条件付き正規分布について証明していきたいと思います。証明は行列変量正規分布で行いますが、多変量正規分布の証明は証明中で\(n=1\)としたものになります。
条件付き正規分布
\(n\times p\)の行列変量正規分布に従う確率変数を\(X = (X_{1}\ X_{2})\sim N_{p}(M,\ U,\ V)\)と分割します。ただし、\(X_{1}\)は\(n\times p_{1}\)の確率変数、\(X_{2}\)は\(n\times p_{2}\)の確率変数であり\(p_{1}+p_{2}=p\)が成り立っています。
\(X\)が行列正規分布に従っていることから、\(X_{1}\)と\(X_{2}\)も行列正規分布に従います。よって
\begin{align}
X_{1}\sim N_{n\times p_{1}}(M_{1},\ U,\ V_{11}),\ \ X_{2} \sim N_{n\times p_{2}}(M_{2},\ U,\ V_{22})
\end{align}
と表すことができます。ただし、\(M\)と\(M_{1},\ M_{2}\)、\(V\)と\(V_{11},\ V_{22}\)とはつぎのような関係があります。X_{1}\sim N_{n\times p_{1}}(M_{1},\ U,\ V_{11}),\ \ X_{2} \sim N_{n\times p_{2}}(M_{2},\ U,\ V_{22})
\end{align}
\begin{align}
M=\left( \begin{array}{cc}
M_{1} & M_{2}
\end{array}\right),\ \ V = \left( \begin{array}{cc}
V_{11} & V_{12}\\
V_{21} & V_{22}
\end{array}\right),\ \ V_{ij}はp_{i}\times p_{j}の行列です
\end{align}
M=\left( \begin{array}{cc}
M_{1} & M_{2}
\end{array}\right),\ \ V = \left( \begin{array}{cc}
V_{11} & V_{12}\\
V_{21} & V_{22}
\end{array}\right),\ \ V_{ij}はp_{i}\times p_{j}の行列です
\end{align}
このとき、次のような関係が成り立ちます。
条件付き行列変量正規分布
\(X_{1}\)が与えられたとき、\(X_{2}\)の条件付き分布も行列変量正規分布に従い、
\begin{align}
X_{2}\,|\,X_{1} \sim N_{n\times p_{2}} \left( M_{2}+(X_{1}-M_{1})V_{11}^{-1}V_{12},\ U\otimes(V_{22}-V_{21}V_{11}^{-1}V_{12}) \right)
\end{align}
となります。つまり、条件付き期待値、条件付き分散が
\begin{align}
\mathrm{E}[X_{2}|X_{1}] &= M_{2}+(X_{1}-M_{1})V_{11}^{-1}V_{12} \\
\mathrm{Var}[X_{2}|X_{1}] &= U\otimes(V_{22}-V_{21}V_{11}^{-1}V_{12})
\end{align}
となる、条件付き行列変量正規分布に従います。
特に\(n=1\)のとき、行列変量正規分布は多変量正規分布となることから、この条件付き正規分布も多変量正規分布だった場合の条件付き正規分布に従うことがわかります。
※行列変量正規分布、多変量正規分布については以下のリンクからお願いします。
証明
行列\(E\)を次のように定義します。
\begin{align}
E = \left(\begin{array}{cc}
I_{p_{1}} & O \\
-V_{21}V_{11}^{-1} & I_{p_{2}}
\end{array}\right)
\end{align}
E = \left(\begin{array}{cc}
I_{p_{1}} & O \\
-V_{21}V_{11}^{-1} & I_{p_{2}}
\end{array}\right)
\end{align}
この行列について、行列式が\(|E|=1\)となり、逆行列が
\begin{align}
E^{-1} = \left(\begin{array}{cc}
I_{p_{1}} & O \\
V_{21}V_{11}^{-1} & I_{p_{2}}
\end{array}\right)
\end{align}
が成り立ちます。このことを用いて、行列変量正規分布の確率密度関数を書き換えていきます。E^{-1} = \left(\begin{array}{cc}
I_{p_{1}} & O \\
V_{21}V_{11}^{-1} & I_{p_{2}}
\end{array}\right)
\end{align}
\begin{align}
f(x) &= \frac{1}{(2\pi)^{\frac{1}{2}np}|U|^{\frac{1}{2}p}|V|^{\frac{1}{2}n}}\exp\left[ -\frac{1}{2}\mathrm{tr} \left\{ U^{-1}(X-M)V^{-1}\,{}^{T}\!(X-M) \right\} \right] \\
&= \frac{1}{(2\pi)^{\frac{1}{2}np}|U|^{\frac{1}{2}p}|V|^{\frac{1}{2}n}}\exp\left[ -\frac{1}{2}\mathrm{tr} \left\{ U^{-1}(X-M)\ {}^{T}\!E\ {}^{T}\!E^{-1}V^{-1}E^{-1}E\,{}^{T}\!(X-M) \right\} \right] \\
&= \frac{1}{(2\pi)^{\frac{1}{2}np}|U|^{\frac{1}{2}p}|V|^{\frac{1}{2}n}}\exp\left[ -\frac{1}{2}\mathrm{tr} \left\{ U^{-1}\{ (X-M)\ {}^{T}\!E\} ( {}^{T}\!EVE)^{-1}\ {}^{T}\{ (X-M)\ {}^{T}\!E \} \right\} \right] \\
\end{align}
ここで、この式をもう少し詳しく分解して計算していきます。ただし\(M_{2}^{\prime}=M_{2}-(X_{1}-M_{1})V_{11}^{-1}V_{12}\)とします。
f(x) &= \frac{1}{(2\pi)^{\frac{1}{2}np}|U|^{\frac{1}{2}p}|V|^{\frac{1}{2}n}}\exp\left[ -\frac{1}{2}\mathrm{tr} \left\{ U^{-1}(X-M)V^{-1}\,{}^{T}\!(X-M) \right\} \right] \\
&= \frac{1}{(2\pi)^{\frac{1}{2}np}|U|^{\frac{1}{2}p}|V|^{\frac{1}{2}n}}\exp\left[ -\frac{1}{2}\mathrm{tr} \left\{ U^{-1}(X-M)\ {}^{T}\!E\ {}^{T}\!E^{-1}V^{-1}E^{-1}E\,{}^{T}\!(X-M) \right\} \right] \\
&= \frac{1}{(2\pi)^{\frac{1}{2}np}|U|^{\frac{1}{2}p}|V|^{\frac{1}{2}n}}\exp\left[ -\frac{1}{2}\mathrm{tr} \left\{ U^{-1}\{ (X-M)\ {}^{T}\!E\} ( {}^{T}\!EVE)^{-1}\ {}^{T}\{ (X-M)\ {}^{T}\!E \} \right\} \right] \\
\end{align}
まず指数部分について変形していきます。
\begin{align}
EV\ {}^{T}E &= \left(\begin{array}{cc}
I_{p_{1}} & O \\
-V_{21}V_{11}^{-1} & I_{p_{2}}
\end{array}\right)\left(\begin{array}{cc}
V_{11} & V_{12} \\
V_{21} & V_{22}
\end{array}\right)\left(\begin{array}{cc}
I_{p_{1}} & -V_{11}^{-1}V_{12} \\
O & I_{p_{2}}
\end{array}\right) \\
&= \left(\begin{array}{cc}
V_{11} & V_{12} \\
O & -V_{21}V_{11}^{-1}V_{12}+V_{22}
\end{array}\right)\left(\begin{array}{cc}
I_{p_{1}} & -V_{11}^{-1}V_{12} \\
O & I_{p_{2}}
\end{array}\right) \\
&= \left(\begin{array}{cc}
V_{11} & O \\
O & -V_{21}V_{11}^{-1}V_{12}+V_{22}
\end{array}\right)
\end{align}
さらに、もうひとつの部分\((X-M)\ {}^{T}\!E\)について計算していくとEV\ {}^{T}E &= \left(\begin{array}{cc}
I_{p_{1}} & O \\
-V_{21}V_{11}^{-1} & I_{p_{2}}
\end{array}\right)\left(\begin{array}{cc}
V_{11} & V_{12} \\
V_{21} & V_{22}
\end{array}\right)\left(\begin{array}{cc}
I_{p_{1}} & -V_{11}^{-1}V_{12} \\
O & I_{p_{2}}
\end{array}\right) \\
&= \left(\begin{array}{cc}
V_{11} & V_{12} \\
O & -V_{21}V_{11}^{-1}V_{12}+V_{22}
\end{array}\right)\left(\begin{array}{cc}
I_{p_{1}} & -V_{11}^{-1}V_{12} \\
O & I_{p_{2}}
\end{array}\right) \\
&= \left(\begin{array}{cc}
V_{11} & O \\
O & -V_{21}V_{11}^{-1}V_{12}+V_{22}
\end{array}\right)
\end{align}
\begin{align}
(X-M)\ {}^{T}\!E &= \left(\begin{array}{cc}
X_{1}-M_{1} & X_{2}-M_{2}
\end{array}\right)\left(\begin{array}{cc}
I_{p_{1}} & -V_{11}^{-1}V_{12} \\
O & I_{p_{2}}
\end{array}\right) \\
&= \left(\begin{array}{cc}
X_{1}-M_{1} & X_{2}-M_{2}^{\prime}
\end{array}\right)
\end{align}
となります。これらをまとめると(X-M)\ {}^{T}\!E &= \left(\begin{array}{cc}
X_{1}-M_{1} & X_{2}-M_{2}
\end{array}\right)\left(\begin{array}{cc}
I_{p_{1}} & -V_{11}^{-1}V_{12} \\
O & I_{p_{2}}
\end{array}\right) \\
&= \left(\begin{array}{cc}
X_{1}-M_{1} & X_{2}-M_{2}^{\prime}
\end{array}\right)
\end{align}
\begin{align}
\{ (X-M)\ {}^{T}\!E\} ( {}^{T}\!EVE)^{-1}\ {}^{T}\{ (X-M)\ {}^{T}\!E \} &= \left(\begin{array}{cc}
X_{1}-M_{1} & X_{2}-M_{2}
\end{array}\right)\left(\begin{array}{cc}
V_{11} & O \\
O & -V_{21}V_{11}^{-1}V_{12}+V_{22}
\end{array}\right)^{-1}\left(\begin{array}{c}
{}^{T}(X_{1}-M_{1}) \\
{}^{T}(X_{2}-M_{2})
\end{array}\right) \\
&= (X_{1}-M_{1})V_{11}^{-1}\ {}^{T}\!(X_{1}-M_{1}) + (X_{2}-M_{2}^{\prime})(V_{22}-V_{21}V_{11}^{-1}V_{12})^{-1}\ {}^{T}\!(X_{2}-M_{2}^{\prime})
\end{align}
が成り立ちます。加えて、分母部分にある\(|U|^{\frac{1}{2}p}|V|^{\frac{1}{2}n}\)についても変形しておくと、\{ (X-M)\ {}^{T}\!E\} ( {}^{T}\!EVE)^{-1}\ {}^{T}\{ (X-M)\ {}^{T}\!E \} &= \left(\begin{array}{cc}
X_{1}-M_{1} & X_{2}-M_{2}
\end{array}\right)\left(\begin{array}{cc}
V_{11} & O \\
O & -V_{21}V_{11}^{-1}V_{12}+V_{22}
\end{array}\right)^{-1}\left(\begin{array}{c}
{}^{T}(X_{1}-M_{1}) \\
{}^{T}(X_{2}-M_{2})
\end{array}\right) \\
&= (X_{1}-M_{1})V_{11}^{-1}\ {}^{T}\!(X_{1}-M_{1}) + (X_{2}-M_{2}^{\prime})(V_{22}-V_{21}V_{11}^{-1}V_{12})^{-1}\ {}^{T}\!(X_{2}-M_{2}^{\prime})
\end{align}
\begin{align}
|U|^{\frac{1}{2}p}|V|^{\frac{1}{2}n} &= |U|^{\frac{1}{2}p}|EV\ {}^{T}\!E|^{\frac{1}{2}n} \\
&= |U|^{\frac{1}{2}(p_{1}+p_{2})}( |V_{11}||V_{22}-V_{21}V_{11}^{-1}V_{12}| )^{\frac{1}{2}n} \\
&= |U|^{\frac{1}{2}p_{1}}|V_{11}|^{n}\times |U|^{\frac{1}{2}p_{2}}|V_{22}-V_{21}V_{11}^{-1}V_{12}|^{\frac{1}{2}n}
\end{align}
となることから、上で変形していた確率密度関数は|U|^{\frac{1}{2}p}|V|^{\frac{1}{2}n} &= |U|^{\frac{1}{2}p}|EV\ {}^{T}\!E|^{\frac{1}{2}n} \\
&= |U|^{\frac{1}{2}(p_{1}+p_{2})}( |V_{11}||V_{22}-V_{21}V_{11}^{-1}V_{12}| )^{\frac{1}{2}n} \\
&= |U|^{\frac{1}{2}p_{1}}|V_{11}|^{n}\times |U|^{\frac{1}{2}p_{2}}|V_{22}-V_{21}V_{11}^{-1}V_{12}|^{\frac{1}{2}n}
\end{align}
\begin{align}
f(x) &= \frac{1}{(2\pi)^{\frac{1}{2}np_{1}}|U|^{\frac{1}{2}p_{1}}|V_{11}|^{\frac{1}{2}n}}\exp\left[ -\frac{1}{2}\mathrm{tr}\left\{ U^{-1}(X_{1}-M_{1})V_{11}^{-1}\ {}^{T}\!(X_{1}-M_{1}) \right\} \right] \\
&\ \ \times \frac{1}{(2\pi)^{\frac{1}{2}np_{2}}|U|^{\frac{1}{2}p_{2}}|V_{22}-V_{21}V_{11}^{-1}V_{12}|^{\frac{1}{2}n}}\exp\left[ -\frac{1}{2}\mathrm{tr}\left\{ U^{-1}(X_{2}-M_{2}^{\prime})(V_{22}-V_{21}V_{11}^{-1}V_{12})^{-1}\ {}^{T}\!(X_{2}-M_{2}^{\prime}) \right\} \right] \\
\end{align}
と変形することができます。＜条件付き確率の定義＞から\(X_{1}\)を与えた下で、\(X_{2}\)の条件付き確率密度関数は、上で変形していた\(X_{1}\)と\(X_{2}\)の同時確率密度関数を、\(X_{1}\)の確率密度関数で割ると、f(x) &= \frac{1}{(2\pi)^{\frac{1}{2}np_{1}}|U|^{\frac{1}{2}p_{1}}|V_{11}|^{\frac{1}{2}n}}\exp\left[ -\frac{1}{2}\mathrm{tr}\left\{ U^{-1}(X_{1}-M_{1})V_{11}^{-1}\ {}^{T}\!(X_{1}-M_{1}) \right\} \right] \\
&\ \ \times \frac{1}{(2\pi)^{\frac{1}{2}np_{2}}|U|^{\frac{1}{2}p_{2}}|V_{22}-V_{21}V_{11}^{-1}V_{12}|^{\frac{1}{2}n}}\exp\left[ -\frac{1}{2}\mathrm{tr}\left\{ U^{-1}(X_{2}-M_{2}^{\prime})(V_{22}-V_{21}V_{11}^{-1}V_{12})^{-1}\ {}^{T}\!(X_{2}-M_{2}^{\prime}) \right\} \right] \\
\end{align}
\begin{align}
f(X_{2}|X_{1}) &= \frac{1}{(2\pi)^{\frac{1}{2}np_{2}}|U|^{\frac{1}{2}p_{2}}|V_{22}-V_{21}V_{11}^{-1}V_{12}|^{\frac{1}{2}n}}\exp\left[ -\frac{1}{2}\mathrm{tr}\left\{ U^{-1}(X_{2}-M_{2}^{\prime})(V_{22}-V_{21}V_{11}^{-1}V_{12})^{-1}\ {}^{T}\!(X_{2}-M_{2}^{\prime}) \right\} \right]
\end{align}
となります。この確率密度関数は期待値\(M_{2}^{\prime}M_{2}+(X_{1}-M_{1})V_{11}^{-1}V_{12}\)分散\(U\otimes(V_{22}-V_{21}V_{11}^{-1}V_{12}) \)の行列変量正規分布の確率密度関数になっていることがわかります。
f(X_{2}|X_{1}) &= \frac{1}{(2\pi)^{\frac{1}{2}np_{2}}|U|^{\frac{1}{2}p_{2}}|V_{22}-V_{21}V_{11}^{-1}V_{12}|^{\frac{1}{2}n}}\exp\left[ -\frac{1}{2}\mathrm{tr}\left\{ U^{-1}(X_{2}-M_{2}^{\prime})(V_{22}-V_{21}V_{11}^{-1}V_{12})^{-1}\ {}^{T}\!(X_{2}-M_{2}^{\prime}) \right\} \right]
\end{align}
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