This paper presents an inverse reinforcement learning (IRL) framework for Bayesian stopping time problems. By observing the actions of a Bayesian decision maker, we provide a necessary and sufficient condition to identify if these actions are consistent with optimizing a cost function; then we construct set valued estimates of the cost function. To achieve this IRL objective, we use novel ideas from Bayesian revealed preferences stemming from microeconomics. To illustrate our IRL scheme,we consider two important examples of stopping time problems, namely, sequential hypothesis testing and Bayesian search. Finally, for finite datasets, we propose an IRL detection algorithm and give finite sample bounds on its error probabilities. Also we discuss how to identify $\epsilon$-optimal Bayesian decision makers and perform IRL.