Indirect Information Measure and Dynamic Learning

I study the robust predictions of optimal dynamic learning strategy when the measure of signal informativeness is an indirect measure from sequential cost minimization. I first show that an indirect information measure is supported by sequential cost minimization iff it satisfies: 1) monotonicity in Blackwell order, 2) sub-additivity in compound experiments and 3) linearity in mixing with no information. In a dynamic learning problem, if the cost of information depend on an indirect information measure and delay cost is fixed, then the optimal solution involves direct Poisson signals: arrival of signals directly suggest the optimal actions, and non-arrival of signal provides no information.