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<title>PRML 第4章 演習 4.11-4.20</title>
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<div id="content">
<h1 class="title">PRML 第4章 演習 4.11-4.20</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4章 演習 4.11-4.20</a>
<ul>
<li><a href="#sec-1-1">4.11</a></li>
<li><a href="#sec-1-2">4.12</a></li>
<li><a href="#sec-1-3">4.14</a></li>
<li><a href="#sec-1-4">4.14</a></li>
<li><a href="#sec-1-5">4.15</a></li>
<li><a href="#sec-1-6">4.16</a></li>
<li><a href="#sec-1-7">4.17</a></li>
<li><a href="#sec-1-8">4.18</a></li>
<li><a href="#sec-1-9"><span class="done DONE">DONE</span> 4.19 [www] プロビット回帰モデルの対数尤度の勾配とヘッセ行列</a></li>
<li><a href="#sec-1-10">4.20</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{\bPhi}{{\rm \bf \Phi}}
\newcommand{\bphi}{\boldsymbol \phi}
\newcommand{\bvphi}{\boldsymbol \varphi}
\newcommand{\E}{{\mathbb{E}}}
\newcommand{\D}{{\cal D}}
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\newcommand{\T}{\mathrm{T}}
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\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4章 演習 4.11-4.20</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">4.11</h3>
</div>
<div id="outline-container-sec-1-2" class="outline-3">
<h3 id="sec-1-2">4.12</h3>
</div>
<div id="outline-container-sec-1-3" class="outline-3">
<h3 id="sec-1-3">4.14</h3>
</div>
<div id="outline-container-sec-1-4" class="outline-3">
<h3 id="sec-1-4">4.14</h3>
</div>
<div id="outline-container-sec-1-5" class="outline-3">
<h3 id="sec-1-5">4.15</h3>
</div>
<div id="outline-container-sec-1-6" class="outline-3">
<h3 id="sec-1-6">4.16</h3>
</div>
<div id="outline-container-sec-1-7" class="outline-3">
<h3 id="sec-1-7">4.17</h3>
</div>
<div id="outline-container-sec-1-8" class="outline-3">
<h3 id="sec-1-8">4.18</h3>
</div>
<div id="outline-container-sec-1-9" class="outline-3">
<h3 id="sec-1-9"><span class="done DONE">DONE</span> 4.19 [www] プロビット回帰モデルの対数尤度の勾配とヘッセ行列</h3>
<div class="outline-text-3" id="text-1-9">
<p>
プロビット回帰モデルの誤差関数は、ロジスティック回帰モデルと同様に、<br />
交差エントロピー誤差関数の形で与えられる。<br />
</p>
\begin{align*}
E(\w) = & - \sum_{n=1}^N \{ t_n \ln y_n + (1 - t_n) \ln (1 - y_n) \} \\
\end{align*}
<p>
ただし<br />
</p>
\begin{align*}
y_n = & \Phi(a_n), & a_n = & \w^\T \bphi_n
\end{align*}
<p>
誤差関数\(E_n\)の\(y_n\)による微分<br />
</p>
\begin{align*}
\p{E_n}{y_n}
= - \p{}{y_n} \{ t_n \ln y_n + (1 - t_n) \ln (1 - y_n) \}
= \f{y_n - t_n}{y_n (1 - y_n)}
\end{align*}
<p>
プロビット活性化関数の微分<br />
</p>
\begin{align*}
\p{\Phi(a)}{a}
= \p{}{a} \int_{-\infty}^a \N(\theta|0,1) \d \theta
= \N(a|0,1)
\end{align*}
<p>
誤差関数の勾配<br />
</p>
\begin{align*}
\nabla E(\w)
= \sum_{n=1}^N \p{E_n}{y_n} \p{y_n}{a_n} \nabla a_n
= \sum_{n=1}^N \f{y_n - t_n}{y_n (1 - y_n)} \N(a_n|0,1) \bphi_n
\end{align*}
\begin{align*}
\p{}{y_n} \f{y_n - t_n}{y_n (1 - y_n)}
= \f{y_n (1 - y_n) - (y_n - t_n) (1 - 2 y_n)}{y_n^2 (1 - y_n)^2}
= \f{y_n^2 + t_n - 2 y_n t_n}{y_n^2 (1 - y_n)^2}
\end{align*}
\begin{align*}
\p{}{a} \N(a|0,1) &
= \f{1}{\sqrt{2 \pi}} \p{}{a} \exp\l(-\f{a^2}{2}\r)
= \f{1}{\sqrt{2 \pi}} (-a) \exp\l(-\f{a^2}{2}\r)
= - a \N(a|0,1)
\end{align*}
<p>
ヘッセ行列<br />
</p>
\begin{align*}
\nabla \nabla E(\w)
= & \nabla \sum_{n=1}^N \f{y_n - t_n}{y_n (1 - y_n)} \N(a_n|0,1) \bphi_n^\T \\
= & \sum_{n=1}^N
\l\{ \l[ \nabla \f{y_n - t_n}{y_n (1 - y_n)} \r] \N(a_n|0,1)
+ \f{y_n - t_n}{y_n (1 - y_n)} \nabla \N(a_n|0,1) \r\} \bphi_n^\T \\
= & \sum_{n=1}^N
\l\{ \l( \p{}{y_n} \f{y_n - t_n}{y_n (1 - y_n)} \r) \p{y_n}{a_n} \nabla a_n \N(a_n|0,1)
+ \f{y_n - t_n}{y_n (1 - y_n)} \l( \p{}{a_n} \N(a_n|0,1) \r) \nabla a_n \r\} \bphi_n^\T \\
= & \sum_{n=1}^N
\l\{ \f{y_n^2 + t_n - 2 y_n t_n}{y_n^2 (1 - y_n)^2} \N(a_n|0,1) \bphi_n
+ \f{y_n - t_n}{y_n (1 - y_n)} (-a_n) \N(a_n|0,1) \bphi_n \r\} \bphi_n^\T \\
= & \sum_{n=1}^N
\l\{ \f{y_n^2 + t_n - 2 y_n t_n}{y_n (1 - y_n)} - a_n (y_n - t_n) \r\}
\f{\N(a_n|0,1) \bphi_n \bphi_n^\T}{y_n (1 - y_n)} \\
\end{align*}
</div>
</div>
<div id="outline-container-sec-1-10" class="outline-3">
<h3 id="sec-1-10">4.20</h3>
</div>
</div>
</div>
<div id="postamble" class="status">
<p class="creator"><a href="http://www.gnu.org/software/emacs/">Emacs</a> 24.4.1 (<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>