Bayesian inverse problems using homotopy

TL;DR

This paper introduces a homotopy-based method for Bayesian inverse problems that transforms the challenge into solving differential equations to find optimal density parameters within the exponential family.

Contribution

It proposes a novel homotopy approach that converts Bayesian inverse problems into differential equations for efficient posterior approximation.

Findings

Effective in numerical examples

Provides explicit differential equations for parameter evolution

Achieves close posterior approximation within exponential family

Abstract

In solving Bayesian inverse problems, it is often desirable to use a common density parameterization to denote the prior and posterior. Typically we seek a density from the same family as the prior which closely approximates the true posterior. As one of the most important classes of distributions in statistics, the exponential family is considered as the parameterization. The optimal parameter values for representing the approximated posterior are achieved by minimizing the deviation between the parameterized density and a homotopy that deforms the prior density into the posterior density. Rather than trying to solve the original problem, it is exactly converted into a corresponding system of explicit ordinary first-order differential equations. Solving this system over a finite 'time' interval yields the desired optimal density parameters. This method is proven to be effective by some numerical examples.