Expectation Propagation for Poisson Data

TL;DR

This paper introduces an expectation propagation method for approximate Bayesian inference in Poisson models, particularly useful for imaging applications like PET, providing explicit formulas and demonstrating effectiveness on 2D images.

Contribution

It develops a novel expectation propagation algorithm for Poisson data with Laplace priors, including explicit update formulas and efficient quadrature techniques.

Findings

Effective Gaussian approximation for Poisson likelihoods

Explicit update formulas derived for iterative inference

Demonstrated on two-dimensional PET imaging data

Abstract

The Poisson distribution arises naturally when dealing with data involving counts, and it has found many applications in inverse problems and imaging. In this work, we develop an approximate Bayesian inference technique based on expectation propagation for approximating the posterior distribution formed from the Poisson likelihood function and a Laplace type prior distribution, e.g., the anisotropic total variation prior. The approach iteratively yields a Gaussian approximation, and at each iteration, it updates the Gaussian approximation to one factor of the posterior distribution by moment matching. We derive explicit update formulas in terms of one-dimensional integrals, and also discuss stable and efficient quadrature rules for evaluating these integrals. The method is showcased on two-dimensional PET images.