  In model-based reinforcement learning, the transition matrix and reward
vector are often estimated from random samples subject to noise. Even if the
estimated model is an unbiased estimate of the true underlying model, the value
function computed from the estimated model is biased. We introduce an operator
shifting method for reducing the error introduced by the estimated model. When
the error is in the residual norm, we prove that the shifting factor is always
positive and upper bounded by $1+O\left(1/n\right)$, where $n$ is the number of
samples used in learning each row of the transition matrix. We also propose a
practical numerical algorithm for implementing the operator shifting.
