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Tempering the Bayes Filter towards Improved Model-Based Estimation

Menno van Zutphen, Domagoj Herceg, Giannis Delimpaltadakis, Duarte J. Antunes

TL;DR

This work tackles state estimation under imperfect models in partially observable systems by introducing the tempered Bayes filter, a three-parameter posterior-tempering framework that includes likelihood tempering, full-posterior tempering, and belief tempering. The authors derive a recursive update that maintains the computational complexity of the classic Bayes filter and provide entropy-regularization and Bayes–MAP interpolation interpretations, plus a tempered Kalman filter for linear-Gaussian cases. They conduct gradient-based analysis to understand when tempering improves performance and demonstrate practical tuning methods. Empirical results in a partially observable grid-world show consistent predictive improvements over the standard Bayes filter, with insights into parameter roles and model-identification effects. Overall, the tempered Bayes framework offers a principled, efficient approach to filtering under model mismatch with broad applicability to HMMs and Kalman filtering contexts.

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 tempered Bayes filter, improving estimation performance through both of the aforementioned, and one newly introduced, 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 in the field of Bayesian inference that likelihood tempering affects the balance between prior and likelihood -- full-posterior tempering tunes the level of entropy in the final belief distribution. 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 achieves improved 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 propagation. Empirical results confirm that our method consistently improves predictive accuracy over the Bayes filter baseline.

Tempering the Bayes Filter towards Improved Model-Based Estimation

TL;DR

This work tackles state estimation under imperfect models in partially observable systems by introducing the tempered Bayes filter, a three-parameter posterior-tempering framework that includes likelihood tempering, full-posterior tempering, and belief tempering. The authors derive a recursive update that maintains the computational complexity of the classic Bayes filter and provide entropy-regularization and Bayes–MAP interpolation interpretations, plus a tempered Kalman filter for linear-Gaussian cases. They conduct gradient-based analysis to understand when tempering improves performance and demonstrate practical tuning methods. Empirical results in a partially observable grid-world show consistent predictive improvements over the standard Bayes filter, with insights into parameter roles and model-identification effects. Overall, the tempered Bayes framework offers a principled, efficient approach to filtering under model mismatch with broad applicability to HMMs and Kalman filtering contexts.

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 tempered Bayes filter, improving estimation performance through both of the aforementioned, and one newly introduced, 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 in the field of Bayesian inference that likelihood tempering affects the balance between prior and likelihood -- full-posterior tempering tunes the level of entropy in the final belief distribution. 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 achieves improved 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 propagation. Empirical results confirm that our method consistently improves predictive accuracy over the Bayes filter baseline.

Paper Structure

This paper contains 33 sections, 6 theorems, 73 equations, 4 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Our contribution
  3. Preliminaries
  4. Hidden Markov Models
  5. Filtering
  6. Bayes Filter
  7. Maximum a Posteriori Filter
  8. Model Mismatch and Estimator Performance
  9. Tempering the Bayes filter
  10. Tempering interpretation
  11. A complete interpretation of posterior tempering
  12. Complete distributional tempering and Gibbs distributions
  13. Recovering the MAP filter
  14. Recursive form
  15. Gradient Analysis
  16. ...and 18 more sections

Key Result

Proposition 1

The tempered posterior, as defined in eq:canonically_tempered_posterior, is the unique maximizing argument of the entropy regularized, rebalanced ELBO objective, as where $\operatorname{KL}(q\|p)$ is the Kullback-Leibler divergence of distribution $q(x_{0:k})$ from the prior distribution $p(x_{0:k})$, and $H(q)$ its Shannon entropy.

Figures (4)

  • Figure 1: Filtering performance as a function of the amount of available data used for identification under unablated (standard) and ablated conditions.
  • Figure 2: Tuned lambda values as a function of the amount of available data under unablated (standard) and ablated conditions.
  • Figure 3: Log of the NLL cost landscape across values of $\lambda$ after training on the 195-data-points data set. The single $\lambda$-parameter value that is kept constant in each of the figures is kept at its associated optimal value.
  • Figure 4: The true, identified and tempered models at the optimal $\lambda^{*}$ at 195 data points.

Theorems & Definitions (6)

  • Proposition 1: Regularized ELBO maximization
  • Lemma 1: Gibbs distribution
  • Lemma 2: $\lambda_{B}=1/\lambda_{P}$ yields $L_{p}$-norm
  • Theorem 1: Recovering the MAP-filter based belief
  • Proposition 2: Tempered Bayes filter recursive form
  • Theorem 2: NLL Derivative w.r.t. $\lambda$