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Multivariate normal distribution
--------------------------------
.. math::
\mathbf{x} &\sim \mathcal{N}(\boldsymbol{\mu}, \mathbf{\Lambda}),
.. math::
\mathbf{x},\boldsymbol{\mu} \in \mathbb{R}^{D},
\quad \mathbf{\Lambda} \in \mathbb{R}^{D \times D},
\quad \mathbf{\Lambda} \text{ symmetric positive definite}
.. math::
\log\mathcal{N}( \mathbf{x} | \boldsymbol{\mu}, \mathbf{\Lambda} )
&=
- \frac{1}{2} \mathbf{x}^{\mathrm{T}} \mathbf{\Lambda} \mathbf{x}
+ \mathbf{x}^{\mathrm{T}} \mathbf{\Lambda} \boldsymbol{\mu}
- \frac{1}{2} \boldsymbol{\mu}^{\mathrm{T}} \mathbf{\Lambda}
\boldsymbol{\mu}
+ \frac{1}{2} \log |\mathbf{\Lambda}|
- \frac{D}{2} \log (2\pi)
.. math::
\mathbf{u} (\mathbf{x})
&=
\left[ \begin{matrix}
\mathbf{x}
\\
\mathbf{xx}^{\mathrm{T}}
\end{matrix} \right]
\\
\boldsymbol{\phi} (\boldsymbol{\mu}, \mathbf{\Lambda})
&=
\left[ \begin{matrix}
\mathbf{\Lambda} \boldsymbol{\mu}
\\
- \frac{1}{2} \mathbf{\Lambda}
\end{matrix} \right]
\\
\boldsymbol{\phi}_{\boldsymbol{\mu}} (\mathbf{x}, \mathbf{\Lambda})
&=
\left[ \begin{matrix}
\mathbf{\Lambda} \mathbf{x}
\\
- \frac{1}{2} \mathbf{\Lambda}
\end{matrix} \right]
\\
\boldsymbol{\phi}_{\mathbf{\Lambda}} (\mathbf{x}, \boldsymbol{\mu})
&=
\left[ \begin{matrix}
- \frac{1}{2} \mathbf{xx}^{\mathrm{T}}
+ \frac{1}{2} \mathbf{x}\boldsymbol{\mu}^{\mathrm{T}}
+ \frac{1}{2} \boldsymbol{\mu}\mathbf{x}^{\mathrm{T}}
- \frac{1}{2} \boldsymbol{\mu\mu}^{\mathrm{T}}
\\
\frac{1}{2}
\end{matrix} \right]
\\
g (\boldsymbol{\mu}, \mathbf{\Lambda})
&=
- \frac{1}{2} \operatorname{tr}(\boldsymbol{\mu\mu}^{\mathrm{T}}
\mathbf{\Lambda} )
+ \frac{1}{2} \log |\mathbf{\Lambda}|
\\
g_{\boldsymbol{\phi}} (\boldsymbol{\phi})
&=
\frac{1}{4} \boldsymbol{\phi}^{\mathrm{T}}_1 \boldsymbol{\phi}^{-1}_2
\boldsymbol{\phi}_1
+ \frac{1}{2} \log | -2 \boldsymbol{\phi}_2 |
\\
f(\mathbf{x})
&= - \frac{D}{2} \log(2\pi)
\\
\overline{\mathbf{u}} (\boldsymbol{\phi})
&=
\left[ \begin{matrix}
- \frac{1}{2} \boldsymbol{\phi}^{-1}_2 \boldsymbol{\phi}_1
\\
\frac{1}{4} \boldsymbol{\phi}^{-1}_2 \boldsymbol{\phi}_1
\boldsymbol{\phi}^{\mathrm{T}}_1 \boldsymbol{\phi}^{-1}_2
- \frac{1}{2} \boldsymbol{\phi}^{-1}_2
\end{matrix} \right]
|
