  We analyze a problem of revealed preference given state-dependent stochastic
choice data in which the payoff to a decision maker (DM) only depends on their
beliefs about posterior means. Often, the DM must also learn about or pay
attention to the state; in applied work on this subject, a convenient
assumption is that the costs of such learning are linearly dependent in the
distribution over posterior means. We provide testable conditions to identify
whether this assumption holds. This allows for the use of information design
techniques to solve the DM's problem.
