  We study mean-field variational inference in a Bayesian linear model when the
sample size n is comparable to the dimension p. In high dimensions, the common
approach of minimizing a Kullback-Leibler divergence from the posterior
distribution, or maximizing an evidence lower bound, may deviate from the true
posterior mean and underestimate posterior uncertainty. We study instead
minimization of the TAP free energy, showing in a high-dimensional asymptotic
framework that it has a local minimizer which provides a consistent estimate of
the posterior marginals and may be used for correctly calibrated posterior
inference. Geometrically, we show that the landscape of the TAP free energy is
strongly convex in an extensive neighborhood of this local minimizer, which
under certain general conditions can be found by an Approximate Message Passing
(AMP) algorithm. We then exhibit an efficient algorithm that linearly converges
to the minimizer within this local neighborhood. In settings where it is
conjectured that no efficient algorithm can find this local neighborhood, we
prove analogous geometric properties for a local minimizer of the TAP free
energy reachable by AMP, and show that posterior inference based on this
minimizer remains correctly calibrated.
