Homotopic Gaussian Mixture Filtering for Applied Bayesian Inference

Abstract

Bayes' rule, although a powerful framework for performing state estimation, is often intractable for real-world, nonlinear dynamic systems. As a result, estimation algorithms typically rely on a simplifying assumption, such as the linearity of the measurement model or Gaussianity of the likelihood function. For nonlinear, non-Gaussian systems, these approximations can introduce statistical inconsistencies into the underlying estimator. To mitigate approximation errors, a homotopic scheme is proposed for Bayesian inference. The approach partitions Bayes' rule into smaller, incremental corrections, over which linear and/or Gaussian assumptions are more accurate. The incremental update is limited to zero, yielding a system of first-order differential equations governing the update from prior to posterior for the weights, means, and covariances of a finite Gaussian mixture approximation. The proposed method is shown to be generalizable to both non-Gaussian likelihoods and likelihoods with non-Euclidean support. The homotopic filter is applied to a dynamic state estimation scenario and noticeable improvements over traditional Bayesian filtering techniques (e.g., the unscented Kalman filter and conventional Gaussian mixture filtering) are observed.