Exercise 21.6 - Alternative derivation of the mean field updates for the Ising model

The energy for the Ising model is given by:

where the first term corresponds to the interaction field and the second to the external field. The variational free energy (VFE) is the expectation of the energy after replacing each $x_i$ with a Bernoulli random variable whose mean is ${\mu }_i$. The dependency of the VEF on ${\mu }_i$ is:

from which we can derive the posterior margin as well as the new mean, which are identical to (21.46)-(21.48). Thus the update (21.50) remains the same.