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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.

Write $\pi (x)=\mu (x\mid y)$. Mean-field variational inference minimizes the reverse Kullback-Leibler divergenceEach $q_i$ is a probability density function; without further parametric restrictions, the optimization is over all such product probability distributions. Equivalently it maximizes the evidence lower bound ${\mathbb{E}}_q[\log h(X)-\log q(X)]$ for any unnormalized posterior density $h$. Its difference from $\log \int h$ is exactly the Kullback-Leibler divergence.

Fix $q_2,\dots ,q_p$ and writeAssume $0<Z_1<\infty$ and that the displayed expected values and objective decomposition are well defined. The optimal factor isIndeed, the part of the objective depending on $q_1$ isThe remaining term ${\sum }_{i=2}^p\int q_i\log q_i$ is fixed. Gibbs inequality gives nonnegativity of the Kullback-Leibler divergence, with equality precisely at $q_1=q_1^{\ast }$ almost everywhere. This proves global optimality of that coordinate update; it does not assert global optimality of a sequence of coordinate updates for the full nonconvex product-family problem.

The printed upper index $k$ in the list of remaining coordinates is inconsistent with the dimension $p$; the natural interpretation is $k=p$. Also, if the exponential expression has zero or infinite normalizing constant, or the expected values are undefined, the usual coordinate formula needs additional support or integrability hypotheses. A restricted parametric factor family need not contain this unrestricted optimal factor.