Tempering the Bayes Filter Towards Improved Model-Based Estimation

Abstract

Model-based filtering is often carried out while subject to an imperfect model, as learning partially-observable stochastic systems remains a challenge. Recent work on Bayesian inference found that tempering the likelihood or full posterior of an imperfect model can improve predictive accuracy, as measured by expected negative log likelihood. In this paper, we develop the discrete-time tempered Bayes filter, improving performance of the belief-distribution estimate through both of the aforementioned, and one newly introduced, tempering modalities. The result admits a recursive implementation with a computational complexity no higher than that of the original Bayes filter. Our analysis reveals that — besides the well-known fact that likelihood tempering affects the balance between prior and likelihood — full-posterior tempering tunes the belief-distribution entropy level. We further find that a region of the tempering space can be understood as interpolating between the Bayes- and MAP filters, recovering these as special cases. Analytical results further establish conditions under which a tempered Bayes filter improves predictive performance. Specializing the results to the linear Gaussian case, we obtain the tempered Kalman filter. In this context, we interpret how the parameters affect the Kalman state estimate and covariance. Empirical results confirm that our method consistently improves predictive accuracy over the Bayes filter baseline.