Before choosing among two actions with state-dependent payoffs, a Bayesian decision-maker with a finite memory sees a sequence of informative signals, ending each period with fixed chance. He summarizes information observed with a finite-state automaton. I characterize the optimal protocol as an equilibrium of a dynamic game of imperfect recall; a new player runs each memory state each period. Players act as if maximizing expected payoffs in a common finite action decision problem. I characterize equilibrium play with many multinomial signals. The optimal protocol rationalizes many behavioral phenomena, like �stickiness,� salience, confirmation bias, and belief polarization.