Estimating Private Beliefs of Bayesian Agents Based on Observed Decisions

We consider sequential stochastic decision problems in which, at each time instant, an agent optimizes its local utility by solving a stochastic program and, subsequently, announces its decision to the world. Given this action, we study the problem of estimating the agent's private belief (i.e., its posterior distribution over the set of states of nature based on its private observations). We demonstrate that it is possible to determine the set of private beliefs that are consistent with public data by leveraging techniques from inverse optimization. We further give a number of useful characterizations of this set; for example, tight bounds by solving a set of linear programs (under concave utility). As an illustrative example, we consider estimating the private belief of an investor in regime-switching portfolio allocation. Finally, our theoretical results are illustrated and evaluated in numerical simulations.