-
Notifications
You must be signed in to change notification settings - Fork 0
/
高斯网络.html
446 lines (408 loc) · 30.6 KB
/
高斯网络.html
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
<!DOCTYPE html>
<html lang="" xml:lang="">
<head>
<meta charset="utf-8" />
<meta http-equiv="X-UA-Compatible" content="IE=edge" />
<title>16 高斯网络 | 机器学习白板系列</title>
<meta name="description" content="这是用R的bookdown功能制作中文图书的模板，输出格式为bookdown::gitbook和bookdown::pdf_book." />
<meta name="generator" content="bookdown 0.26 and GitBook 2.6.7" />
<meta property="og:title" content="16 高斯网络 | 机器学习白板系列" />
<meta property="og:type" content="book" />
<meta property="og:description" content="这是用R的bookdown功能制作中文图书的模板，输出格式为bookdown::gitbook和bookdown::pdf_book." />
<meta name="twitter:card" content="summary" />
<meta name="twitter:title" content="16 高斯网络 | 机器学习白板系列" />
<meta name="twitter:description" content="这是用R的bookdown功能制作中文图书的模板，输出格式为bookdown::gitbook和bookdown::pdf_book." />
<meta name="author" content="庄闪闪" />
<meta name="viewport" content="width=device-width, initial-scale=1" />
<meta name="apple-mobile-web-app-capable" content="yes" />
<meta name="apple-mobile-web-app-status-bar-style" content="black" />
<link rel="prev" href="条件随机场.html"/>
<link rel="next" href="贝叶斯线性回归.html"/>
<script src="libs/jquery-3.6.0/jquery-3.6.0.min.js"></script>
<script src="https://cdn.jsdelivr.net/npm/fuse.js@6.4.6/dist/fuse.min.js"></script>
<link href="libs/gitbook-2.6.7/css/style.css" rel="stylesheet" />
<link href="libs/gitbook-2.6.7/css/plugin-table.css" rel="stylesheet" />
<link href="libs/gitbook-2.6.7/css/plugin-bookdown.css" rel="stylesheet" />
<link href="libs/gitbook-2.6.7/css/plugin-highlight.css" rel="stylesheet" />
<link href="libs/gitbook-2.6.7/css/plugin-search.css" rel="stylesheet" />
<link href="libs/gitbook-2.6.7/css/plugin-fontsettings.css" rel="stylesheet" />
<link href="libs/gitbook-2.6.7/css/plugin-clipboard.css" rel="stylesheet" />
<link href="libs/anchor-sections-1.1.0/anchor-sections.css" rel="stylesheet" />
<link href="libs/anchor-sections-1.1.0/anchor-sections-hash.css" rel="stylesheet" />
<script src="libs/anchor-sections-1.1.0/anchor-sections.js"></script>
<script type="text/x-mathjax-config">
MathJax.Hub.Config({
jax: ["input/TeX","output/SVG"],
extensions: ["tex2jax.js","MathMenu.js","MathZoom.js"],
TeX: {
extensions: ["AMSmath.js","AMSsymbols.js","noErrors.js","noUndefined.js"]
}
});
</script>
<script type="text/javascript"
src="../../../MathJax/MathJax.js">
</script>
<style type="text/css">
pre > code.sourceCode { white-space: pre; position: relative; }
pre > code.sourceCode > span { display: inline-block; line-height: 1.25; }
pre > code.sourceCode > span:empty { height: 1.2em; }
.sourceCode { overflow: visible; }
code.sourceCode > span { color: inherit; text-decoration: inherit; }
pre.sourceCode { margin: 0; }
@media screen {
div.sourceCode { overflow: auto; }
}
@media print {
pre > code.sourceCode { white-space: pre-wrap; }
pre > code.sourceCode > span { text-indent: -5em; padding-left: 5em; }
}
pre.numberSource code
{ counter-reset: source-line 0; }
pre.numberSource code > span
{ position: relative; left: -4em; counter-increment: source-line; }
pre.numberSource code > span > a:first-child::before
{ content: counter(source-line);
position: relative; left: -1em; text-align: right; vertical-align: baseline;
border: none; display: inline-block;
-webkit-touch-callout: none; -webkit-user-select: none;
-khtml-user-select: none; -moz-user-select: none;
-ms-user-select: none; user-select: none;
padding: 0 4px; width: 4em;
color: #aaaaaa;
}
pre.numberSource { margin-left: 3em; border-left: 1px solid #aaaaaa; padding-left: 4px; }
div.sourceCode
{ }
@media screen {
pre > code.sourceCode > span > a:first-child::before { text-decoration: underline; }
}
code span.al { color: #ff0000; font-weight: bold; } /* Alert */
code span.an { color: #60a0b0; font-weight: bold; font-style: italic; } /* Annotation */
code span.at { color: #7d9029; } /* Attribute */
code span.bn { color: #40a070; } /* BaseN */
code span.bu { } /* BuiltIn */
code span.cf { color: #007020; font-weight: bold; } /* ControlFlow */
code span.ch { color: #4070a0; } /* Char */
code span.cn { color: #880000; } /* Constant */
code span.co { color: #60a0b0; font-style: italic; } /* Comment */
code span.cv { color: #60a0b0; font-weight: bold; font-style: italic; } /* CommentVar */
code span.do { color: #ba2121; font-style: italic; } /* Documentation */
code span.dt { color: #902000; } /* DataType */
code span.dv { color: #40a070; } /* DecVal */
code span.er { color: #ff0000; font-weight: bold; } /* Error */
code span.ex { } /* Extension */
code span.fl { color: #40a070; } /* Float */
code span.fu { color: #06287e; } /* Function */
code span.im { } /* Import */
code span.in { color: #60a0b0; font-weight: bold; font-style: italic; } /* Information */
code span.kw { color: #007020; font-weight: bold; } /* Keyword */
code span.op { color: #666666; } /* Operator */
code span.ot { color: #007020; } /* Other */
code span.pp { color: #bc7a00; } /* Preprocessor */
code span.sc { color: #4070a0; } /* SpecialChar */
code span.ss { color: #bb6688; } /* SpecialString */
code span.st { color: #4070a0; } /* String */
code span.va { color: #19177c; } /* Variable */
code span.vs { color: #4070a0; } /* VerbatimString */
code span.wa { color: #60a0b0; font-weight: bold; font-style: italic; } /* Warning */
</style>
</head>
<body>
<div class="book without-animation with-summary font-size-2 font-family-1" data-basepath=".">
<div class="book-summary">
<nav role="navigation">
<ul class="summary">
<li class="chapter" data-level="" data-path="index.html"><a href="index.html"><i class="fa fa-check"></i>简介</a></li>
<li class="chapter" data-level="1" data-path="introduction.html"><a href="introduction.html"><i class="fa fa-check"></i><b>1</b> Introduction</a>
<ul>
<li class="chapter" data-level="1.1" data-path="introduction.html"><a href="introduction.html#频率派的观点"><i class="fa fa-check"></i><b>1.1</b> 频率派的观点</a></li>
<li class="chapter" data-level="1.2" data-path="introduction.html"><a href="introduction.html#贝叶斯派的观点"><i class="fa fa-check"></i><b>1.2</b> 贝叶斯派的观点</a></li>
<li class="chapter" data-level="1.3" data-path="introduction.html"><a href="introduction.html#小结"><i class="fa fa-check"></i><b>1.3</b> 小结</a></li>
</ul></li>
<li class="chapter" data-level="2" data-path="mathbasics.html"><a href="mathbasics.html"><i class="fa fa-check"></i><b>2</b> MathBasics</a>
<ul>
<li class="chapter" data-level="2.1" data-path="mathbasics.html"><a href="mathbasics.html#高斯分布"><i class="fa fa-check"></i><b>2.1</b> 高斯分布</a>
<ul>
<li class="chapter" data-level="2.1.1" data-path="mathbasics.html"><a href="mathbasics.html#一维情况-mle"><i class="fa fa-check"></i><b>2.1.1</b> 一维情况 MLE</a></li>
<li class="chapter" data-level="2.1.2" data-path="mathbasics.html"><a href="mathbasics.html#多维情况"><i class="fa fa-check"></i><b>2.1.2</b> 多维情况</a></li>
</ul></li>
</ul></li>
<li class="chapter" data-level="3" data-path="线性回归.html"><a href="线性回归.html"><i class="fa fa-check"></i><b>3</b> 线性回归</a>
<ul>
<li class="chapter" data-level="3.1" data-path="线性回归.html"><a href="线性回归.html#最小二乘法"><i class="fa fa-check"></i><b>3.1</b> 最小二乘法</a></li>
<li class="chapter" data-level="3.2" data-path="线性回归.html"><a href="线性回归.html#噪声为高斯分布的-mle"><i class="fa fa-check"></i><b>3.2</b> 噪声为高斯分布的 MLE</a></li>
<li class="chapter" data-level="3.3" data-path="线性回归.html"><a href="线性回归.html#权重先验也为高斯分布的-map"><i class="fa fa-check"></i><b>3.3</b> 权重先验也为高斯分布的 MAP</a></li>
<li class="chapter" data-level="3.4" data-path="线性回归.html"><a href="线性回归.html#正则化"><i class="fa fa-check"></i><b>3.4</b> 正则化</a>
<ul>
<li class="chapter" data-level="3.4.1" data-path="线性回归.html"><a href="线性回归.html#l1-lasso"><i class="fa fa-check"></i><b>3.4.1</b> L1 Lasso</a></li>
<li class="chapter" data-level="3.4.2" data-path="线性回归.html"><a href="线性回归.html#l2-ridge"><i class="fa fa-check"></i><b>3.4.2</b> L2 Ridge</a></li>
</ul></li>
<li class="chapter" data-level="3.5" data-path="线性回归.html"><a href="线性回归.html#小结-1"><i class="fa fa-check"></i><b>3.5</b> 小结</a></li>
</ul></li>
<li class="chapter" data-level="4" data-path="线性分类.html"><a href="线性分类.html"><i class="fa fa-check"></i><b>4</b> 线性分类</a>
<ul>
<li class="chapter" data-level="4.1" data-path="线性分类.html"><a href="线性分类.html#两分类-硬分类-感知机算法"><i class="fa fa-check"></i><b>4.1</b> 两分类-硬分类-感知机算法</a></li>
<li class="chapter" data-level="4.2" data-path="线性分类.html"><a href="线性分类.html#两分类-硬分类-线性判别分析-lda"><i class="fa fa-check"></i><b>4.2</b> 两分类-硬分类-线性判别分析 LDA</a></li>
<li class="chapter" data-level="4.3" data-path="线性分类.html"><a href="线性分类.html#两分类-软分类-概率判别模型-logistic-回归"><i class="fa fa-check"></i><b>4.3</b> 两分类-软分类-概率判别模型-Logistic 回归</a></li>
<li class="chapter" data-level="4.4" data-path="线性分类.html"><a href="线性分类.html#两分类-软分类-概率生成模型-高斯判别分析-gda"><i class="fa fa-check"></i><b>4.4</b> 两分类-软分类-概率生成模型-高斯判别分析 GDA</a></li>
<li class="chapter" data-level="4.5" data-path="线性分类.html"><a href="线性分类.html#两分类-软分类-概率生成模型-朴素贝叶斯"><i class="fa fa-check"></i><b>4.5</b> 两分类-软分类-概率生成模型-朴素贝叶斯</a></li>
<li class="chapter" data-level="4.6" data-path="线性分类.html"><a href="线性分类.html#小结-2"><i class="fa fa-check"></i><b>4.6</b> 小结</a></li>
</ul></li>
<li class="chapter" data-level="5" data-path="降维.html"><a href="降维.html"><i class="fa fa-check"></i><b>5</b> 降维</a>
<ul>
<li class="chapter" data-level="5.1" data-path="降维.html"><a href="降维.html#线性降维-主成分分析-pca"><i class="fa fa-check"></i><b>5.1</b> 线性降维-主成分分析 PCA</a>
<ul>
<li class="chapter" data-level="5.1.1" data-path="降维.html"><a href="降维.html#损失函数"><i class="fa fa-check"></i><b>5.1.1</b> 损失函数</a></li>
<li class="chapter" data-level="5.1.2" data-path="降维.html"><a href="降维.html#svd-与-pcoa"><i class="fa fa-check"></i><b>5.1.2</b> SVD 与 PCoA</a></li>
<li class="chapter" data-level="5.1.3" data-path="降维.html"><a href="降维.html#p-pca"><i class="fa fa-check"></i><b>5.1.3</b> p-PCA</a></li>
</ul></li>
<li class="chapter" data-level="5.2" data-path="降维.html"><a href="降维.html#小结-3"><i class="fa fa-check"></i><b>5.2</b> 小结</a></li>
</ul></li>
<li class="chapter" data-level="6" data-path="支撑向量机.html"><a href="支撑向量机.html"><i class="fa fa-check"></i><b>6</b> 支撑向量机</a>
<ul>
<li class="chapter" data-level="6.1" data-path="支撑向量机.html"><a href="支撑向量机.html#约束优化问题"><i class="fa fa-check"></i><b>6.1</b> 约束优化问题</a></li>
<li class="chapter" data-level="6.2" data-path="支撑向量机.html"><a href="支撑向量机.html#hard-margin-svm"><i class="fa fa-check"></i><b>6.2</b> Hard-margin SVM</a></li>
<li class="chapter" data-level="6.3" data-path="支撑向量机.html"><a href="支撑向量机.html#soft-margin-svm"><i class="fa fa-check"></i><b>6.3</b> Soft-margin SVM</a></li>
<li class="chapter" data-level="6.4" data-path="支撑向量机.html"><a href="支撑向量机.html#kernel-method"><i class="fa fa-check"></i><b>6.4</b> Kernel Method</a></li>
<li class="chapter" data-level="6.5" data-path="支撑向量机.html"><a href="支撑向量机.html#小结-4"><i class="fa fa-check"></i><b>6.5</b> 小结</a></li>
</ul></li>
<li class="chapter" data-level="7" data-path="指数族分布.html"><a href="指数族分布.html"><i class="fa fa-check"></i><b>7</b> 指数族分布</a>
<ul>
<li class="chapter" data-level="7.1" data-path="指数族分布.html"><a href="指数族分布.html#一维高斯分布"><i class="fa fa-check"></i><b>7.1</b> 一维高斯分布</a></li>
<li class="chapter" data-level="7.2" data-path="指数族分布.html"><a href="指数族分布.html#充分统计量和对数配分函数的关系"><i class="fa fa-check"></i><b>7.2</b> 充分统计量和对数配分函数的关系</a></li>
<li class="chapter" data-level="7.3" data-path="指数族分布.html"><a href="指数族分布.html#充分统计量和极大似然估计"><i class="fa fa-check"></i><b>7.3</b> 充分统计量和极大似然估计</a></li>
<li class="chapter" data-level="7.4" data-path="指数族分布.html"><a href="指数族分布.html#最大熵"><i class="fa fa-check"></i><b>7.4</b> 最大熵</a></li>
</ul></li>
<li class="chapter" data-level="8" data-path="概率图模型.html"><a href="概率图模型.html"><i class="fa fa-check"></i><b>8</b> 概率图模型</a>
<ul>
<li class="chapter" data-level="8.1" data-path="概率图模型.html"><a href="概率图模型.html#有向图-贝叶斯网络"><i class="fa fa-check"></i><b>8.1</b> 有向图-贝叶斯网络</a></li>
<li class="chapter" data-level="8.2" data-path="概率图模型.html"><a href="概率图模型.html#无向图-马尔可夫网络马尔可夫随机场"><i class="fa fa-check"></i><b>8.2</b> 无向图-马尔可夫网络（马尔可夫随机场）</a></li>
<li class="chapter" data-level="8.3" data-path="概率图模型.html"><a href="概率图模型.html#两种图的转换-道德图"><i class="fa fa-check"></i><b>8.3</b> 两种图的转换-道德图</a></li>
<li class="chapter" data-level="8.4" data-path="概率图模型.html"><a href="概率图模型.html#更精细的分解-因子图"><i class="fa fa-check"></i><b>8.4</b> 更精细的分解-因子图</a></li>
<li class="chapter" data-level="8.5" data-path="概率图模型.html"><a href="概率图模型.html#推断"><i class="fa fa-check"></i><b>8.5</b> 推断</a>
<ul>
<li class="chapter" data-level="8.5.1" data-path="概率图模型.html"><a href="概率图模型.html#推断-变量消除ve"><i class="fa fa-check"></i><b>8.5.1</b> 推断-变量消除（VE）</a></li>
<li class="chapter" data-level="8.5.2" data-path="概率图模型.html"><a href="概率图模型.html#推断-信念传播bp"><i class="fa fa-check"></i><b>8.5.2</b> 推断-信念传播（BP）</a></li>
<li class="chapter" data-level="8.5.3" data-path="概率图模型.html"><a href="概率图模型.html#推断-max-product-算法"><i class="fa fa-check"></i><b>8.5.3</b> 推断-Max-Product 算法</a></li>
</ul></li>
</ul></li>
<li class="chapter" data-level="9" data-path="期望最大.html"><a href="期望最大.html"><i class="fa fa-check"></i><b>9</b> 期望最大</a>
<ul>
<li class="chapter" data-level="9.1" data-path="期望最大.html"><a href="期望最大.html#广义-em"><i class="fa fa-check"></i><b>9.1</b> 广义 EM</a></li>
<li class="chapter" data-level="9.2" data-path="期望最大.html"><a href="期望最大.html#em-的推广"><i class="fa fa-check"></i><b>9.2</b> EM 的推广</a></li>
</ul></li>
<li class="chapter" data-level="10" data-path="高斯混合模型.html"><a href="高斯混合模型.html"><i class="fa fa-check"></i><b>10</b> 高斯混合模型</a>
<ul>
<li class="chapter" data-level="10.1" data-path="高斯混合模型.html"><a href="高斯混合模型.html#极大似然估计"><i class="fa fa-check"></i><b>10.1</b> 极大似然估计</a></li>
<li class="chapter" data-level="10.2" data-path="高斯混合模型.html"><a href="高斯混合模型.html#em-求解-gmm"><i class="fa fa-check"></i><b>10.2</b> EM 求解 GMM</a></li>
</ul></li>
<li class="chapter" data-level="11" data-path="变分推断.html"><a href="变分推断.html"><i class="fa fa-check"></i><b>11</b> 变分推断</a>
<ul>
<li class="chapter" data-level="11.1" data-path="变分推断.html"><a href="变分推断.html#基于平均场假设的变分推断"><i class="fa fa-check"></i><b>11.1</b> 基于平均场假设的变分推断</a></li>
<li class="chapter" data-level="11.2" data-path="变分推断.html"><a href="变分推断.html#sgvi"><i class="fa fa-check"></i><b>11.2</b> SGVI</a></li>
</ul></li>
<li class="chapter" data-level="12" data-path="马尔可夫链蒙特卡洛.html"><a href="马尔可夫链蒙特卡洛.html"><i class="fa fa-check"></i><b>12</b> 马尔可夫链蒙特卡洛</a>
<ul>
<li class="chapter" data-level="12.1" data-path="马尔可夫链蒙特卡洛.html"><a href="马尔可夫链蒙特卡洛.html#蒙特卡洛方法"><i class="fa fa-check"></i><b>12.1</b> 蒙特卡洛方法</a></li>
<li class="chapter" data-level="12.2" data-path="马尔可夫链蒙特卡洛.html"><a href="马尔可夫链蒙特卡洛.html#mcmc"><i class="fa fa-check"></i><b>12.2</b> MCMC</a></li>
<li class="chapter" data-level="12.3" data-path="马尔可夫链蒙特卡洛.html"><a href="马尔可夫链蒙特卡洛.html#平稳分布"><i class="fa fa-check"></i><b>12.3</b> 平稳分布</a></li>
<li class="chapter" data-level="12.4" data-path="马尔可夫链蒙特卡洛.html"><a href="马尔可夫链蒙特卡洛.html#隐马尔可夫模型"><i class="fa fa-check"></i><b>12.4</b> 隐马尔可夫模型</a></li>
<li class="chapter" data-level="12.5" data-path="马尔可夫链蒙特卡洛.html"><a href="马尔可夫链蒙特卡洛.html#hmm"><i class="fa fa-check"></i><b>12.5</b> HMM</a>
<ul>
<li class="chapter" data-level="12.5.1" data-path="马尔可夫链蒙特卡洛.html"><a href="马尔可夫链蒙特卡洛.html#evaluation"><i class="fa fa-check"></i><b>12.5.1</b> Evaluation</a></li>
<li class="chapter" data-level="12.5.2" data-path="马尔可夫链蒙特卡洛.html"><a href="马尔可夫链蒙特卡洛.html#learning"><i class="fa fa-check"></i><b>12.5.2</b> Learning</a></li>
<li class="chapter" data-level="12.5.3" data-path="马尔可夫链蒙特卡洛.html"><a href="马尔可夫链蒙特卡洛.html#decoding"><i class="fa fa-check"></i><b>12.5.3</b> Decoding</a></li>
</ul></li>
<li class="chapter" data-level="12.6" data-path="马尔可夫链蒙特卡洛.html"><a href="马尔可夫链蒙特卡洛.html#小结-5"><i class="fa fa-check"></i><b>12.6</b> 小结</a></li>
</ul></li>
<li class="chapter" data-level="13" data-path="线性动态系统.html"><a href="线性动态系统.html"><i class="fa fa-check"></i><b>13</b> 线性动态系统</a></li>
<li class="chapter" data-level="14" data-path="粒子滤波.html"><a href="粒子滤波.html"><i class="fa fa-check"></i><b>14</b> 粒子滤波</a>
<ul>
<li class="chapter" data-level="14.1" data-path="粒子滤波.html"><a href="粒子滤波.html#sis"><i class="fa fa-check"></i><b>14.1</b> SIS</a></li>
</ul></li>
<li class="chapter" data-level="15" data-path="条件随机场.html"><a href="条件随机场.html"><i class="fa fa-check"></i><b>15</b> 条件随机场</a>
<ul>
<li class="chapter" data-level="15.1" data-path="条件随机场.html"><a href="条件随机场.html#crf-的-pdf"><i class="fa fa-check"></i><b>15.1</b> CRF 的 PDF</a></li>
<li class="chapter" data-level="15.2" data-path="条件随机场.html"><a href="条件随机场.html#边缘概率"><i class="fa fa-check"></i><b>15.2</b> 边缘概率</a></li>
<li class="chapter" data-level="15.3" data-path="条件随机场.html"><a href="条件随机场.html#参数估计"><i class="fa fa-check"></i><b>15.3</b> 参数估计</a></li>
<li class="chapter" data-level="15.4" data-path="条件随机场.html"><a href="条件随机场.html#译码"><i class="fa fa-check"></i><b>15.4</b> 译码</a></li>
</ul></li>
<li class="chapter" data-level="16" data-path="高斯网络.html"><a href="高斯网络.html"><i class="fa fa-check"></i><b>16</b> 高斯网络</a>
<ul>
<li class="chapter" data-level="16.1" data-path="高斯网络.html"><a href="高斯网络.html#高斯贝叶斯网络-gbn"><i class="fa fa-check"></i><b>16.1</b> 高斯贝叶斯网络 GBN</a></li>
<li class="chapter" data-level="16.2" data-path="高斯网络.html"><a href="高斯网络.html#高斯马尔可夫网络-gmn"><i class="fa fa-check"></i><b>16.2</b> 高斯马尔可夫网络 GMN</a></li>
</ul></li>
<li class="chapter" data-level="17" data-path="贝叶斯线性回归.html"><a href="贝叶斯线性回归.html"><i class="fa fa-check"></i><b>17</b> 贝叶斯线性回归</a>
<ul>
<li class="chapter" data-level="17.1" data-path="贝叶斯线性回归.html"><a href="贝叶斯线性回归.html#推断-1"><i class="fa fa-check"></i><b>17.1</b> 推断</a></li>
<li class="chapter" data-level="17.2" data-path="贝叶斯线性回归.html"><a href="贝叶斯线性回归.html#预测"><i class="fa fa-check"></i><b>17.2</b> 预测</a></li>
</ul></li>
<li class="chapter" data-level="18" data-path="高斯过程回归.html"><a href="高斯过程回归.html"><i class="fa fa-check"></i><b>18</b> 高斯过程回归</a>
<ul>
<li class="chapter" data-level="18.1" data-path="高斯过程回归.html"><a href="高斯过程回归.html#核贝叶斯线性回归"><i class="fa fa-check"></i><b>18.1</b> 核贝叶斯线性回归</a></li>
<li class="chapter" data-level="18.2" data-path="高斯过程回归.html"><a href="高斯过程回归.html#函数空间的观点"><i class="fa fa-check"></i><b>18.2</b> 函数空间的观点</a></li>
</ul></li>
<li class="chapter" data-level="19" data-path="受限玻尔兹曼机.html"><a href="受限玻尔兹曼机.html"><i class="fa fa-check"></i><b>19</b> 受限玻尔兹曼机</a>
<ul>
<li class="chapter" data-level="19.1" data-path="受限玻尔兹曼机.html"><a href="受限玻尔兹曼机.html#推断-2"><i class="fa fa-check"></i><b>19.1</b> 推断</a>
<ul>
<li class="chapter" data-level="19.1.1" data-path="受限玻尔兹曼机.html"><a href="受限玻尔兹曼机.html#phv"><i class="fa fa-check"></i><b>19.1.1</b> <span class="math inline">\(p(h|v)\)</span></a></li>
<li class="chapter" data-level="19.1.2" data-path="受限玻尔兹曼机.html"><a href="受限玻尔兹曼机.html#pv"><i class="fa fa-check"></i><b>19.1.2</b> <span class="math inline">\(p(v)\)</span></a></li>
</ul></li>
</ul></li>
<li class="chapter" data-level="20" data-path="谱聚类.html"><a href="谱聚类.html"><i class="fa fa-check"></i><b>20</b> 谱聚类</a></li>
<li class="chapter" data-level="21" data-path="前馈神经网络.html"><a href="前馈神经网络.html"><i class="fa fa-check"></i><b>21</b> 前馈神经网络</a>
<ul>
<li class="chapter" data-level="21.1" data-path="前馈神经网络.html"><a href="前馈神经网络.html#from-pla-to-dl"><i class="fa fa-check"></i><b>21.1</b> From PLA to DL</a></li>
<li class="chapter" data-level="21.2" data-path="前馈神经网络.html"><a href="前馈神经网络.html#非线性问题"><i class="fa fa-check"></i><b>21.2</b> 非线性问题</a></li>
</ul></li>
<li class="chapter" data-level="22" data-path="配分函数.html"><a href="配分函数.html"><i class="fa fa-check"></i><b>22</b> 配分函数</a>
<ul>
<li class="chapter" data-level="22.1" data-path="配分函数.html"><a href="配分函数.html#包含配分函数的-mle"><i class="fa fa-check"></i><b>22.1</b> 包含配分函数的 MLE</a></li>
<li class="chapter" data-level="22.2" data-path="配分函数.html"><a href="配分函数.html#对比散度-cd-learning"><i class="fa fa-check"></i><b>22.2</b> 对比散度-CD Learning</a></li>
<li class="chapter" data-level="22.3" data-path="配分函数.html"><a href="配分函数.html#rbm-的学习问题"><i class="fa fa-check"></i><b>22.3</b> RBM 的学习问题</a></li>
</ul></li>
<li class="chapter" data-level="23" data-path="近似推断.html"><a href="近似推断.html"><i class="fa fa-check"></i><b>23</b> 近似推断</a></li>
</ul>
</nav>
</div>
<div class="book-body">
<div class="body-inner">
<div class="book-header" role="navigation">
<h1>
<i class="fa fa-circle-o-notch fa-spin"></i><a href="./">机器学习白板系列</a>
</h1>
</div>
<div class="page-wrapper" tabindex="-1" role="main">
<div class="page-inner">
<section class="normal" id="section-">
<div id="高斯网络" class="section level1 hasAnchor" number="16">
<h1><span class="header-section-number">16</span> 高斯网络<a href="高斯网络.html#高斯网络" class="anchor-section" aria-label="Anchor link to header"></a></h1>
<p>高斯图模型（高斯网络）是一种随机变量为连续的有向或者无向图。有向图版本的高斯图是高斯贝叶斯网络，无向版本的叫高斯马尔可夫网络。</p>
<p>高斯网络的每一个节点都是高斯分布：<span class="math inline">\(\mathcal{N}(\mu_i,\Sigma_i)\)</span>，于是所有节点的联合分布就是一个高斯分布，均值为 <span class="math inline">\(\mu\)</span>，方差为 <span class="math inline">\(\Sigma\)</span>。</p>
<p>对于边缘概率，我们有下面三个结论：</p>
<ol style="list-style-type: decimal">
<li><p>对于方差矩阵，可以得到独立性条件：<span class="math inline">\(x_i\perp x_j\Leftrightarrow\sigma_{ij}=0\)</span>，这个叫做全局独立性。</p></li>
<li><p>我们看方差矩阵的逆（精度矩阵或信息矩阵）：<span class="math inline">\(\Lambda=\Sigma^{-1}=(\lambda_{ij})_{pp}\)</span>，有定理：</p>
<blockquote>
<p><span class="math inline">\(x_i\perp x_j|(X-\{x_i,x_j\})\Leftrightarrow\lambda_{ij}=0\)</span></p>
</blockquote>
<p>因此，我们使用精度矩阵来表示条件独立性。</p></li>
<li><p>对于任意一个无向图中的节点 <span class="math inline">\(x_i\)</span>，<span class="math inline">\(x_i|(X-x_i)\sim \mathcal{N}(\sum\limits_{j\ne i}\frac{\lambda_{ij}}{\lambda_{ii}}x_j,\lambda_{ii}^{-1})\)</span></p>
<p>也就是其他所有分量的线性组合，即所有与它有链接的分量的线性组合。</p></li>
</ol>
<div id="高斯贝叶斯网络-gbn" class="section level2 hasAnchor" number="16.1">
<h2><span class="header-section-number">16.1</span> 高斯贝叶斯网络 GBN<a href="高斯网络.html#高斯贝叶斯网络-gbn" class="anchor-section" aria-label="Anchor link to header"></a></h2>
<p>高斯贝叶斯网络可以看成是 LDS 的一个推广，LDS 的假设是相邻时刻的变量之间的依赖关系，因此是一个局域模型，而高斯贝叶斯网络，每一个节点的父亲节点不一定只有一个，因此可以看成是一个全局的模型。根据有向图的因子分解：
<span class="math display">\[
p(x)=\prod\limits_{i=1}^pp(x_i|x_{Parents(i)})
\]</span>
对里面每一项，假设每一个特征是一维的，可以写成线性组合：
<span class="math display">\[
p(x_i|x_{Parents(i)})=\mathcal{N}(x_i|\mu_i+W_i^Tx_{Parents(i)},\sigma^2_i)
\]</span>
将随机变量写成：
<span class="math display">\[
x_i=\mu_i+\sum\limits_{j\in x_{Parents(i)}}w_{ij}(x_j-\mu_j)+\sigma_i\varepsilon_i,\varepsilon_i\sim \mathcal{N}(0,1)
\]</span>
写成矩阵形式，并且对 <span class="math inline">\(w\)</span> 进行扩展：
<span class="math display">\[
x-\mu=W(x-\mu)+S\varepsilon
\]</span>
其中，<span class="math inline">\(S=diag(\sigma_i)\)</span>。所以有：<span class="math inline">\(x-\mu=(\mathbb{I}-W)^{-1}S\varepsilon\)</span></p>
<p>由于：
<span class="math display">\[
Cov(x)=Cov(x-\mu)
\]</span>
可以得到协方差矩阵。</p>
</div>
<div id="高斯马尔可夫网络-gmn" class="section level2 hasAnchor" number="16.2">
<h2><span class="header-section-number">16.2</span> 高斯马尔可夫网络 GMN<a href="高斯网络.html#高斯马尔可夫网络-gmn" class="anchor-section" aria-label="Anchor link to header"></a></h2>
<p>对于无向图版本的高斯网络，可以写成：
<span class="math display">\[
p(x)=\frac{1}{Z}\prod\limits_{i=1}^p\phi_i(x_i)\prod\limits_{i,j\in X}\phi_{i,j}(x_i,x_j)
\]</span>
为了将高斯分布和这个式子结合，我们写出高斯分布和变量相关的部分：
<span class="math display">\[
\begin{align}p(x)&\propto \exp(-\frac{1}{2}(x-\mu)^T\Sigma^{-1}(x-\mu))\nonumber\\
&=\exp(-\frac{1}{2}(x^T\Lambda x-2\mu^T\Lambda x+\mu^T\Lambda\mu))\nonumber\\
&=\exp(-\frac{1}{2}x^T\Lambda x+(\Lambda\mu)^Tx)
\end{align}
\]</span>
可以看到，这个式子与无向图分解中的两个部分对应，我们记 <span class="math inline">\(h=\Lambda\mu\)</span>为 Potential Vector。其中和 <span class="math inline">\(x_i\)</span> 相关的为：<span class="math inline">\(x_i:-\frac{1}{2}\lambda_{ii}x_i^2+h_ix_i\)</span>，与 <span class="math inline">\(x_i,x_j\)</span> 相关的是：<span class="math inline">\(x_i,x_j:-\lambda_{ij}x_ix_j\)</span>，这里利用了精度矩阵为对称矩阵的性质。我们看到，这里也可以看出，<span class="math inline">\(x_i,x_j\)</span> 构成的一个势函数，只和 <span class="math inline">\(\lambda_{ij}\)</span> 有关，于是 $x_ix_j|(X-{x_i,x_j})_{ij}=0 $。</p>
</div>
</div>
</section>
</div>
</div>
</div>
<a href="条件随机场.html" class="navigation navigation-prev " aria-label="Previous page"><i class="fa fa-angle-left"></i></a>
<a href="贝叶斯线性回归.html" class="navigation navigation-next " aria-label="Next page"><i class="fa fa-angle-right"></i></a>
</div>
</div>
<script src="libs/gitbook-2.6.7/js/app.min.js"></script>
<script src="libs/gitbook-2.6.7/js/clipboard.min.js"></script>
<script src="libs/gitbook-2.6.7/js/plugin-search.js"></script>
<script src="libs/gitbook-2.6.7/js/plugin-sharing.js"></script>
<script src="libs/gitbook-2.6.7/js/plugin-fontsettings.js"></script>
<script src="libs/gitbook-2.6.7/js/plugin-bookdown.js"></script>
<script src="libs/gitbook-2.6.7/js/jquery.highlight.js"></script>
<script src="libs/gitbook-2.6.7/js/plugin-clipboard.js"></script>
<script>
gitbook.require(["gitbook"], function(gitbook) {
gitbook.start({
"sharing": {
"github": false,
"facebook": true,
"twitter": true,
"linkedin": false,
"weibo": false,
"instapaper": false,
"vk": false,
"whatsapp": false,
"all": ["facebook", "twitter", "linkedin", "weibo", "instapaper"]
},
"fontsettings": {
"theme": "white",
"family": "sans",
"size": 2
},
"edit": {
"link": null,
"text": null
},
"history": {
"link": null,
"text": null
},
"view": {
"link": null,
"text": null
},
"download": ["CBook.pdf"],
"search": {
"engine": "fuse",
"options": null
},
"toc": {
"collapse": "subsection"
}
});
});
</script>
<!-- dynamically load mathjax for compatibility with self-contained -->
<script>
(function () {
var script = document.createElement("script");
script.type = "text/javascript";
var src = "true";
if (src === "" || src === "true") src = "https://mathjax.rstudio.com/latest/MathJax.js?config=TeX-MML-AM_CHTML";
if (location.protocol !== "file:")
if (/^https?:/.test(src))
src = src.replace(/^https?:/, '');
script.src = src;
document.getElementsByTagName("head")[0].appendChild(script);
})();
</script>
</body>
</html>