  Frequently, population studies feature pyramidally-organized data represented
using Hierarchical Bayesian Models (HBM) enriched with plates. These models can
become prohibitively large in settings such as neuroimaging, where a sample is
composed of a functional MRI signal measured on 300 brain locations, across 4
measurement sessions, and 30 subjects, resulting in around 1 million latent
parameters.Such high dimensionality hampers the usage of modern, expressive
flow-based techniques.To infer parameter posterior distributions in this
challenging class of problems, we designed a novel methodology that
automatically produces a variational family dual to a target HBM. This
variational family, represented as a neural network, consists in the
combination of an attention-based hierarchical encoder feeding summary
statistics to a set of normalizing flows. Our automatically-derived neural
network exploits exchangeability in the plate-enriched HBM and factorizes its
parameter space. The resulting architecture reduces by orders of magnitude its
parameterization with respect to that of a typical flow-based representation,
while maintaining expressivity.Our method performs inference on the specified
HBM in an amortized setup: once trained, it can readily be applied to a new
data sample to compute the parameters' full posterior.We demonstrate the
capability and scalability of our method on simulated data, as well as a
challenging high-dimensional brain parcellation experiment. We also open up
several questions that lie at the intersection between normalizing flows, SBI,
structured Variational Inference, and inference amortization.
