Denote the pdf of the \(\mathscr{N}(\mathbf{0}, \boldsymbol{\Sigma})\) distribution by \(\varphi \Sigma(\cdot)\), and let

\[ \mathscr{D}\left(\boldsymbol{\mu}_{0} \boldsymbol{\Sigma}_{0} \mid \boldsymbol{\mu}_{1}, \boldsymbol{\Sigma}_{1}\right)=\int_{\mathbb{R}^{d}} \varphi_{\Sigma_{0}}\left(\boldsymbol{x}-\boldsymbol{\mu}_{0}\right) \ln \frac{\varphi_{\Sigma_{0}}\left(\boldsymbol{x}-\boldsymbol{\mu}_{0}\right)}{\varphi_{\Sigma_{1}}\left(\boldsymbol{x}-\boldsymbol{\mu}_{1}\right)} \mathrm{d} \boldsymbol{x} \]

be the Kullback-Leibler divergence between the densities of the \(\mathscr{N}\left(\boldsymbol{\mu}_{0}, \boldsymbol{\Sigma}_{0}\right)\) and \(\mathscr{N}\left(\boldsymbol{\mu}_{1}, \boldsymbol{\Sigma}_{1}\right)\) distributions on \(\mathbb{R}^{d}\). Show that

\(2 \mathscr{D}\left(\boldsymbol{\mu}_{0} \boldsymbol{\Sigma}_{0} \mid \boldsymbol{\mu}_{1}, \boldsymbol{\Sigma}_{1}\right)=\operatorname{tr}\left(\boldsymbol{\Sigma}_{1}^{-1} \boldsymbol{\Sigma}_{0}\right)-\ln \left|\boldsymbol{\Sigma}_{1}^{-1} \boldsymbol{\Sigma}_{0}\right|+\left(\boldsymbol{\mu}_{1}-\boldsymbol{\mu}_{0}\right)^{\top} \boldsymbol{\Sigma}_{1}^{-1}\left(\boldsymbol{\mu}_{1}-\boldsymbol{\mu}_{0}\right)-d\)

Hence, deduce the formula in (B.22).