We consider a decision maker who first chooses a costly information structure, and then chooses an action after receiving a signal. The choice of action is observed, but the choice of information is not. Moreover, the state of nature is considered unobservable as well. In such a situation, the choice of action appears random and cannot be conditioned on the states from an outside analyst's point of view. We show that, given only stochastic choice from menus of actions, an analyst can identify the agent's taste (risk attitude), prior belief, and information cost function. In addition, we discuss the behavioral implications of our model that are weaker than some key properties of random expected utility models because of the endogeneity and hence menu-dependence of private information. Finally, we provide necessary and sufficient conditions for stochastic choice to be rationalized by our model.

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