The Wald Problem and the Relation of Sequential Sampling and Ex-Ante Information Costs

We consider the two state sequential sampling problem of Wald (1945). We show that any distribution of posteriors can be attained and that the ex-ante cost of attaining any probability distribution over posteriors equals the expected change of the log likelihood ratio. With many states, not all distributions can be attained but we provide a characterization of those that can be attained and show that the characterization of ex-ante cost generalizes naturally to those that can be attained