# Probabilistic Inference in Reinforcement Learning Done Right

arXiv: https://arxiv.org/abs/2311.13294 · HTML: https://mlsanity.com/paper/2311.13294/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- Builds on 57 of its 85 references that are in the map.
- First posted on arXiv 22 Nov 2023.

## Builds on (most cited first, 30 of 57)

- [Soft Actor-Critic: Off-Policy Maximum Entropy Deep Reinforcement Learning with a Stochastic Actor](https://mlsanity.com/paper/1801.01290/) (2018)
- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [Asynchronous Methods for Deep Reinforcement Learning](https://mlsanity.com/paper/1602.01783/) (2016)
- [Curiosity-driven Exploration by Self-supervised Prediction](https://mlsanity.com/paper/1705.05363/) (2017)
- [Prioritized Experience Replay](https://mlsanity.com/paper/1511.05952/) (2015)
- [Exploration by Random Network Distillation](https://mlsanity.com/paper/1810.12894/) (2018)
- [IMPALA: Scalable Distributed Deep-RL with Importance Weighted Actor-Learner Architectures](https://mlsanity.com/paper/1802.01561/) (2018)
- [Reinforcement Learning with Deep Energy-Based Policies](https://mlsanity.com/paper/1702.08165/) (2017)
- [CVXPY: A Python-Embedded Modeling Language for Convex Optimization](https://mlsanity.com/paper/1603.00943/) (2016)
- [Unifying Count-Based Exploration and Intrinsic Motivation](https://mlsanity.com/paper/1606.01868/) (2016)
- [Deep Exploration via Bootstrapped DQN](https://mlsanity.com/paper/1602.04621/) (2016)
- [Minimax Regret Bounds for Reinforcement Learning](https://mlsanity.com/paper/1703.05449/) (2017)
- [Is Q-learning Provably Efficient?](https://mlsanity.com/paper/1807.03765/) (2018)
- [Reinforcement Learning and Control as Probabilistic Inference: Tutorial and Review](https://mlsanity.com/paper/1805.00909/) (2018)
- [A Tutorial on Thompson Sampling](https://mlsanity.com/paper/1707.02038/) (2017)
- [Maximum a Posteriori Policy Optimisation](https://mlsanity.com/paper/1806.06920/) (2018)
- [A Theory of Regularized Markov Decision Processes](https://mlsanity.com/paper/1901.11275/) (2019)
- [Count-Based Exploration with Neural Density Models](https://mlsanity.com/paper/1703.01310/) (2017)
- [Provably Efficient Maximum Entropy Exploration](https://mlsanity.com/paper/1812.02690/) (2018)
- [(More) Efficient Reinforcement Learning via Posterior Sampling](https://mlsanity.com/paper/1306.0940/) (2013)
- [Randomized Prior Functions for Deep Reinforcement Learning](https://mlsanity.com/paper/1806.03335/) (2018)
- [Efficient Exploration via State Marginal Matching](https://mlsanity.com/paper/1906.05274/) (2019)
- [Bridging the Gap Between Value and Policy Based Reinforcement Learning](https://mlsanity.com/paper/1702.08892/) (2017)
- [A unified view of entropy-regularized Markov decision processes](https://mlsanity.com/paper/1705.07798/) (2017)
- [An Information-Theoretic Analysis of Thompson Sampling](https://mlsanity.com/paper/1403.5341/) (2014)
- [Learning to Optimize via Information-Directed Sampling](https://mlsanity.com/paper/1403.5556/) (2014)
- [Deep Exploration via Randomized Value Functions](https://mlsanity.com/paper/1703.07608/) (2017)
- [Behaviour Suite for Reinforcement Learning](https://mlsanity.com/paper/1908.03568/) (2019)
- [Taming the Noise in Reinforcement Learning via Soft Updates](https://mlsanity.com/paper/1512.08562/) (2015)
- [Almost Optimal Model-Free Reinforcement Learning via Reference-Advantage Decomposition](https://mlsanity.com/paper/2004.10019/) (2020)

## Built on by (most cited first, 11 of 11)

- [Maximum Likelihood Reinforcement Learning](https://mlsanity.com/paper/2602.02710/) (2026)
- [MaxInfoRL: Boosting exploration in reinforcement learning through information gain maximization](https://mlsanity.com/paper/2412.12098/) (2024)
- [Toward Efficient Exploration by Large Language Model Agents](https://mlsanity.com/paper/2504.20997/) (2025)
- [Thompson Sampling via Fine-Tuning of LLMs](https://mlsanity.com/paper/2510.13328/) (2025)
- [IL-SOAR : Imitation Learning with Soft Optimistic Actor cRitic](https://mlsanity.com/paper/2502.19859/) (2025)
- [LITE: Efficiently Estimating Gaussian Probability of Maximality](https://mlsanity.com/paper/2501.13535/) (2025)
- [Convergence Theorems for Entropy-Regularized and Distributional Reinforcement Learning](https://mlsanity.com/paper/2510.08526/) (2025)
- [What Type of Inference is Active Inference?](https://mlsanity.com/paper/2606.04935/) (2026)
- [Structured Cooperative Multi-Agent Reinforcement Learning: a Bayesian Network Perspective](https://mlsanity.com/paper/2510.09937/) (2025)
- [Provably avoiding over-optimization in Direct Preference Optimization without knowing the data distribution](https://mlsanity.com/paper/2602.06239/) (2026)
- [Evolving Robustness--Exploration Trade-off in Online Reinforcement Learning via Quantile Bayesian Risk MDPs](https://mlsanity.com/paper/2605.24345/) (2026)

## Abstract (arXiv)

A popular perspective in Reinforcement learning (RL) casts the problem as probabilistic inference on a graphical model of the Markov decision process (MDP). The core object of study is the probability of each state-action pair being visited under the optimal policy. Previous approaches to approximate this quantity can be arbitrarily poor, leading to algorithms that do not implement genuine statistical inference and consequently do not perform well in challenging problems. In this work, we undertake a rigorous Bayesian treatment of the posterior probability of state-action optimality and clarify how it flows through the MDP. We first reveal that this quantity can indeed be used to generate a policy that explores efficiently, as measured by regret. Unfortunately, computing it is intractable, so we derive a new variational Bayesian approximation yielding a tractable convex optimization problem and establish that the resulting policy also explores efficiently. We call our approach VAPOR and show that it has strong connections to Thompson sampling, K-learning, and maximum entropy exploration. We conclude with some experiments demonstrating the performance advantage of a deep RL version of VAPOR.