A Bayesian Update Method for Exponential Family Projection Filters with Non-Conjugate Likelihoods

Abstract

The projection filter is one of the approximations to the solution of the optimal filtering problem. It approximates the filtering density by projecting the dynamics of the square-root filtering density onto the tangent space of the square-root parametric density manifold. While the projection filters for exponential and mixture families with continuous measurement processes have been well studied, the continuous-discrete projection filtering algorithm for non-conjugate priors has received less attention. In this paper, we introduce a simple Riemannian optimization method to be used for the Bayesian update step in the continuous-discrete projection filter for exponential families. Specifically, we show that the Bayesian update can be formulated as an optimization problem of $\alpha$-R\'enyi divergence, where the corresponding Riemannian gradient can be easily computed. We demonstrate the effectiveness of the proposed method via two highly non-Gaussian Bayesian update problems.