In This Exercise We Will Prove Theorem 11 11 The Proof Relies On The Following Propo 2528127

In this exercise, we will prove theorem 11.11. The proof relies on the following proposition, which characterizes the derivatives of an expectation relative to a Gibbs distribution.

Theorem 11.11

If Q(X) = 1/ZQ j ψj, then the potential ψj is a stationary point of the energy functional if and only if

The proof is straightforward algebraic manipulation and is left as an exercise. This theorem establishes a characterization of the fixed point as the difference between the expected value of logarithm of the original potentials and the expected value of the logarithm of the approximating potentials. The last term in equation is the energy functional F[Φ, Q], which is independent of the assignment cj; thus, as we discussed in the proof of corollary 11.5, we can absorb this term into the normalization constant of the distribution and ignore it.