Before revision: In this paper, I first showed that an indirect information measure is supported by expected learning cost minimizationif and only if it satisfies: 1. monotonicity in Blackwell order, 2. sub-additivity in compound experiment and 3. linearity in mixing with no information. Then I studied a dynamic information acquisition problem with flexible design of information dynamics, costly waiting and costly information. When flow information measure satisfies the three conditions, dynamic problem can be solved in two steps: solving a static rational inattention problem, and implementing optimal learning dynamics. The optimal solution involves stationary Poisson direct signals: arrival of signal directly suggests optimal action, and non-arrival of signal provides no information .
After revision: In this paper, I first showed that an indirect information measure is supported by expected learning cost minimizationif and only if it satisfies: 1. monotonicity in Blackwell order, 2. sub-additivity condition, together with monotonicity equivalently characterizes the class of indirect cost functions generated from any direct information cost. Adding an extra (uniform) posterior separability condition equivalently characterizes the indirect cost generated from any direct cost favoring incremental evidences. We also provide the necessary and sufficient condition when prior independent direct cost generates posterior separable indirect cost .