Adjoint operators as summary functions in amortized Bayesian inference frameworks

An important concern in seismic inverse problems is the large and varying size of observed data. The large size can cause computational cost concerns and its varying size (such as when changing receiver geometries) implies the need to rerun inference algorithms from scratch for each new observation. Motivated by these two problems, we take inspiration from the statistics literature which commonly relies on summary statistic of observed data. Summary statistic compress the observed data leaving only information needed for inference. In this work, we argue that the adjoint operator provides a natural candidate for a summary function in the context of physics-based inverse problems. We first mathematically show that for certain general assumptions transforming data under the adjoint operator defines a new conditional distribution which preserves the expectations of the original posterior. We validate our hypothesis by evaluating our framework in a learned amortized inference algorithm. Our seismic and medical synthetic experiments show computational gains and increased quality of point estimates using our framework. We discuss statistical metrics that show our learned posterior is well calibrated therefore justifying its use in uncertainty quantification.