= Mean-field variational inference
{title2=$q(x)=\prod_iq_i(x_i)$}
<Mean-field variational inference> restricts the trial <probability density function> to $q(x)=\prod_iq_i(x_i)$. With the other factors fixed, $q_j^*(x_j)\propto\exp(\mathbb E_{q_{-j}}\log\pi(x_j,X_{-j}))$, provided this expression has a finite positive <normalization constant>. Subtracting the objective at $q_j^*$ leaves $\operatorname{KL}(q_j\Vert q_j^*)\geq0$, proving the update is optimal when the terms are well defined.
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