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<div id="content">
<h1 class="title">PRML 第2章 演習 2.51-2.61</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 2.51-2.61</a>
<ul>
<li><a href="#sec-1-1">2.51 [www] 三角関数の公式</a></li>
<li><a href="#sec-1-2">2.52 フォン・ミーゼス分布が\(m→∞\)の極限でガウス分布になることの証明</a></li>
<li><a href="#sec-1-3">2.53 フォン・ミーゼス分布における\(θ\)の最尤推定</a></li>
<li><a href="#sec-1-4">2.54 フォン・ミーゼス分布の最大値と最小値</a></li>
<li><a href="#sec-1-5">2.55 フォン・ミーゼス分布の集中度の最尤推定</a></li>
<li><a href="#sec-1-6">2.56 [www] ベータ分布、ガンマ分布、フォン・ミーゼス分布の指数型分布族の一般形への変形</a></li>
<li><a href="#sec-1-7"><span class="todo TODO">TODO</span> 2.57 多変量ガウス分布の指数型分布族の一般形への変形</a></li>
<li><a href="#sec-1-8">2.58 指数型分布族で\(\ln g(\η)\)の2階微分が\(\u(\x)\)の共分散になることの証明</a></li>
<li><a href="#sec-1-9">2.59 \(f(x)\)が正規化されていれば密度も正規化されていることの証明</a></li>
<li><a href="#sec-1-10">2.60 [www] ヒストグラム型の密度モデルの最尤推定</a></li>
<li><a href="#sec-1-11">2.61 K近傍密度モデルが変速分布であることの証明</a></li>
</ul>
</li>
</ul>
</div>
</div>
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\newcommand{\Φ}{\mathbf{Φ}}
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<div id="outline-container-sec-1" class="outline-2">
<h2 id="sec-1">PRML 第2章 演習 2.51-2.61</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">2.51 [www] 三角関数の公式</h3>
</div>
<div id="outline-container-sec-1-2" class="outline-3">
<h3 id="sec-1-2">2.52 フォン・ミーゼス分布が\(m→∞\)の極限でガウス分布になることの証明</h3>
</div>
<div id="outline-container-sec-1-3" class="outline-3">
<h3 id="sec-1-3">2.53 フォン・ミーゼス分布における\(θ\)の最尤推定</h3>
</div>
<div id="outline-container-sec-1-4" class="outline-3">
<h3 id="sec-1-4">2.54 フォン・ミーゼス分布の最大値と最小値</h3>
</div>
<div id="outline-container-sec-1-5" class="outline-3">
<h3 id="sec-1-5">2.55 フォン・ミーゼス分布の集中度の最尤推定</h3>
</div>
<div id="outline-container-sec-1-6" class="outline-3">
<h3 id="sec-1-6">2.56 [www] ベータ分布、ガンマ分布、フォン・ミーゼス分布の指数型分布族の一般形への変形</h3>
</div>
<div id="outline-container-sec-1-7" class="outline-3">
<h3 id="sec-1-7"><span class="todo TODO">TODO</span> 2.57 多変量ガウス分布の指数型分布族の一般形への変形</h3>
<div class="outline-text-3" id="text-1-7">
<p>
指数型分布族の一般形<br />
</p>
\begin{align*}
p(\x|\η) = h(\x) g(\η) \exp\{ \η^T \u(\x) \} \\
\end{align*}
<p>
多変量ガウス分布<br />
</p>
\begin{align*}
\N(\x|\μ,\Σ)
= & \f{1}{(2π)^{D/2}} \f{1}{|\Σ|^{1/2}}
\exp\l\{ -\f{1}{2} (\x - \μ)^T \Σ^{-1} (\x - \μ) \r\} \\
= & \f{1}{(2π)^{D/2}} \f{1}{|\Σ|^{1/2}}
\exp\l\{ -\f{1}{2} ( \x^T \Σ^{-1} \x - \x^T \Σ^{-1} \μ
- \μ^T \Σ^{-1} \x + \μ^T \Σ^{-1} \μ) \r\} \\
= & \f{1}{(2π)^{D/2}} \f{1}{|\Σ|^{1/2}}
\exp\l\{ - \f{1}{2} \x^T \Σ^{-1} \x + \μ^T \Σ^{-1} \x - \f{1}{2} \μ^T \Σ^{-1} \μ \r\} \\
\end{align*}
\begin{align*}
\η = & \l( \begin{array}{c}
\μ^T \\
-\f{1}{2} \Σ^{-1} \\
\end{array} \r) \\
\u(\x) = & \l( \begin{array}{c}
\Σ^{-1} \x \\
\Σ \x^T \Σ^{-1} \x \\
\end{array} \r) \\
h(\x) = & \f{1}{(2π)^{D/2}} \\
g(\η) = & -2 |\η_2|^{1/2} \exp\l( \η_1 \η_2 \η_1^T \r)
\end{align*}
</div>
</div>
<div id="outline-container-sec-1-8" class="outline-3">
<h3 id="sec-1-8">2.58 指数型分布族で\(\ln g(\η)\)の2階微分が\(\u(\x)\)の共分散になることの証明</h3>
</div>
<div id="outline-container-sec-1-9" class="outline-3">
<h3 id="sec-1-9">2.59 \(f(x)\)が正規化されていれば密度も正規化されていることの証明</h3>
</div>
<div id="outline-container-sec-1-10" class="outline-3">
<h3 id="sec-1-10">2.60 [www] ヒストグラム型の密度モデルの最尤推定</h3>
</div>
<div id="outline-container-sec-1-11" class="outline-3">
<h3 id="sec-1-11">2.61 K近傍密度モデルが変速分布であることの証明</h3>
</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>