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true |
Joram Soch |
BCCN Berlin |
joram.soch@bccn-berlin.de |
2022-05-14 16:58:00 -0700 |
Probability density function of the normal-Wishart distribution |
Probability Distributions |
Matrix-variate continuous distributions |
Normal-Wishart distribution |
Probability density function |
|
P323 |
nw-pdf |
JoramSoch |
Theorem: Let $X$ and $Y$ follow a normal-Wishart distribution:
$$ \label{eq:nw}
X,Y \sim \mathrm{NW}(M, U, V, \nu) ; .
$$
Then, the joint probability density function of $X$ and $Y$ is
$$ \label{eq:nw-pdf}
\begin{split}
p(X,Y) = ; & \frac{1}{\sqrt{(2\pi)^{np} |U|^p |V|^{\nu}}} \cdot \frac{\sqrt{2^{-\nu p}}}{\Gamma_p \left( \frac{\nu}{2} \right)} \cdot |Y|^{(\nu+n-p-1)/2} \cdot \\
& \exp\left[-\frac{1}{2} \mathrm{tr}\left( Y \left[ (X-M)^\mathrm{T} , U^{-1} (X-M) + V^{-1} \right] \right) \right] ; .
\end{split}
$$
Proof: The normal-Wishart distribution is defined as $X$ conditional on $Y$ following a matrix-normal distribution and $Y$ following a Wishart distribution:
$$ \label{eq:matn-wish}
\begin{split}
X \vert Y &\sim \mathcal{MN}(M, U, Y^{-1}) \\
Y &\sim \mathcal{W}(V, \nu) ; .
\end{split}
$$
Thus, using the probability density function of the matrix-normal distribution and the probability density function of the Wishart distribution, we have the following probabilities:
$$ \label{eq:matn-wish-pdf}
\begin{split}
p(X \vert Y) &= \mathcal{MN}(X; M, U, Y^{-1}) \\
&= \sqrt{\frac{|Y|^n}{(2\pi)^{np} |U|^p}} \cdot \exp\left[-\frac{1}{2} \mathrm{tr}\left( Y (X-M)^\mathrm{T} , U^{-1} (X-M) \right) \right] \\
p(Y) &= \mathcal{W}(Y; V, \nu) \\
&= \frac{1}{\Gamma_p \left( \frac{\nu}{2} \right)} \cdot \frac{1}{\sqrt{2^{\nu p} |V|^{\nu}}} \cdot |Y|^{(\nu-p-1)/2} \cdot \exp\left[ -\frac{1}{2} \mathrm{tr}\left( V^{-1} Y \right) \right] ; .
\end{split}
$$
The law of conditional probability implies that
$$ \label{eq:prob-cond}
p(X,Y) = p(X \vert Y) , p(Y) ; ,
$$
such that the normal-Wishart density function becomes:
$$ \label{eq:nw-pdf-qed}
\begin{split}
p(X,Y) = ; & \mathcal{MN}(X; M, U, Y^{-1}) \cdot \mathcal{W}(Y; V, \nu) \\
= ; & \sqrt{\frac{|Y|^n}{(2\pi)^{np} |U|^p}} \cdot \exp\left[-\frac{1}{2} \mathrm{tr}\left( Y (X-M)^\mathrm{T} , U^{-1} (X-M) \right) \right] \cdot \\
& \frac{1}{\Gamma_p \left( \frac{\nu}{2} \right)} \cdot \frac{1}{\sqrt{2^{\nu p} |V|^{\nu}}} \cdot |Y|^{(\nu-p-1)/2} \cdot \exp\left[ -\frac{1}{2} \mathrm{tr}\left( V^{-1} Y \right) \right] \\
= ; & \frac{1}{\sqrt{(2\pi)^{np} |U|^p |V|^{\nu}}} \cdot \frac{\sqrt{2^{-\nu p}}}{\Gamma_p \left( \frac{\nu}{2} \right)} \cdot |Y|^{(\nu+n-p-1)/2} \cdot \\
& \exp\left[-\frac{1}{2} \mathrm{tr}\left( Y \left[ (X-M)^\mathrm{T} , U^{-1} (X-M) + V^{-1} \right] \right) \right] ; .
\end{split}
$$