For a Markov decision problem in which unknown transition probabilities serve as hidden state variables, we study the quality of two approximations to the decision rule of a Bayesian who each period updates his subjective distribu- tion over the transition probabilities by Bayes' law. The first is the usual ratio- nal expectations approximation that assumes that the decision maker knows the transition probabilities. The second approximation is a version of Kreps' (1998) anticipated utility model in which decision makers update using Bayes' law but optimize in a way that is myopic with respect to their updating of probabili- ties. For a range of consumption smoothing examples, the anticipated utility approximation outperforms the rational expectations approximation. The anticipated utility and Bayesian models augment market prices of risk relative to the rational expectations approximation.