This paper proposes a simple linear Bayesian approach to reinforcement learning. We show that
with an appropriate basis, a Bayesian linear Gaussian model is sufficient for accurately estimating
the system dynamics, and in particular when we allow for correlated noise. Policies are estimated
by first sampling a transition model from the current posterior, and then performing approximate dynamic programming on the sampled model. This form of approximate Thompson sampling results in
good exploration in unknown environments. The approach can also be seen as a Bayesian generalisation of least-squares policy iteration, where the empirical transition matrix is replaced with a sample from the posterio