translate(716,-150)"><use transform="scale(0.707)" xlink:href="#E21-MJMATHI-69" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E21-MJMAIN-2212" x="345" y="0"></use><use transform="scale(0.707)" xlink:href="#E21-MJMAIN-31" x="1124" y="0"></use></g></g><use xlink:href="#E21-MJMAIN-29" x="6951" y="0"></use></g></g></g></g></g></g></svg></span></span><script type="math/tex; mode=display" id="MathJax-Element-21">
PP(W) = \sqrt[N] {\prod_{i=1}^{N} \frac{1}{P(w_i|w_1...w_{i-1})}}
</script></div><p>降低困惑度等同于提高句子概率，对于Bigram</p><div contenteditable="false" class="mathjax-block md-end-block" id="mathjax-c148" cid="c148" mdtype="math_block"><span class="MathJax_Preview"></span><span class="MathJax_SVG_Display" style="text-align: center;"><span class="MathJax_SVG" id="MathJax-Element-22-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="28.71ex" height="7.83ex" viewBox="0 -2112.9 12361.2 3371.4" role="img" focusable="false" style="vertical-align: -2.923ex;"><defs><path stroke-width="1" id="E22-MJMATHI-50" d="M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 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y="0"></use><use xlink:href="#E22-MJMAIN-3D" x="3608" y="0"></use><g transform="translate(4664,0)"><use transform="scale(0.5)" xlink:href="#E22-MJMATHI-4E" x="273" y="838"></use><g transform="translate(0,1962)"><use xlink:href="#E22-MJSZ4-E001" x="0" y="-606"></use><g transform="translate(0,-1368.9703475289352) scale(1,1.2747130152219412)"><use xlink:href="#E22-MJSZ4-E000"></use></g><use xlink:href="#E22-MJSZ4-23B7" x="0" y="-2275"></use></g><rect stroke="none" width="6640" height="60" x="1056" y="1902"></rect><g transform="translate(1056,0)"><use xlink:href="#E22-MJSZ2-220F" x="0" y="0"></use><g transform="translate(64,-1090)"><use transform="scale(0.707)" xlink:href="#E22-MJMATHI-69" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E22-MJMAIN-3D" x="345" y="0"></use><use transform="scale(0.707)" xlink:href="#E22-MJMAIN-31" x="1124" y="0"></use></g><use transform="scale(0.707)" xlink:href="#E22-MJMATHI-4E" x="459" y="1627"></use><g transform="translate(1278,0)"><g 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type="math/tex; mode=display" id="MathJax-Element-22">
PP(W) = \sqrt[N] {\prod_{i=1}^{N} \frac{1}{P(w_i|w_{i-1})}}
</script></div><p>例子：假设一个句子只包含随机数字，句子的困惑度：</p><div contenteditable="false" class="mathjax-block md-end-block" id="mathjax-c151" cid="c151" mdtype="math_block"><span class="MathJax_Preview"></span><span class="MathJax_SVG_Display" style="text-align: center;"><span class="MathJax_SVG" id="MathJax-Element-23-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="53.992ex" height="3.044ex" viewBox="0 -1007.2 23246.6 1310.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E23-MJMATHI-50" d="M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 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PP(W) = P(w_1w_2...w_N)^{-1/N}=((1/10)^{N})^{-1/N}=10
</script></div><h2><a name='header-c153' class='md-header-anchor '></a>5 语言模型泛化</h2><h3><a name='header-c154' class='md-header-anchor '></a>5.1 平滑问题</h3><p>N元语言模型只在训练集和测试集相似的情况下起作用，但在实际情况中，往往在测试集中存在OOV words(out of vocabulary words)，即会出现测试集中某句子概率为0的情况（如果该句子包含OOV词汇），这样既不符合常理，困惑度也无法计算（分母上会出现零），处理方法：</p><ol><li>将稀有词汇都用<UNK>代替，即出现频率低于某阈值的词，测试的时候OOV words就用<UNK>词的概率代替</li><li>加一平滑，又称Laplace平滑</li></ol><p>原估计<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-44-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="22.096ex" height="4.445ex" viewBox="-38.5 -1208.2 9513.6 1913.9" role="img" focusable="false" style="vertical-align: -1.639ex; margin-left: -0.089ex;"><defs><path stroke-width="1" id="E44-MJMATHI-70" d="M23 287Q24 290 25 295T30 317T40 348T55 381T75 411T101 433T134 442Q209 442 230 378L240 387Q302 442 358 442Q423 442 460 395T497 281Q497 173 421 82T249 -10Q227 -10 210 -4Q199 1 187 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x="6596" y="0"></use><use xlink:href="#E46-MJMAIN-2B" x="7207" y="0"></use><g transform="translate(8208,0)"><use xlink:href="#E46-MJMATHI-3BB" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E46-MJMAIN-33" x="825" y="-213"></use></g><use xlink:href="#E46-MJMATHI-70" x="9246" y="0"></use><use xlink:href="#E46-MJMAIN-28" x="9749" y="0"></use><g transform="translate(10139,0)"><use xlink:href="#E46-MJMATHI-77" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E46-MJMATHI-6E" x="1013" y="-213"></use></g><use xlink:href="#E46-MJMAIN-29" x="11380" y="0"></use><use xlink:href="#E46-MJSZ1-2211" x="13835" y="0"></use><use xlink:href="#E46-MJMATHI-3BB" x="15059" y="0"></use><use xlink:href="#E46-MJMAIN-3D" x="15920" y="0"></use><use xlink:href="#E46-MJMAIN-31" x="16976" y="0"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-46">\hat p(w_n|w_{n-1}w_{n-2})=\lambda_1p(w_n|w_{n-1}w_{n-2})\\+\lambda_2p(w_n|w_{n-1})+\lambda_3p(w_n) \hspace{1cm} 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277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E49-MJMATHI-3BB" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-49">\lambda</script>，使用验证集：选择<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-49-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.355ex" height="1.994ex" viewBox="0 -755.9 583.5 858.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E49-MJMATHI-3BB" d="M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 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logP(w_1...w_n|M(\lambda_1...\lambda_k))=\sum_ilogP_{M(\lambda_1...\lambda_k)}(w_i|w_{i-1})
</script></div><h3><a name='header-c195' class='md-header-anchor '></a>5.3 极大规模 N-gram model</h3><h4><a name='header-c196' class='md-header-anchor '></a>5.3.1 预处理</h4><p>只存储文本中出现次数大于阈值的n元组。</p><h4><a name='header-c199' class='md-header-anchor '></a>5.3.2 优化</h4><ol><li><p>采用trie树的数据结构，可以优化时间复杂度为<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-50-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="10.964ex" height="2.928ex" viewBox="0 -806.1 4720.5 1260.5" role="img" focusable="false" style="vertical-align: -1.055ex;"><defs><path stroke-width="1" id="E50-MJMATHI-4F" d="M740 435Q740 320 676 213T511 42T304 -22Q207 -22 138 35T51 201Q50 209 50 244Q50 346 98 438T227 601Q351 704 476 704Q514 704 524 703Q621 689 680 617T740 435ZM637 476Q637 565 591 615T476 665Q396 665 322 605Q242 542 200 428T157 216Q157 126 200 73T314 19Q404 19 485 98T608 313Q637 408 637 476Z"></path><path stroke-width="1" 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y="0"></use><use xlink:href="#E50-MJMATHI-6F" x="1451" y="0"></use><g transform="translate(1937,0)"><use xlink:href="#E50-MJMATHI-67" x="0" y="0"></use><g transform="translate(477,-187)"><use transform="scale(0.707)" xlink:href="#E50-MJMAIN-7C" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E50-MJMATHI-56" x="278" y="0"></use><use transform="scale(0.707)" xlink:href="#E50-MJMAIN-7C" x="1048" y="0"></use></g></g><use xlink:href="#E50-MJMATHI-6D" x="3452" y="0"></use><use xlink:href="#E50-MJMAIN-29" x="4330" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-50">O(log_{|V|}m)</script> |V|为字母个数，如英文的话即26个</p><p><img src='http://e.hiphotos.baidu.com/baike/c0%3Dbaike80%2C5%2C5%2C80%2C26/sign=27e168080fb30f242197e451a9fcba26/d62a6059252dd42a745cc2c2033b5bb5c9eab806.jpg' alt='' /></p></li><li><p>使用Bloom Filter</p></li><li><p>利用郝夫曼树对词进行编码，将词作为索引值而不是字符串进行存储，能将所有词编码成包含在2个字节内的索引值</p></li><li><p>优化概率值存储，概率值原使用的数据类型是（float），用4-8bit来代替原来8Byte的存储内容</p></li></ol><h4><a name='header-c215' class='md-header-anchor '></a>5.3.3 平滑方法</h4><p>stupid backoff方法</p><div contenteditable="false" class="mathjax-block md-end-block" id="mathjax-c217" cid="c217" mdtype="math_block"><span class="MathJax_Preview"></span><span class="MathJax_SVG_Display"><span class="MathJax_SVG" id="MathJax-Element-25-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="68.872ex" height="10.865ex" viewBox="0 -1761.1 29652.9 4678.1" role="img" focusable="false" style="vertical-align: -6.775ex;"><defs><path stroke-width="1" id="E25-MJMATHI-53" d="M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 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S(w_i|w_{i-k+1}^{i-1})=\frac {count(w_{i-k+1}^{i})}{count(w_{i-k+1}^{i-1})} \ \ \ \ \ \ if \ count(w_{i-k+1}^{i}))>0 \\=0.4S(w_i|w_{i-k+2}^{i-1}) \ \ \ \ \ Otherwise
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S(w_i)=\frac {count(w_i)}{N}
</script></div><h2><a name='header-c222' class='md-header-anchor '></a>6 高级语言模型</h2><ol><li><p>判别模型：将n-gram训练的权重用于提高外部任务的性能（语言模型中引入外部任务）</p></li><li><p>基于语义分析的模型，如下例子</p><p>The computer which is bought by her is crashed</p><p> <img src='parseTree.png' alt='parseTree' /></p></li></ol><p>基于语法分析树，crashed和computer的距离很近，而在N-gram中，这中依赖几乎不可能建立</p><ol><li><p>缓存模型</p><p>最近使用的词更有可能出现：</p><div contenteditable="false" class="mathjax-block md-end-block" id="mathjax-c241" cid="c241" mdtype="math_block"><span class="MathJax_Preview"></span><span class="MathJax_SVG_Display" style="text-align: center;"><span class="MathJax_SVG" id="MathJax-Element-27-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="65.509ex" height="6.079ex" viewBox="0 -1560 28205.2 2617.5" role="img" focusable="false" style="vertical-align: -2.456ex;"><defs><path stroke-width="1" id="E27-MJMATHI-50" d="M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 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y="0"></use></g></g></g></g></svg></span></span><script type="math/tex; mode=display" id="MathJax-Element-27">
P_{CACHE}(w|history) = \lambda P(w_i|w_{i-2}w_{i-1})+(1-\lambda)\frac{c(w\in history)}{|history|}
</script></div></li></ol><p></p><p></p><h2><a name='header-c248' class='md-header-anchor '></a>7 N-gram模型的缺陷</h2><ol><li>数据稀疏问题：利用平滑技术解决</li><li>空间占用大</li><li>长距离依赖问题</li><li>多义性</li><li>同义性－如鸡肉和狗肉属于同一类词，<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-51-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="7.089ex" height="2.928ex" viewBox="-38.5 -856.4 3052.3 1260.5" role="img" focusable="false" style="vertical-align: -0.938ex; margin-left: -0.089ex;"><defs><path stroke-width="1" id="E51-MJMATHI-70" d="M23 287Q24 290 25 295T30 317T40 348T55 381T75 411T101 433T134 442Q209 442 230 378L240 387Q302 442 358 442Q423 442 460 395T497 281Q497 173 421 82T249 -10Q227 -10 210 -4Q199 1 187 11T168 28L161 36Q160 35 139 -51T118 -138Q118 -144 126 -145T163 -148H188Q194 -155 194 -157T191 -175Q188 -187 185 -190T172 -194Q170 -194 161 -194T127 -193T65 -192Q-5 -192 -24 -194H-32Q-39 -187 -39 -183Q-37 -156 -26 -148H-6Q28 -147 33 -136Q36 -130 94 103T155 350Q156 355 156 364Q156 405 131 405Q109 405 94 377T71 316T59 280Q57 278 43 278H29Q23 284 23 287ZM178 102Q200 26 252 26Q282 26 310 49T356 107Q374 141 392 215T411 325V331Q411 405 350 405Q339 405 328 402T306 393T286 380T269 365T254 350T243 336T235 326L232 322Q232 321 229 308T218 264T204 212Q178 106 178 102Z"></path><path stroke-width="1" id="E51-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="1" id="E51-MJMAIN-7C" d="M139 -249H137Q125 -249 119 -235V251L120 737Q130 750 139 750Q152 750 159 735V-235Q151 -249 141 -249H139Z"></path><path stroke-width="1" id="E51-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E51-MJMATHI-70" x="0" y="0"></use><use xlink:href="#E51-MJMAIN-28" x="503" y="0"></use><g transform="translate(893,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" stroke="none" transform="scale(50.259) matrix(1 0 0 -1 0 0)">肉</text></g><use xlink:href="#E51-MJMAIN-7C" x="1619" y="0"></use><g transform="translate(1897,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" stroke="none" transform="scale(50.259) matrix(1 0 0 -1 0 0)">鸡</text></g><use xlink:href="#E51-MJMAIN-29" x="2624" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-51">p(肉|鸡)</script>应当等于<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-52-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="7.089ex" height="2.928ex" viewBox="-38.5 -856.4 3052.3 1260.5" role="img" focusable="false" style="vertical-align: -0.938ex; margin-left: -0.089ex;"><defs><path stroke-width="1" id="E52-MJMATHI-70" d="M23 287Q24 290 25 295T30 317T40 348T55 381T75 411T101 433T134 442Q209 442 230 378L240 387Q302 442 358 442Q423 442 460 395T497 281Q497 173 421 82T249 -10Q227 -10 210 -4Q199 1 187 11T168 28L161 36Q160 35 139 -51T118 -138Q118 -144 126 -145T163 -148H188Q194 -155 194 -157T191 -175Q188 -187 185 -190T172 -194Q170 -194 161 -194T127 -193T65 -192Q-5 -192 -24 -194H-32Q-39 -187 -39 -183Q-37 -156 -26 -148H-6Q28 -147 33 -136Q36 -130 94 103T155 350Q156 355 156 364Q156 405 131 405Q109 405 94 377T71 316T59 280Q57 278 43 278H29Q23 284 23 287ZM178 102Q200 26 252 26Q282 26 310 49T356 107Q374 141 392 215T411 325V331Q411 405 350 405Q339 405 328 402T306 393T286 380T269 365T254 350T243 336T235 326L232 322Q232 321 229 308T218 264T204 212Q178 106 178 102Z"></path><path stroke-width="1" id="E52-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="1" id="E52-MJMAIN-7C" d="M139 -249H137Q125 -249 119 -235V251L120 737Q130 750 139 750Q152 750 159 735V-235Q151 -249 141 -249H139Z"></path><path stroke-width="1" id="E52-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E52-MJMATHI-70" x="0" y="0"></use><use xlink:href="#E52-MJMAIN-28" x="503" y="0"></use><g transform="translate(893,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" stroke="none" transform="scale(50.259) matrix(1 0 0 -1 0 0)">肉</text></g><use xlink:href="#E52-MJMAIN-7C" x="1619" y="0"></use><g transform="translate(1897,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" stroke="none" transform="scale(50.259) matrix(1 0 0 -1 0 0)">狗</text></g><use xlink:href="#E52-MJMAIN-29" x="2624" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-52">p(肉|狗)</script>，而在训练集中学习到概率可能相差悬殊</li></ol><p>版权声明：本文为复旦nlp组施展根据课堂内容整理的原创文章，转载请注明出处。</p></div>
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