Variational Reformulation of Bayesian Inverse Problems

TL;DR

This paper introduces a variational reformulation of Bayesian inverse problems, transforming Bayesian inference into an optimization problem that balances theoretical rigor with computational efficiency.

Contribution

It presents a novel variational approach that reformulates Bayesian inverse problems as an optimization task, merging Bayesian accuracy with practical computational methods.

Findings

Provides a new variational framework for Bayesian inverse problems

Achieves computational efficiency comparable to classical optimization methods

Maintains the theoretical advantages of Bayesian inference

Abstract

The classical approach to inverse problems is based on the optimization of a misfit function. Despite its computational appeal, such an approach suffers from many shortcomings, e.g., non-uniqueness of solutions, modeling prior knowledge, etc. The Bayesian formalism to inverse problems avoids most of the difficulties encountered by the optimization approach, albeit at an increased computational cost. In this work, we use information theoretic arguments to cast the Bayesian inference problem in terms of an optimization problem. The resulting scheme combines the theoretical soundness of fully Bayesian inference with the computational efficiency of a simple optimization.