Variational Bayesian methods: Difference between revisions

Revision as of 09:49, 18 November 2024

Revision as of 09:52, 18 November 2024

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In this case, the global minimizer <math>Q^{*}(\mathbf{Z}) = q^{*}(\mathbf{Z}_1\mid\mathbf{Z}_2)q^{*}(\mathbf{Z}_2) = q^{*}(\mathbf{Z}_2\mid\mathbf{Z}_1)q^{*}(\mathbf{Z}_1),</math> with <math>\mathbf{Z}=\{\mathbf{Z_1},\mathbf{Z_2}\},</math> can be found as follows:<ref name=Tran2018/>

:<math> q^{*}(\mathbf{Z}_2)

\begin{array}{rl}

= \frac{P(\mathbf{X})}{\zeta(\mathbf{X})}\frac{P(\mathbf{Z}_2\mid\mathbf{X})}{\exp(D_{\mathrm{KL}}(q^{*}(\mathbf{Z}_1\mid\mathbf{Z}_2)\parallel P(\mathbf{Z}_1\mid\mathbf{Z}_2,\mathbf{X})))}

q^{*}(\mathbf{Z}_2)

= \frac{1}{\zeta(\mathbf{X})}\exp\mathbb{E}_{q^{*}(\mathbf{Z}_1\mid\mathbf{Z}_2)}\left(\log\frac{P(\mathbf{Z},\mathbf{X})}{q^{*}(\mathbf{Z}_1\mid\mathbf{Z}_2)}\right),</math>

&= \frac{P(\mathbf{X})}{\zeta(\mathbf{X})}\frac{P(\mathbf{Z}_2\mid\mathbf{X})}{\exp(D_{\mathrm{KL}}(q^{*}(\mathbf{Z}_1\mid\mathbf{Z}_2)\parallel P(\mathbf{Z}_1\mid\mathbf{Z}_2,\mathbf{X})))} \\

&= \frac{1}{\zeta(\mathbf{X})}\exp\mathbb{E}_{q^{*}(\mathbf{Z}_1\mid\mathbf{Z}_2)}\left(\log\frac{P(\mathbf{Z},\mathbf{X})}{q^{*}(\mathbf{Z}_1\mid\mathbf{Z}_2)}\right),</math>

\end{array}</math>

in which the normalizing constant is:

:<math>\zeta(\mathbf{X})

\begin{array}{rl}

=P(\mathbf{X})\int_{\mathbf{Z}_2}\frac{P(\mathbf{Z}_2\mid\mathbf{X})}{\exp(D_{\mathrm{KL}}(q^{*}(\mathbf{Z}_1\mid\mathbf{Z}_2)\parallel P(\mathbf{Z}_1\mid\mathbf{Z}_2,\mathbf{X})))}

\zeta(\mathbf{X})

= \int_{\mathbf{Z}_{2}}\exp\mathbb{E}_{q^{*}(\mathbf{Z}_1\mid\mathbf{Z}_2)}\left(\log\frac{P(\mathbf{Z},\mathbf{X})}{q^{*}(\mathbf{Z}_1\mid\mathbf{Z}_2)}\right).</math>

&=P(\mathbf{X})\int_{\mathbf{Z}_2}\frac{P(\mathbf{Z}_2\mid\mathbf{X})}{\exp(D_{\mathrm{KL}}(q^{*}(\mathbf{Z}_1\mid\mathbf{Z}_2)\parallel P(\mathbf{Z}_1\mid\mathbf{Z}_2,\mathbf{X})))} \\

&= \int_{\mathbf{Z}_{2}}\exp\mathbb{E}_{q^{*}(\mathbf{Z}_1\mid\mathbf{Z}_2)}\left(\log\frac{P(\mathbf{Z},\mathbf{X})}{q^{*}(\mathbf{Z}_1\mid\mathbf{Z}_2)}\right).</math>

\end{array}</math>

The term <math>\zeta(\mathbf{X})</math> is often called the [[model evidence|evidence]] lower bound ('''ELBO''') in practice, since <math>P(\mathbf{X})\geq\zeta(\mathbf{X})=\exp(\mathcal{L}(Q^{*}))</math>,<ref name=Tran2018/> as shown above.