  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
