-
Notifications
You must be signed in to change notification settings - Fork 8
/
ex_02_21-30.html
454 lines (433 loc) · 15.3 KB
/
ex_02_21-30.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
<?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 2.21-2.30</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 2.21-2.30</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 2.21-2.30</a>
<ul>
<li><a href="#sec-1-1"><span class="todo TODO">TODO</span> 2.21 \(D×D\)の実対称行列の独立なパラメータの数</a></li>
<li><a href="#sec-1-2"><span class="todo TODO">TODO</span> 2.22 [www] 対称行列の逆行列も対称であることの証明</a></li>
<li><a href="#sec-1-3"><span class="todo TODO">TODO</span> 2.23 マハラノビス距離が定数になる超楕円体の内部の体積</a></li>
<li><a href="#sec-1-4"><span class="done DONE">DONE</span> 2.24 [www] 分割された行列の逆行列に関する公式の証明</a></li>
<li><a href="#sec-1-5"><span class="todo TODO">TODO</span> 2.25 多変量ガウス分布の変数集合を3分割したときの周辺化された条件付き分布</a></li>
<li><a href="#sec-1-6"><span class="todo TODO">TODO</span> 2.26 Woodbury行列反転公式の証明</a></li>
<li><a href="#sec-1-7"><span class="todo TODO">TODO</span> 2.27 2つの独立な確率ベクトルの和の平均が、個別の平均の和となることの証明</a></li>
<li><a href="#sec-1-8"><span class="todo TODO">TODO</span> 2.28 [www] 2分割された平均ベクトルと共分散行列から周辺分布と条件付き分布を求める</a></li>
<li><a href="#sec-1-9"><span class="done DONE">DONE</span> 2.29 精度行列(2.104)の逆行列が共分散行列(2.105)になることの証明</a></li>
<li><a href="#sec-1-10"><span class="done DONE">DONE</span> 2.30 ガウス分布の同時分布の平均値</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 2.21-2.30</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="todo TODO">TODO</span> 2.21 \(D×D\)の実対称行列の独立なパラメータの数</h3>
</div>
<div id="outline-container-sec-1-2" class="outline-3">
<h3 id="sec-1-2"><span class="todo TODO">TODO</span> 2.22 [www] 対称行列の逆行列も対称であることの証明</h3>
</div>
<div id="outline-container-sec-1-3" class="outline-3">
<h3 id="sec-1-3"><span class="todo TODO">TODO</span> 2.23 マハラノビス距離が定数になる超楕円体の内部の体積</h3>
</div>
<div id="outline-container-sec-1-4" class="outline-3">
<h3 id="sec-1-4"><span class="done DONE">DONE</span> 2.24 [www] 分割された行列の逆行列に関する公式の証明</h3>
<div class="outline-text-3" id="text-1-4">
<p>
(2.76)の右辺に\(
\l(\begin{array}{cc}
A & B \\
C & D \\
\end{array}\r)
\)を掛ける。<br />
</p>
\begin{align*}
X = &
\l(\begin{array}{cc}
A & B \\
C & D \\
\end{array}\r)
\l(\begin{array}{cc}
M & -MBD^{-1} \\
-D^{-1}CM & D^{-1} + D^{-1}CMBD^{-1} \\
\end{array}\r) \\
= &
\l(\begin{array}{cc}
AM - BD^{-1}CM & -AMBD^{-1} + B(D^{-1} + D^{-1}CMBD^{-1}) \\
CM - DD^{-1}CM & -CMBD^{-1} + D(D^{-1} + D^{-1}CMBD^{-1}) \\
\end{array}\r)
\end{align*}
\begin{align*}
X_{11} = & AM - BD^{-1}CM \\
= & (A - BD^{-1}C)M \\
= & (A - BD^{-1}C)(A - BD^{-1}C)^{-1} \\
= & I \\
\end{align*}
\begin{align*}
X_{12} = & -AMBD^{-1} + B(D^{-1} + D^{-1}CMBD^{-1}) \\
= & -AMBD^{-1} + BD^{-1} + BD^{-1}CMBD^{-1} \\
= & -(A - BD^{-1}C)MBD^{-1} + BD^{-1} \\
= & -(A - BD^{-1}C)(A - BD^{-1}C)^{-1}BD^{-1} + BD^{-1} \\
= & -BD^{-1} + BD^{-1} \\
= & O \\
\end{align*}
\begin{align*}
X_{21} = & CM - DD^{-1}CM \\
= & O \\
\end{align*}
\begin{align*}
X_{22} = & -CMBD^{-1} + D(D^{-1} + D^{-1}CMBD^{-1}) \\
= & -CMBD^{-1} + I + CMBD^{-1} \\
= & I \\
\end{align*}
</div>
</div>
<div id="outline-container-sec-1-5" class="outline-3">
<h3 id="sec-1-5"><span class="todo TODO">TODO</span> 2.25 多変量ガウス分布の変数集合を3分割したときの周辺化された条件付き分布</h3>
</div>
<div id="outline-container-sec-1-6" class="outline-3">
<h3 id="sec-1-6"><span class="todo TODO">TODO</span> 2.26 Woodbury行列反転公式の証明</h3>
</div>
<div id="outline-container-sec-1-7" class="outline-3">
<h3 id="sec-1-7"><span class="todo TODO">TODO</span> 2.27 2つの独立な確率ベクトルの和の平均が、個別の平均の和となることの証明</h3>
</div>
<div id="outline-container-sec-1-8" class="outline-3">
<h3 id="sec-1-8"><span class="todo TODO">TODO</span> 2.28 [www] 2分割された平均ベクトルと共分散行列から周辺分布と条件付き分布を求める</h3>
</div>
<div id="outline-container-sec-1-9" class="outline-3">
<h3 id="sec-1-9"><span class="done DONE">DONE</span> 2.29 精度行列(2.104)の逆行列が共分散行列(2.105)になることの証明</h3>
<div class="outline-text-3" id="text-1-9">
<p>
分割行列の逆行列の公式<br />
</p>
\begin{align*}
\l(\begin{array}{cc}
A & B \\
C & D \\
\end{array}\r)^{-1}
=
\l(\begin{array}{cc}
M & -MBD^{-1} \\
-D^{-1}CM & D^{-1} + CMBD^{-1} \\
\end{array}\r)^{-1}
\end{align*}
<p>
ただし<br />
</p>
\begin{align*}
M = (A - BD^{-1}C)^{-1}
\end{align*}
<p>
(2.104)<br />
</p>
\begin{align*}
R = \l(\begin{array}{cc}
Λ + A^TLA & -A^TL \\
-LA & L \\
\end{array}\r)^{-1}
\end{align*}
\begin{align*}
(R^{-1})_{11}
= & M \\
= & (Λ + A^TLA - (-A^TL)L^{-1}(-LA))^{-1} \\
= & (Λ + A^TLA - A^TLL^{-1}LA)^{-1} \\
= & (Λ + A^TLA - A^TLA)^{-1} \\
= & Λ^{-1} \\
\end{align*}
\begin{align*}
(R^{-1})_{12}
= & -MBD^{-1} \\
= & -(Λ^{-1})(-A^TL)(L)^{-1} \\
= & Λ^{-1}A^TLL^{-1} \\
= & Λ^{-1}A^T \\
\end{align*}
\begin{align*}
(R^{-1})_{21}
= & -D^{-1}CM \\
= & -(L)^{-1}(-LA)(Λ^{-1}) \\
= & L^{-1}LAΛ^{-1} \\
= & AΛ^{-1} \\
\end{align*}
\begin{align*}
(R^{-1})_{22}
= & D^{-1} + CMBD^{-1} \\
= & (L)^{-1} + (-LA)(Λ^{-1})(-A^TL)(L)^{-1} \\
= & L^{-1} + LAΛ^{-1}A^TLL^{-1} \\
= & L^{-1} + LAΛ^{-1}A^T \\
\end{align*}
</div>
</div>
<div id="outline-container-sec-1-10" class="outline-3">
<h3 id="sec-1-10"><span class="done DONE">DONE</span> 2.30 ガウス分布の同時分布の平均値</h3>
<div class="outline-text-3" id="text-1-10">
<p>
(2.107)<br />
</p>
\begin{align*}
E[z] = R^{-1} \l( \begin{array}{c}
Λμ - A^TLb \\
Lb \\
\end{array} \r)
\end{align*}
<p>
(2.105)<br />
</p>
\begin{align*}
R^{-1} = \l( \begin{array}{cc}
Λ^{-1} & Λ^{-1}A^T \\
AΛ^{-1} & L^{-1} + AΛ^{-1}A^T \\
\end{array} \r)
\end{align*}
\begin{align*}
E[z]_1 = & Λ^{-1}(Λμ - A^TLb) + Λ^{-1}A^TLb \\
= & Λ^{-1}Λμ - Λ^{-1}A^TLb + Λ^{-1}A^TLb \\
= & μ \\
\end{align*}
\begin{align*}
E[z]_2 = & AΛ^{-1}(Λμ - A^TLb) + (L^{-1} + AΛ^{-1}A^T)Lb \\
= & AΛ^{-1}Λμ - AΛ^{-1}A^TLb + L^{-1}Lb + AΛ^{-1}A^TLb \\
= & Aμ - AΛ^{-1}A^TLb + b + AΛ^{-1}A^TLb \\
= & Aμ+ b \\
\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>