  Inferring the parameters of a stochastic model based on experimental
observations is central to the scientific method. A particularly challenging
setting is when the model is strongly indeterminate, i.e. when distinct sets of
parameters yield identical observations. This arises in many practical
situations, such as when inferring the distance and power of a radio source (is
the source close and weak or far and strong?) or when estimating the amplifier
gain and underlying brain activity of an electrophysiological experiment. In
this work, we present hierarchical neural posterior estimation (HNPE), a novel
method for cracking such indeterminacy by exploiting additional information
conveyed by an auxiliary set of observations sharing global parameters. Our
method extends recent developments in simulation-based inference (SBI) based on
normalizing flows to Bayesian hierarchical models. We validate quantitatively
our proposal on a motivating example amenable to analytical solutions and then
apply it to invert a well known non-linear model from computational
neuroscience, using both simulated and real EEG data.
