Bayesian reinforcement learning (BRL) offers a decision-theoretic solution for reinforcement learning. While "model-based" BRL algorithms have focused either on maintaining a posterior distribution on models or value functions and combining this with approximate dynamic programming or tree search, previous Bayesian "model-free" value function distribution approaches implicitly make strong assumptions or approximations. We describe a novel Bayesian framework, Inferential Induction, for correctly inferring value function distributions from data, which leads to the development of a new class of BRL algorithms. We design an algorithm, Bayesian Backwards Induction, with this framework. We experimentally demonstrate that the proposed algorithm is competitive with respect to the state of the art.