# Bayesian Bellman Operators

arXiv: https://arxiv.org/abs/2106.05012 · HTML: https://mlsanity.com/paper/2106.05012/

## Key facts

- Cited by 16 papers in the map (data through 14 Oct 2026).
- Builds on 29 of its 91 references that are in the map.
- First posted on arXiv 9 Jun 2021.

## Builds on (most cited first, 29 of 29)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Soft Actor-Critic: Off-Policy Maximum Entropy Deep Reinforcement Learning with a Stochastic Actor](https://mlsanity.com/paper/1801.01290/) (2018)
- [Dropout as a Bayesian Approximation: Representing Model Uncertainty in Deep Learning](https://mlsanity.com/paper/1506.02142/) (2015)
- [Auto-Encoding Variational Bayes](https://mlsanity.com/paper/1312.6114/) (2013)
- [Simple and Scalable Predictive Uncertainty Estimation using Deep Ensembles](https://mlsanity.com/paper/1612.01474/) (2016)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [A Survey on Bias and Fairness in Machine Learning](https://mlsanity.com/paper/1908.09635/) (2019)
- [Soft Actor-Critic Algorithms and Applications](https://mlsanity.com/paper/1812.05905/) (2018)
- [DeepMind Control Suite](https://mlsanity.com/paper/1801.00690/) (2018)
- [A Distributional Perspective on Reinforcement Learning](https://mlsanity.com/paper/1707.06887/) (2017)
- [Deep Reinforcement Learning for Autonomous Driving: A Survey](https://mlsanity.com/paper/2002.00444/) (2020)
- [Reinforcement Learning in Healthcare: A Survey](https://mlsanity.com/paper/1908.08796/) (2019)
- [Noisy Networks for Exploration](https://mlsanity.com/paper/1706.10295/) (2017)
- [Randomized Prior Functions for Deep Reinforcement Learning](https://mlsanity.com/paper/1806.03335/) (2018)
- [VariBAD: A Very Good Method for Bayes-Adaptive Deep RL via Meta-Learning](https://mlsanity.com/paper/1910.08348/) (2019)
- [Learning Continuous Control Policies by Stochastic Value Gradients](https://mlsanity.com/paper/1510.09142/) (2015)
- [Deep Exploration via Randomized Value Functions](https://mlsanity.com/paper/1703.07608/) (2017)
- [A Deep Reinforcement Learning Framework for the Financial Portfolio Management Problem](https://mlsanity.com/paper/1706.10059/) (2017)
- [Better Exploration with Optimistic Actor-Critic](https://mlsanity.com/paper/1910.12807/) (2019)
- [The Uncertainty Bellman Equation and Exploration](https://mlsanity.com/paper/1709.05380/) (2017)
- [Stochastic Variance Reduction Methods for Policy Evaluation](https://mlsanity.com/paper/1702.07944/) (2017)
- [Market Making via Reinforcement Learning](https://mlsanity.com/paper/1804.04216/) (2018)
- [A Kernel Loss for Solving the Bellman Equation](https://mlsanity.com/paper/1905.10506/) (2019)
- [Leverage the Average: an Analysis of KL Regularization in RL](https://mlsanity.com/paper/2003.14089/) (2020)
- [BBQ-Networks: Efficient Exploration in Deep Reinforcement Learning for Task-Oriented Dialogue Systems](https://mlsanity.com/paper/1711.05715/) (2017)
- [Randomized Value Functions via Multiplicative Normalizing Flows](https://mlsanity.com/paper/1806.02315/) (2018)
- [Geometric Insights into the Convergence of Nonlinear TD Learning](https://mlsanity.com/paper/1905.12185/) (2019)
- [Two Timescale Stochastic Approximation with Controlled Markov noise and Off-policy temporal difference learning](https://mlsanity.com/paper/1503.09105/) (2015)
- [Fourier Policy Gradients](https://mlsanity.com/paper/1802.06891/) (2018)

## Built on by (most cited first, 16 of 16)

- [MaxInfoRL: Boosting exploration in reinforcement learning through information gain maximization](https://mlsanity.com/paper/2412.12098/) (2024)
- [An Adaptive Deep RL Method for Non-Stationary Environments with Piecewise Stable Context](https://mlsanity.com/paper/2212.12735/) (2022)
- [Model-Based Uncertainty in Value Functions](https://mlsanity.com/paper/2302.12526/) (2023)
- [Why Target Networks Stabilise Temporal Difference Methods](https://mlsanity.com/paper/2302.12537/) (2023)
- [Accelerating Value Iteration with Anchoring](https://mlsanity.com/paper/2305.16569/) (2023)
- [Diverse Projection Ensembles for Distributional Reinforcement Learning](https://mlsanity.com/paper/2306.07124/) (2023)
- [Uncertainty-based Offline Variational Bayesian Reinforcement Learning for Robustness under Diverse Data Corruptions](https://mlsanity.com/paper/2411.00465/) (2024)
- [Iterated Q-Network: Beyond One-Step Bellman Updates in Deep Reinforcement Learning](https://mlsanity.com/paper/2403.02107/) (2024)
- [Deep Exploration with PAC-Bayes](https://mlsanity.com/paper/2402.03055/) (2024)
- [Bayesian Exploration Networks](https://mlsanity.com/paper/2308.13049/) (2023)
- [Model-Based Epistemic Variance of Values for Risk-Aware Policy Optimization](https://mlsanity.com/paper/2312.04386/) (2023)
- [Parameterized Projected Bellman Operator](https://mlsanity.com/paper/2312.12869/) (2023)
- [Isoperimetry is All We Need: Langevin Posterior Sampling for RL with Sublinear Regret](https://mlsanity.com/paper/2412.20824/) (2024)
- [Universal Value-Function Uncertainties](https://mlsanity.com/paper/2505.21119/) (2025)
- [Concurrent Learning with Aggregated States via Randomized Least Squares Value Iteration](https://mlsanity.com/paper/2501.13394/) (2025)
- [Bayesian learning for the stochastic shortest path problem](https://mlsanity.com/paper/2606.04845/) (2026)

## Abstract (arXiv)

We introduce a novel perspective on Bayesian reinforcement learning (RL); whereas existing approaches infer a posterior over the transition distribution or Q-function, we characterise the uncertainty in the Bellman operator. Our Bayesian Bellman operator (BBO) framework is motivated by the insight that when bootstrapping is introduced, model-free approaches actually infer a posterior over Bellman operators, not value functions. In this paper, we use BBO to provide a rigorous theoretical analysis of model-free Bayesian RL to better understand its relationshipto established frequentist RL methodologies. We prove that Bayesian solutions are consistent with frequentist RL solutions, even when approximate inference isused, and derive conditions for which convergence properties hold. Empirically, we demonstrate that algorithms derived from the BBO framework have sophisticated deep exploration properties that enable them to solve continuous control tasks at which state-of-the-art regularised actor-critic algorithms fail catastrophically