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As the ''log-[[model evidence|evidence]]'' <math>\log P(\mathbf{X})</math> is fixed with respect to <math>Q</math>, maximizing the final term <math>\mathcal{L}(Q)</math> minimizes the KL divergence of <math>Q</math> from <math>P</math>. By appropriate choice of <math>Q</math>, <math>\mathcal{L}(Q)</math> becomes tractable to compute and to maximize. Hence we have both an analytical approximation <math>Q</math> for the posterior <math>P(\mathbf{Z}\mid \mathbf{X})</math>, and a lower bound <math>\mathcal{L}(Q)</math> for the log-evidence <math>\log P(\mathbf{X})</math> (since the KL-divergence is non-negative).
The lower bound <math>\mathcal{L}(Q)</math> is known as the (negative) '''variational free energy''' in analogy with [[thermodynamic free energy]] because it can also be expressed as a negative energy <math>\operatorname{E}_{Q}[\log P(\mathbf{Z},\mathbf{X})]</math> plus the [[Entropy (information theory) |entropy]] of <math>Q</math>. The term <math>\mathcal{L}(Q)</math> is also known as '''Evidence Lower
=== Proofs ===
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