variational free energy

Is a functional of the mental model (probability distribution) $Q(x)$ used and the (choice of) data observed $y$.

Has three different, equivalent decompositions

$$\begin{array}{rl}F[Q,y]& =-{\mathbb{E}}_{Q(x)}[\ln P(y,x)]-H[Q(x)]\\ & =D_{KL}[Q(x)||P(x)]-{\mathbb{E}}_{Q(x)}[\ln P(y|x)]\\ & =D_{KL}[Q(x)||P(x|y)]-\ln P(y)\end{array}$$

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References

@parr2022b