Mean-field variational inference ($q(x)={\prod }_i{q}_i(x_i)$)

Mean-field variational inference restricts the trial probability density function to $q(x)={\prod }_i{q}_i(x_i)$. With the other factors fixed, $q_j^{\ast }(x_j)\propto \exp ({\mathbb{E}}_{q_{-j}}\log \pi (x_j,X_{-j}))$, provided this expression has a finite positive normalization constant. Subtracting the objective at $q_j^{\ast }$ leaves $\mathrm{KL}(q_j\Vert q_j^{\ast })\ge 0$, proving the update is optimal when the terms are well defined.