#harmonic-mean

The true posterior is $p(x_1, x_2) = \mathcal{N}(x_1; 0, \sigma_1^2)\,\mathcal{N}(x_2; 0, \sigma_2^2)$ with $\sigma_1^2 = 1$ and $\sigma_2^2 = 4$. The variational family is isotropic, $q(x_1, x_2) = \mathcal{N}(x_1; 0, \lambda^2)\,\mathcal{N}(x_2; 0, \lambda^2)$. Which value of $\lambda^2$ minimises $\mathrm{KL}(q \,\|\, p)$?