# Tempering the Bayes Filter towards Improved Model-Based Estimation

arXiv: https://arxiv.org/abs/2512.02823 · HTML: https://mlsanity.com/paper/2512.02823/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 23 of its 45 references that are in the map.
- First posted on arXiv 2 Dec 2025.

## Builds on (most cited first, 23 of 23)

- [Bayesian Deep Learning and a Probabilistic Perspective of Generalization](https://mlsanity.com/paper/2002.08791/) (2020)
- [What Are Bayesian Neural Network Posteriors Really Like?](https://mlsanity.com/paper/2104.14421/) (2021)
- [A General Framework for Updating Belief Distributions](https://mlsanity.com/paper/1306.6430/) (2013)
- [Practical Deep Learning with Bayesian Principles](https://mlsanity.com/paper/1906.02506/) (2019)
- [Bayesian Neural Network Priors Revisited](https://mlsanity.com/paper/2102.06571/) (2021)
- [Inconsistency of Bayesian Inference for Misspecified Linear Models, and a Proposal for Repairing It](https://mlsanity.com/paper/1412.3730/) (2014)
- [A statistical theory of cold posteriors in deep neural networks](https://mlsanity.com/paper/2008.05912/) (2020)
- [On Uncertainty, Tempering, and Data Augmentation in Bayesian Classification](https://mlsanity.com/paper/2203.16481/) (2022)
- [Data augmentation in Bayesian neural networks and the cold posterior effect](https://mlsanity.com/paper/2106.05586/) (2021)
- [Cold Posteriors and Aleatoric Uncertainty](https://mlsanity.com/paper/2008.00029/) (2020)
- [Disentangling the Roles of Curation, Data-Augmentation and the Prior in the Cold Posterior Effect](https://mlsanity.com/paper/2106.06596/) (2021)
- [Outlier-robust Kalman Filtering through Generalised Bayes](https://mlsanity.com/paper/2405.05646/) (2024)
- [Variational Tempering](https://mlsanity.com/paper/1411.1810/) (2014)
- [Wasserstein Distributionally Robust Kalman Filtering](https://mlsanity.com/paper/1809.08830/) (2018)
- [On the Robustness to Misspecification of α-Posteriors and Their Variational Approximations](https://mlsanity.com/paper/2104.08324/) (2021)
- [How Tempering Fixes Data Augmentation in Bayesian Neural Networks](https://mlsanity.com/paper/2205.13900/) (2022)
- [Eigenvalue Corrected Noisy Natural Gradient](https://mlsanity.com/paper/1811.12565/) (2018)
- [Can a Confident Prior Replace a Cold Posterior?](https://mlsanity.com/paper/2403.01272/) (2024)
- [Hidden Markov Neural Networks](https://mlsanity.com/paper/2004.06963/) (2020)
- [The fine print on tempered posteriors](https://mlsanity.com/paper/2309.05292/) (2023)
- [Learning Hidden Markov Models for Linear Gaussian Systems with Applications to Event-based State Estimation](https://mlsanity.com/paper/2007.04587/) (2020)
- [Robust Data-Driven Moving Horizon Estimation for Linear Discrete-Time Systems](https://mlsanity.com/paper/2210.09017/) (2022)
- [End-to-End Training of a Neural HMM with Label and Transition Probabilities](https://mlsanity.com/paper/2310.02724/) (2023)

## Built on by (most cited first, 1 of 1)

- [UbiQVision: Quantifying Uncertainty in XAI for Image Recognition](https://mlsanity.com/paper/2512.20288/) (2025)

## Abstract (arXiv)

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.