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:<math>Q(\mathbf{Z}) = \prod_{i=1}^M q_i(\mathbf{Z}_i\mid \mathbf{X})</math>
It can be shown using the [[calculus of variations]] (hence the name "variational Bayes") that the "best" distribution <math>q_j^{*}</math> for each of the factors <math>q_j</math> (in terms of the distribution minimizing the KL divergence, as described above) satisfies<ref>{{cite web|last=Nguyen|first=Duy|title= AN IN DEPTH INTRODUCTION TO VARIATIONAL BAYES NOTE|url=https://papers.ssrn.com/sol3/papers.cfm?abstract_id=4541076|access-date=15 August 2023}}</ref>:
:<math>q_j^{*}(\mathbf{Z}_j\mid \mathbf{X}) = \frac{e^{\operatorname{E}_{q^*_{-j}} [\ln p(\mathbf{Z}, \mathbf{X})]}}{\int e^{\operatorname{E}_{q^*_{-j}} [\ln p(\mathbf{Z}, \mathbf{X})]}\, d\mathbf{Z}_j}</math>
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