Expectation Propagation for Poisson Data
Provides a new inference tool for inverse problems and imaging where Poisson data arises, but the method is incremental as it adapts existing EP framework to a specific likelihood-prior combination.
Developed an expectation propagation method for approximate Bayesian inference with Poisson likelihood and Laplace-type priors, achieving efficient Gaussian posterior approximations demonstrated on 2D PET images.
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.