A Variational Approach to Robust Bayesian Filtering

A major challenge of applied Bayesian filtering is deriving estimates that are robust to misspecifications in the underlying statistical models, particularly when Bayes’ rule does not directly yield analytical posterior probability densities. Variational approaches present a promising alternative to “traditional,” closed-form Bayesian inference, wherein an approximate posterior is defined by minimizing the statistical dissimilarity to the conventional Bayesian posterior. There are, however, numerous realistic hurdles to defining an ideal dissimilarity measure, from which posteriors are derived using calculus of variations. This work utilizes a recently proposed framework, known as generalized variational inference (GVI), to define robust and tractable approximations of Bayes’ rule. The GVI framework is presented and accompanied by a novel sensitivity analysis and gradient-based solution method that is applicable for particle, Gaussian, and Gaussian mixture posterior representations. The proposed GVI filter is applied to a dynamic state estimation scenario with inaccurate measurement modeling and, via Monte Carlo analysis, demonstrates both statistical consistency and improvements over conventional robust filtering methods.