= Solution
\b[The printed minimization of the <evidence lower bound> has no solution.] To distinguish that literal result from the usual <variational inference> optimum, put $S=\sigma_1^{-2}+\sigma_2^{-2}>0$. The preceding <Kullback-Leibler divergence> satisfies
$$
\mathcal J'(\phi)=S\phi-\frac2\phi,\qquad \mathcal J''(\phi)=S+\frac2{\phi^2}>0.
$$
Its unique <global minimum> occurs at $\phi^2=2/S$. Since $\mathcal L=\log p(\mathcal D)-\mathcal J$, this is the unique <global maximum> of the <evidence lower bound>:
$$
\boxed{\phi_{\rm ELBO\,max}=\sqrt{\frac{2\sigma_1^2\sigma_2^2}{\sigma_1^2+\sigma_2^2}}}.
$$
The optimal common <variance> is the <harmonic mean> of the two posterior <variances>. However, $\mathcal J(\phi)\to+\infty$ as either $\phi\downarrow0$ or $\phi\to\infty$. Therefore
$$
\boxed{\inf_{\phi>0}\mathcal L(\phi)=-\infty,\qquad\text{no minimizing }\phi\text{ exists}}.
$$
The source PDF confirms the minimization wording; replacing it by maximization would give the finite value above, consistently with part (i).
Back to article page
