forked from directorscut82/osaka_prml_reading
-
Notifications
You must be signed in to change notification settings - Fork 0
/
ex_02_41-50.html
664 lines (625 loc) · 24.8 KB
/
ex_02_41-50.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
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
<?xml version="1.0" encoding="utf-8"?>
<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en">
<head>
<title>PRML 第2章 演習 2.41-2.50</title>
<meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
<meta name="generator" content="Org-mode" />
<style type="text/css">
<!--/*--><![CDATA[/*><!--*/
.title { text-align: center; }
.todo { font-family: monospace; color: red; }
.done { color: green; }
.tag { background-color: #eee; font-family: monospace;
padding: 2px; font-size: 80%; font-weight: normal; }
.timestamp { color: #bebebe; }
.timestamp-kwd { color: #5f9ea0; }
.right { margin-left: auto; margin-right: 0px; text-align: right; }
.left { margin-left: 0px; margin-right: auto; text-align: left; }
.center { margin-left: auto; margin-right: auto; text-align: center; }
.underline { text-decoration: underline; }
#postamble p, #preamble p { font-size: 90%; margin: .2em; }
p.verse { margin-left: 3%; }
pre {
border: 1px solid #ccc;
box-shadow: 3px 3px 3px #eee;
padding: 8pt;
font-family: monospace;
overflow: auto;
margin: 1.2em;
}
pre.src {
position: relative;
overflow: visible;
padding-top: 1.2em;
}
pre.src:before {
display: none;
position: absolute;
background-color: white;
top: -10px;
right: 10px;
padding: 3px;
border: 1px solid black;
}
pre.src:hover:before { display: inline;}
pre.src-sh:before { content: 'sh'; }
pre.src-bash:before { content: 'sh'; }
pre.src-emacs-lisp:before { content: 'Emacs Lisp'; }
pre.src-R:before { content: 'R'; }
pre.src-perl:before { content: 'Perl'; }
pre.src-java:before { content: 'Java'; }
pre.src-sql:before { content: 'SQL'; }
table { border-collapse:collapse; }
caption.t-above { caption-side: top; }
caption.t-bottom { caption-side: bottom; }
td, th { vertical-align:top; }
th.right { text-align: center; }
th.left { text-align: center; }
th.center { text-align: center; }
td.right { text-align: right; }
td.left { text-align: left; }
td.center { text-align: center; }
dt { font-weight: bold; }
.footpara:nth-child(2) { display: inline; }
.footpara { display: block; }
.footdef { margin-bottom: 1em; }
.figure { padding: 1em; }
.figure p { text-align: center; }
.inlinetask {
padding: 10px;
border: 2px solid gray;
margin: 10px;
background: #ffffcc;
}
#org-div-home-and-up
{ text-align: right; font-size: 70%; white-space: nowrap; }
textarea { overflow-x: auto; }
.linenr { font-size: smaller }
.code-highlighted { background-color: #ffff00; }
.org-info-js_info-navigation { border-style: none; }
#org-info-js_console-label
{ font-size: 10px; font-weight: bold; white-space: nowrap; }
.org-info-js_search-highlight
{ background-color: #ffff00; color: #000000; font-weight: bold; }
/*]]>*/-->
</style>
<script type="text/javascript">
/*
@licstart The following is the entire license notice for the
JavaScript code in this tag.
Copyright (C) 2012-2013 Free Software Foundation, Inc.
The JavaScript code in this tag is free software: you can
redistribute it and/or modify it under the terms of the GNU
General Public License (GNU GPL) as published by the Free Software
Foundation, either version 3 of the License, or (at your option)
any later version. The code is distributed WITHOUT ANY WARRANTY;
without even the implied warranty of MERCHANTABILITY or FITNESS
FOR A PARTICULAR PURPOSE. See the GNU GPL for more details.
As additional permission under GNU GPL version 3 section 7, you
may distribute non-source (e.g., minimized or compacted) forms of
that code without the copy of the GNU GPL normally required by
section 4, provided you include this license notice and a URL
through which recipients can access the Corresponding Source.
@licend The above is the entire license notice
for the JavaScript code in this tag.
*/
<!--/*--><![CDATA[/*><!--*/
function CodeHighlightOn(elem, id)
{
var target = document.getElementById(id);
if(null != target) {
elem.cacheClassElem = elem.className;
elem.cacheClassTarget = target.className;
target.className = "code-highlighted";
elem.className = "code-highlighted";
}
}
function CodeHighlightOff(elem, id)
{
var target = document.getElementById(id);
if(elem.cacheClassElem)
elem.className = elem.cacheClassElem;
if(elem.cacheClassTarget)
target.className = elem.cacheClassTarget;
}
/*]]>*///-->
</script>
<script type="text/javascript" src="http://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script>
<script type="text/javascript">
<!--/*--><![CDATA[/*><!--*/
MathJax.Hub.Config({
// Only one of the two following lines, depending on user settings
// First allows browser-native MathML display, second forces HTML/CSS
// config: ["MMLorHTML.js"], jax: ["input/TeX"],
jax: ["input/TeX", "output/HTML-CSS"],
extensions: ["tex2jax.js","TeX/AMSmath.js","TeX/AMSsymbols.js",
"TeX/noUndefined.js"],
tex2jax: {
inlineMath: [ ["\\(","\\)"] ],
displayMath: [ ['$$','$$'], ["\\[","\\]"], ["\\begin{displaymath}","\\end{displaymath}"] ],
skipTags: ["script","noscript","style","textarea","pre","code"],
ignoreClass: "tex2jax_ignore",
processEscapes: false,
processEnvironments: true,
preview: "TeX"
},
showProcessingMessages: true,
displayAlign: "left",
displayIndent: "2em",
"HTML-CSS": {
scale: 100,
availableFonts: ["STIX","TeX"],
preferredFont: "TeX",
webFont: "TeX",
imageFont: "TeX",
showMathMenu: true,
},
MMLorHTML: {
prefer: {
MSIE: "MML",
Firefox: "MML",
Opera: "HTML",
other: "HTML"
}
}
});
/*]]>*///-->
</script>
</head>
<body>
<div id="content">
<h1 class="title">PRML 第2章 演習 2.41-2.50</h1>
<div id="table-of-contents">
<h2>Table of Contents</h2>
<div id="text-table-of-contents">
<ul>
<li><a href="#sec-1">PRML 第2章 演習 2.41-2.50</a>
<ul>
<li><a href="#sec-1-1"><span class="done DONE">DONE</span> 2.41 ガンマ分布が正規化されていることの証明</a></li>
<li><a href="#sec-1-2"><span class="done DONE">DONE</span> 2.42 ガンマ分布の平均、分散、モード</a></li>
<li><a href="#sec-1-3"><span class="todo TODO">TODO</span> 2.43 \(q\)次形式に一般化したガウス関数</a></li>
<li><a href="#sec-1-4"><span class="todo TODO">TODO</span> 2.44 1変数ガウス分布のベイズ推定で事後分布がガウス-ガンマ分布になることの証明</a></li>
<li><a href="#sec-1-5"><span class="done DONE">DONE</span> 2.45 ウィシャート分布が多変量ガウス分布の精度行列の共役事前分布であることの証明</a></li>
<li><a href="#sec-1-6"><span class="done DONE">DONE</span> 2.46 [www] ガウス-ガンマ分布の周辺分布がスチューデントのt分布になることの証明</a></li>
<li><a href="#sec-1-7"><span class="done DONE">DONE</span> 2.47 [www] t分布が\(ν→∞\)の極限でガウス分布になることの証明</a></li>
<li><a href="#sec-1-8"><span class="todo TODO">TODO</span> 2.48 スチューデントのt分布の多変量形式の導出とそれが正規化されていることの証明</a></li>
<li><a href="#sec-1-9"><span class="todo TODO">TODO</span> 2.49 多変量スチューデントt分布の平均、共分散、モード</a></li>
<li><a href="#sec-1-10"><span class="done DONE">DONE</span> 2.50 多変量スチューデントt分布が\(ν→∞\)の極限で多変量ガウス分布になることの証明</a></li>
</ul>
</li>
</ul>
</div>
</div>
\begin{align*}
\newcommand{\l}{\left}
\newcommand{\r}{\right}
\newcommand{\f}{\frac}
\newcommand{\p}[2]{\frac{\partial #1}{\partial #2}}
\newcommand{\A}{\mathbf{A}}
\newcommand{\B}{\mathbf{B}}
\newcommand{\C}{\mathbf{C}}
\newcommand{\D}{\mathbf{D}}
\newcommand{\G}{\mathbf{G}}
\newcommand{\I}{\mathbf{I}}
\newcommand{\L}{\mathbf{L}}
\newcommand{\M}{\mathbf{M}}
\newcommand{\R}{\mathbf{R}}
\newcommand{\S}{\mathbf{S}}
\newcommand{\TT}{\mathbf{T}}
\newcommand{\W}{\mathbf{W}}
\newcommand{\X}{\mathbf{X}}
\newcommand{\Y}{\mathbf{Y}}
\newcommand{\b}{\mathbf{b}}
\newcommand{\e}{\mathbf{e}}
\newcommand{\m}{\mathbf{m}}
\newcommand{\t}{\mathbf{t}}
\newcommand{\u}{\mathbf{u}}
\newcommand{\v}{\mathbf{v}}
\newcommand{\w}{\mathbf{w}}
\newcommand{\x}{\mathbf{x}}
\newcommand{\y}{\mathbf{y}}
\newcommand{\tt}{\mathbf{\mathsf{t}}}
\newcommand{\xx}{\mathbf{\mathsf{x}}}
\newcommand{\yy}{\mathbf{\mathsf{y}}}
\newcommand{\Λ}{\mathbf{Λ}}
\newcommand{\α}{\mathbf{α}}
\newcommand{\ε}{\mathbf{ε}}
\newcommand{\μ}{\mathbf{μ}}
\newcommand{\η}{\mathbf{η}}
\newcommand{\Φ}{\mathbf{Φ}}
\newcommand{\Σ}{\mathbf{Σ}}
\newcommand{\bPhi}{{\rm \bf \Phi}}
\newcommand{\bphi}{\boldsymbol \phi}
\newcommand{\bvphi}{\boldsymbol \varphi}
\newcommand{\E}{{\mathbb{E}}}
\newcommand{\D}{{\cal D}}
\newcommand{\N}{{\cal N}}
\newcommand{\d}{\mathrm{d}}
\newcommand{\T}{\mathrm{T}}
\newcommand{\Tr}{\mathrm{Tr}}
\newcommand{\var}{\mathrm{var}}
\newcommand{\cov}{\mathrm{cov}}
\newcommand{\mode}{\mathrm{mode}}
\newcommand{\Bern}{\mathrm{Bern}}
\newcommand{\Beta}{\mathrm{Beta}}
\newcommand{\Bin}{\mathrm{Bin}}
\newcommand{\Dir}{\mathrm{Dir}}
\newcommand{\Gam}{\mathrm{Gam}}
\newcommand{\St}{\mathrm{St}}
\newcommand{\ML}{\mathrm{ML}}
\end{align*}
<div id="outline-container-sec-1" class="outline-2">
<h2 id="sec-1">PRML 第2章 演習 2.41-2.50</h2>
<div class="outline-text-2" id="text-1">
</div><div id="outline-container-sec-1-1" class="outline-3">
<h3 id="sec-1-1"><span class="done DONE">DONE</span> 2.41 ガンマ分布が正規化されていることの証明</h3>
<div class="outline-text-3" id="text-1-1">
<p>
ガンマ分布<br />
</p>
\begin{align*}
\Gam(λ|a,b) = & \f{1}{Γ(a)} b^a λ^{a-1} \exp(-bλ) \\
\end{align*}
<p>
ガンマ関数の定義<br />
</p>
\begin{align*}
Γ(x) = & \int_0^\infty u^{x-1} e^{-u} du \\
\end{align*}
<p>
ガンマ分布を\(λ\)で積分する。<br />
</p>
\begin{align*}
& \int_0^\infty \Gam(λ|a,b) dλ \\
= & \f{1}{Γ(a)} b^a \int_0^infty λ^{a-1} \exp(-bλ) dλ \\
\end{align*}
<p>
\(u = bλ\)と置く。<br />
</p>
\begin{align*}
= & \f{1}{Γ(a)} b^a \int_0^\infty \f{u^{a-1}}{b^{a-1}} \exp(-u) b du \\
= & \f{1}{Γ(a)} b^a \f{1}{b^a} \int_0^\infty u^{a-1} \exp(-u) du \\
= & \f{1}{Γ(a)} b^a \f{1}{b^a} Γ(a) \\
= & 1 \\
\end{align*}
</div>
</div>
<div id="outline-container-sec-1-2" class="outline-3">
<h3 id="sec-1-2"><span class="done DONE">DONE</span> 2.42 ガンマ分布の平均、分散、モード</h3>
<div class="outline-text-3" id="text-1-2">
</div><div id="outline-container-sec-1-2-1" class="outline-4">
<h4 id="sec-1-2-1"><span class="done DONE">DONE</span> 平均</h4>
<div class="outline-text-4" id="text-1-2-1">
\begin{align*}
\E[λ] = & \int_0^\infty λ \Gam(λ|a,b) dλ \\
= & \f{1}{Γ(a)} b^a \int_0^\infty λ λ^{a-1} \exp(-bλ) dλ \\
= & \f{1}{Γ(a)} b^a \int_0^\infty λ^a \exp(-bλ) dλ \\
= & \f{1}{Γ(a)} b^a \int_0^\infty \f{u^a}{b^a} \exp(-u) \f{1}{b} du \\
= & \f{1}{Γ(a)} b^a \f{1}{b^{a+1}} \int_0^\infty u^a \exp(-u) du \\
= & \f{1}{Γ(a)} \f{1}{b} Γ(a+1) \\
= & \f{1}{Γ(a)} \f{1}{b} a Γ(a) \\
= & \f{a}{b} \\
\end{align*}
</div>
</div>
<div id="outline-container-sec-1-2-2" class="outline-4">
<h4 id="sec-1-2-2"><span class="done DONE">DONE</span> 分散</h4>
<div class="outline-text-4" id="text-1-2-2">
\begin{align*}
var[λ] = & \E[λ^2] - \E[λ]^2 \\
\end{align*}
\begin{align*}
\E[λ^2] = & \int_0^\infty λ^2 \Gam(λ|a,b) dλ \\
= & \f{1}{Γ(a)} b^a \int_0^\infty λ^2 λ^{a-1} \exp(-bλ) dλ \\
= & \f{1}{Γ(a)} b^a \int_0^\infty λ^{a+1} \exp(-bλ) dλ \\
= & \f{1}{Γ(a)} b^a \int_0^\infty \f{u^{a+1}}{b^{a+1}} \exp(-u) \f{1}{b} du \\
= & \f{1}{Γ(a)} b^a \f{1}{b^{a+2}} \int_0^\infty u^{a+1} \exp(-u) du \\
= & \f{1}{Γ(a)} \f{1}{b^2} Γ(a+2) \\
= & \f{1}{Γ(a)} \f{1}{b^2} (a+1) a Γ(a) \\
= & \f{a(a+1)}{b^2} \\
\end{align*}
\begin{align*}
var[λ] = & \E[λ^2] - \E[λ]^2 \\
= & \f{a(a+1)}{b^2} - \f{a^2}{b^2} \\
= & \f{a}{b^2} \\
\end{align*}
</div>
</div>
<div id="outline-container-sec-1-2-3" class="outline-4">
<h4 id="sec-1-2-3"><span class="done DONE">DONE</span> モード</h4>
<div class="outline-text-4" id="text-1-2-3">
\begin{align*}
\p{}{λ} \Gam(λ|a,b) = & 0 \\
\f{1}{Γ(a)} b^a \p{}{λ} [ λ^{a-1} \exp(-bλ) ] = & 0 \\
\p{}{λ} [ λ^{a-1} \exp(-bλ) ] = & 0 \\
(a-1) λ^{a-2} \exp(-bλ) - b λ^{a-1} \exp(-bλ) = & 0 \\
(a-1) - b λ = & 0 \\
λ = & \f{a-1}{b} \\
\end{align*}
</div>
</div>
</div>
<div id="outline-container-sec-1-3" class="outline-3">
<h3 id="sec-1-3"><span class="todo TODO">TODO</span> 2.43 \(q\)次形式に一般化したガウス関数</h3>
<div class="outline-text-3" id="text-1-3">
\begin{align*}
p(x|σ^2,q) = \f{q}{2 (2σ^2)^{1/q} Γ(1/q)} \exp\l( -\f{|x|^q}{2σ^2} \r) \\
\end{align*}
</div>
<div id="outline-container-sec-1-3-1" class="outline-4">
<h4 id="sec-1-3-1"><span class="done DONE">DONE</span> 正規化されていることの証明</h4>
<div class="outline-text-4" id="text-1-3-1">
\begin{align*}
& \int_{-\infty}^\infty p(x|σ^2,q) dx \\
= & \f{q}{2 (2σ^2)^{1/q} Γ(1/q)} \int_{-\infty}^\infty \exp\l( -\f{|x|^q}{2σ^2} \r) dx \\
= & \f{q}{(2σ^2)^{1/q} Γ(1/q)} \int_0^\infty \exp\l( -\f{x^q}{2σ^2} \r) dx \\
\end{align*}
<p>
ここで以下の変数変換を行う。<br />
</p>
\begin{align*}
y = & \f{x^q}{2σ^2} \\
x = & {(2σ^2 y)}^{1/q} \\
\f{dx}{dy} = & \f{{(2σ^2)}^{1/q} y^{1/q-1}}{q} \\
\end{align*}
<p>
すると<br />
</p>
\begin{align*}
= & \f{q}{(2σ^2)^{1/q} Γ(1/q)} \f{{(2σ^2)}^{1/q}}{q} \int_0^\infty y^{1/q-1} \exp(-y) dy \\
= & \f{q}{(2σ^2)^{1/q} Γ(1/q)} \f{{(2σ^2)}^{1/q}}{q} Γ(1/q) \\
= & 1 \\
\end{align*}
</div>
</div>
<div id="outline-container-sec-1-3-2" class="outline-4">
<h4 id="sec-1-3-2"><span class="done DONE">DONE</span> \(q=2\)のときガウス分布になることの証明</h4>
<div class="outline-text-4" id="text-1-3-2">
\begin{align*}
p(x|σ^2,q=2) = & \f{2}{2 (2σ^2)^{1/2} Γ(1/2)} \exp\l( -\f{|x|^2}{2σ^2} \r) \\
= & \f{1}{(2πσ^2)^{1/2}} \exp\l( -\f{x^2}{2σ^2} \r) \\
\end{align*}
<p>
\(Γ(1/2)\)の計算は以下の通り。<br />
</p>
\begin{align*}
Γ(1/2) = & \int_0^\infty u^{1/2-1} \exp(-u) du \\
= & \int_0^\infty u^{-1/2} \exp(-u) du \\
\end{align*}
<p>
\(v^2=u\)と置くと<br />
</p>
\begin{align*}
= & \int_0^\infty v^{-1} \exp(-v^2) 2vdv \\
= & 2 \int_0^\infty \exp(-v^2) dv \\
= & \int_{-\infty}^\infty \exp(-v^2) dv \\
= & \sqrt{π}
\end{align*}
<p>
\(\int_{-\infty}^\infty \exp(-v^2) dv\)はガウス積分と呼ばれる。<br />
計算の仕方は演習1.7を参照。<br />
</p>
</div>
</div>
<div id="outline-container-sec-1-3-3" class="outline-4">
<h4 id="sec-1-3-3"><span class="todo TODO">TODO</span> 対数尤度関数</h4>
<div class="outline-text-4" id="text-1-3-3">
\begin{align*}
t = & y(\x,\w) + ε \\
ε = & t - y(\x,\w) = p(x|σ^2,q) \\
\end{align*}
<p>
尤度関数<br />
</p>
\begin{align*}
p(\tt|\X,\w,σ^2) = \prod_{n=1}^N
\end{align*}
</div>
</div>
</div>
<div id="outline-container-sec-1-4" class="outline-3">
<h3 id="sec-1-4"><span class="todo TODO">TODO</span> 2.44 1変数ガウス分布のベイズ推定で事後分布がガウス-ガンマ分布になることの証明</h3>
<div class="outline-text-3" id="text-1-4">
<p>
平均\(μ\)と分散\(τ^{-1}\)が与えられたときに観測値集合\(\xx=\{x_1,...,x_N\}\)が生じる確率である<br />
尤度関数は以下のように書ける。<br />
</p>
\begin{align*}
p(\xx|μ,τ^{-1}) = \prod_{n=1}^N \N(x_n|μ,τ^{-1}) \\
\end{align*}
<p>
平均\(μ\)と分散\(τ^{-1}\)の事前分布にガウス-ガンマ分布を選ぶ。<br />
</p>
\begin{align*}
p(μ,τ^{-1}) = \N(μ|μ_0,τ^{-1}) \Gam(τ|a,b) \\
\end{align*}
<p>
事後分布は尤度関数と事前分布の積に比例する。<br />
</p>
\begin{align*}
p(μ,τ^{-1}|\xx) \propto & p(\xx|μ,τ^{-1}) p(μ,τ^{-1}) \\
= & \prod_{n=1}^N \N(x_n|μ,τ^{-1}) \N(μ|μ_0,τ^{-1}) \Gam(τ|a,b) \\
\end{align*}
</div>
</div>
<div id="outline-container-sec-1-5" class="outline-3">
<h3 id="sec-1-5"><span class="done DONE">DONE</span> 2.45 ウィシャート分布が多変量ガウス分布の精度行列の共役事前分布であることの証明</h3>
<div class="outline-text-3" id="text-1-5">
<p>
本文p.97で1変数ガウス分布の場合について行なった議論を<br />
多変量ガウス分布の場合に一般化する。<br />
</p>
<p>
精度行列\(\Λ\)が与えられたときに観測値集合\(\X=\{x_1,...,x_N\}\)が生じる確率である<br />
尤度関数は次のように書ける。<br />
</p>
\begin{align*}
p(\X|\Λ) \propto & \prod_{n=1}^N \N(\x_n|\μ,\Λ^{-1}) \\
= & \prod_{n=1}^N \f{1}{(2π)^{D/2}} \f{1}{|\Λ^{-1}|^{1/2}}
\exp\l\{ -\f{1}{2} (\x_n - \μ)^T \Λ (\x_n - \μ) \r\} \\
= & \prod_{n=1}^N \f{1}{(2π)^{D/2}} |\Λ|^{1/2}
\exp\l\{ -\f{1}{2} (\x_n - \μ)^T \Λ (\x_n - \μ) \r\}
& |\A^{-1}| = |\A|^{-1} \\
= & \f{1}{(2π)^{ND/2}} |\Λ|^{N/2}
\exp\l\{ -\f{1}{2} \sum_{n=1}^N (\x_n - \μ)^T \Λ (\x_n - \μ) \r\} \\
\end{align*}
<p>
精度行列\(\Λ\)の事前分布としてウィシャート分布を選ぶ。<br />
</p>
\begin{align*}
\mathcal{W}(\Λ|\W,ν) = & B|\Λ|^{(ν-D-1)/2} \exp\l( -\f{1}{2} \Tr(\W^{-1} \Λ) \r) \\
B(\W,ν) = & |\W|^{-ν/2} \l( 2^{νD/2}π^{D(D-1)/4} \prod_{i=1}^D Γ\l(\f{ν+1-i}{2}\r) \r)^{-1} \\
\end{align*}
<p>
精度行列\(\Λ\)の事後分布は尤度関数と事前分布の積に比例する。<br />
</p>
\begin{align*}
p(\Λ|\X)
\propto & p(\X|\Λ) \mathcal{W}(\Λ|\W,ν) \\
\propto & |\Λ|^{N/2} \exp\l\{ -\f{1}{2} \sum_{n=1}^N (\x_n - \μ)^T \Λ (\x_n - \μ) \r\}
|\Λ|^{(ν-D-1)/2} \exp\l( -\f{1}{2} \Tr(\W^{-1} \Λ) \r) \\
= & |\Λ|^{(ν+N-D-1)/2} \exp\l[ -\f{1}{2} \l\{ \sum_{n=1}^N (\x_n - \μ)^T \Λ (\x_n - \μ)
+ \Tr(\W^{-1} \Λ) \r\} \r] \\
= & |\Λ|^{(ν+N-D-1)/2} \exp\l[ -\f{1}{2}
\l\{ \Tr\l( \sum_{n=1}^N (\x_n - \μ)(\x_n - \μ)^T \Λ \r)
+ \Tr(\W^{-1} \Λ) \r\} \r] \\
= & |\Λ|^{(ν+N-D-1)/2} \exp\l[ -\f{1}{2}
\Tr\l\{ \l( \sum_{n=1}^N (\x_n - \μ)(\x_n - \μ)^T + \W^{-1} \r) \Λ \r\} \r] \\
= & |\Λ|^{(ν_N-D-1)/2} \exp\l[ -\f{1}{2} \Tr(\W_N^{-1} \Λ) \r] \\
\propto & \mathcal{W}(\Λ|\W_N,ν_N) \\
\end{align*}
<p>
事後分布が再びウィシャート分布の形になったので、<br />
ウィシャート分布が多変量ガウス分布の精度行列の共役事前分布であることが言える。<br />
ここで<br />
</p>
\begin{align*}
ν_N = & ν + N \\
\W_N^{-1} = & \sum_{n=1}^N (\x_n - \μ)(\x_n - \μ)^T + \W^{-1} \\
\end{align*}
<p>
ここで以下の性質を用いた。<br />
</p>
\begin{align*}
\Tr(\X \A) = & \sum_i (\X \A)_{ii} = \sum_i \sum_k X_{ik} A_{ki} \\
\x^T \A \x = & \sum_i \sum_k x_i A_{ik} x_k \\
\end{align*}
<p>
よって、\(\X = \x \x^T\)すなわち\(X_{ik} = x_i x_k\)と置けば、<br />
\(\Tr(\X \A)=\x^T \A \x\)となる。<br />
</p>
</div>
</div>
<div id="outline-container-sec-1-6" class="outline-3">
<h3 id="sec-1-6"><span class="done DONE">DONE</span> 2.46 [www] ガウス-ガンマ分布の周辺分布がスチューデントのt分布になることの証明</h3>
<div class="outline-text-3" id="text-1-6">
\begin{align*}
p(x|μ,a,b) = & \int_0^∞ \N(x|μ,τ^{-1}) \Gam(τ|a,b) dτ \\
= & \int_0^∞ \f{b^a e^{-bτ} τ^{a-1}}{Γ(a)} \l(\f{τ}{2π}\r)^{1/2}
\exp\l\{ -\f{τ}{2} (x - μ)^2 \r\} dτ \\
= & \f{b^a}{Γ(a)} \l(\f{1}{2π}\r)^{1/2}
\int_0^∞ τ^{a-1/2} \exp\l\{ -τ \l( b + \f{(x - μ)^2}{2} \r) \r\} dτ \\
\end{align*}
\begin{align*}
z = & τ \l[ b + \f{(x - μ)^2}{2} \r] \\
τ = & \f{z}{b + (x - μ)^2/2} \\
\f{dτ}{dz} = & \f{1}{b + (x - μ)^2/2} \\
\end{align*}
\begin{align*}
p(x|μ,a,b) = & \f{b^a}{Γ(a)} \l(\f{1}{2π}\r)^{1/2}
\int_0^∞ \l[ \f{z}{b + (x - μ)^2/2} \r]^{a-1/2}
\exp(z) \f{1}{b + (x - μ)^2/2} dz \\
= & \f{b^a}{Γ(a)} \l(\f{1}{2π}\r)^{1/2} \l[ b + \f{(x - μ)^2}{2} \r]^{-a-1/2}
\int_0^∞ z^{a-1/2} \exp(z) dz \\
= & \f{b^a}{Γ(a)} \l(\f{1}{2π}\r)^{1/2}
\l[ b + \f{(x - μ)^2}{2} \r]^{-a-1/2} Γ(a+1/2) \\
\end{align*}
\begin{align*}
ν = & 2a & a = & ν/2 \\
λ = & a/b & b = & a/λ = ν/2λ \\
\end{align*}
\begin{align*}
p(x|μ,a,b) = & \f{(ν/2λ)^{ν/2}}{Γ(ν/2)} \l(\f{1}{2π}\r)^{1/2}
\l[ ν/2λ + \f{(x - μ)^2}{2} \r]^{-ν/2-1/2} Γ(ν/2+1/2) \\
= & \f{Γ(ν/2+1/2)}{Γ(ν/2)} (2λ/ν)^{(-ν/2-1/2)+1/2} \l(\f{1}{2π}\r)^{1/2}
\l[ ν/2λ + \f{(x - μ)^2}{2} \r]^{-ν/2-1/2} \\
= & \f{Γ(ν/2+1/2)}{Γ(ν/2)} \l(\f{λ}{πν}\r)^{1/2}
\l[ 1 + \f{λ(x - μ)^2}{ν} \r]^{-ν/2-1/2} \\
\end{align*}
</div>
</div>
<div id="outline-container-sec-1-7" class="outline-3">
<h3 id="sec-1-7"><span class="done DONE">DONE</span> 2.47 [www] t分布が\(ν→∞\)の極限でガウス分布になることの証明</h3>
<div class="outline-text-3" id="text-1-7">
\begin{align*}
\St(x|μ,λ,ν) \propto & \l[ 1 + \f{λ(x - μ)^2}{ν} \r]^{-(ν+1)/2} \\
= & \exp\l( -\f{ν+1}{2} \ln\l[ 1 + \f{λ(x - μ)^2}{ν} \r] \r) \\
\end{align*}
<p>
対数関数は1の周辺で以下のようにテーラー展開できる。<br />
</p>
\begin{align*}
\ln(1 + ε) = ε + O(ε^2) \\
\end{align*}
<p>
\(ν→∞\)のとき、上の展開を適用できる。<br />
</p>
\begin{align*}
& \exp\l( -\f{ν+1}{2} \ln\l[ 1 + \f{λ(x - μ)^2}{ν} \r] \r) \\
= & \exp\l( -\f{ν+1}{2} \l[ \f{λ(x - μ)^2}{ν} + O(ν^{-2}) \r] \r) \\
= & \exp\l( - \f{λ(x - μ)^2}{2} + O(ν^{-1}) \r) \\
\end{align*}
</div>
</div>
<div id="outline-container-sec-1-8" class="outline-3">
<h3 id="sec-1-8"><span class="todo TODO">TODO</span> 2.48 スチューデントのt分布の多変量形式の導出とそれが正規化されていることの証明</h3>
<div class="outline-text-3" id="text-1-8">
\begin{align*}
\St(\x|\μ,\Λ,ν)
= & \int_0^∞ \N(\x|\μ,(η\Λ)^{-1}) \Gam(η|ν/2,ν/2) dη \\
= & \int_0^∞ \f{1}{(2π)^{D/2}} \f{1}{|(η\Λ)^{-1}|^{1/2}} \exp\l( -\f{η Δ^2}{2} \r) \\
& \f{1}{Γ(ν/2)} (ν/2)^{ν/2} η^{ν/2-1} \exp(-νη/2) dη \\
= & \f{1}{Γ(ν/2)}\f{(ν/2)^{ν/2}|\Λ|^{1/2}}{(2π)^{D/2}}
\int_0^∞ η^{ν/2-1/2} \exp\l[ -\f{η(Δ^2 + ν)}{2} \r] dη \\
\end{align*}
\begin{align*}
z = \f{η(Δ^2 + ν)}{2} &
η = \f{2z}{Δ^2 + ν} &
\f{dη}{dz} = & \f{2}{Δ^2 + ν} \\
\end{align*}
\begin{align*}
\St(\x|\μ,\Λ,ν)
= & \f{1}{Γ(ν/2)}\f{(ν/2)^{ν/2}|\Λ|^{1/2}}{(2π)^{D/2}}
\int_0^∞ \l( \f{2z}{Δ^2 + ν} \r)^{ν/2-1/2} \exp(z) \f{2}{Δ^2 + ν} dz \\
= & \f{1}{Γ(ν/2)}\f{(ν/2)^{ν/2}|\Λ|^{1/2}}{(2π)^{D/2}} \l( \f{2}{Δ^2 + ν} \r)^{ν/2+1/2}
\int_0^∞ z^{ν/2-1/2} \exp(z) dz \\
= & \f{Γ(ν/2+1/2)}{Γ(ν/2)}\f{(ν/2)^{ν/2}|\Λ|^{1/2}}{(2π)^{D/2}} \l( \f{2}{Δ^2 + ν} \r)^{ν/2+1/2} \\
\end{align*}
</div>
</div>
<div id="outline-container-sec-1-9" class="outline-3">
<h3 id="sec-1-9"><span class="todo TODO">TODO</span> 2.49 多変量スチューデントt分布の平均、共分散、モード</h3>
</div>
<div id="outline-container-sec-1-10" class="outline-3">
<h3 id="sec-1-10"><span class="done DONE">DONE</span> 2.50 多変量スチューデントt分布が\(ν→∞\)の極限で多変量ガウス分布になることの証明</h3>
<div class="outline-text-3" id="text-1-10">
\begin{align*}
\St(\x|\μ,\Λ,ν)
= & \f{Γ(D/2+ν/2)}{Γ(ν/2)} \f{|\Λ|^{1/2}}{(πν)^{D/2}} \l[ 1 + \f{Δ^2}{ν} \r]^{-D/2-ν/2} \\
\propto & \l[ 1 + \f{Δ^2}{ν} \r]^{-D/2-ν/2} \\
= & \exp\l( - \f{D+ν}{2} \ln\l[ 1 + \f{Δ^2}{ν} \r] \r) \\
= & \exp\l( - \f{D+ν}{2} \l[ \f{Δ^2}{ν} + O(ν^{-2}) \r] \r) \\
= & \exp\l( - \f{Δ^2}{2} + O(ν^{-1}) \r) \\
\end{align*}
</div>
</div>
</div>
</div>
<div id="postamble" class="status">
<p class="creator"><a href="http://www.gnu.org/software/emacs/">Emacs</a> 24.4.4 (<a href="http://orgmode.org">Org</a> mode 8.2.10)</p>
<p class="validation"><a href="http://validator.w3.org/check?uri=referer">Validate</a></p>
</div>
</body>
</html>