  We present an iterative framework to improve the amortized approximations of
posterior distributions in the context of Bayesian inverse problems, which is
inspired by loop-unrolled gradient descent methods and is theoretically
grounded in maximally informative summary statistics. Amortized variational
inference is restricted by the expressive power of the chosen variational
distribution and the availability of training data in the form of joint data
and parameter samples, which often lead to approximation errors such as the
amortization gap. To address this issue, we propose an iterative framework that
refines the current amortized posterior approximation at each step. Our
approach involves alternating between two steps: (1) constructing a training
dataset consisting of pairs of summarized data residuals and parameters, where
the summarized data residual is generated using a gradient-based summary
statistic, and (2) training a conditional generative model -- a normalizing
flow in our examples -- on this dataset to obtain a probabilistic update of the
unknown parameter. This procedure leads to iterative refinement of the
amortized posterior approximations without the need for extra training data. We
validate our method in a controlled setting by applying it to a stylized
problem, and observe improved posterior approximations with each iteration.
Additionally, we showcase the capability of our method in tackling
realistically sized problems by applying it to transcranial ultrasound, a
high-dimensional, nonlinear inverse problem governed by wave physics, and
observe enhanced posterior quality through better image reconstruction with the
posterior mean.
