An Information Metric of Similarity of Expectations

Abstract

Although its specification in economic models with uncertainty is critical to the results obtained, agents' information has not been defined in a way allowing isolation of its effects on other variables. In addition, information needs to be allowed to be optimally chosen by agents when, for example, endogenous shifts in information are a major part of the model. Both problems require a space of information with some appropriate mathematical structure depending only on the uncertainty in the model. Given a probability space, the space of information is the set of all sub- fields of events. The structure defined here is a metric derived from the topology defined to be the weakest one such that for any integrable random variable, the function mapping information to the expected value of the random variable conditional on that information is continuous. This metric is shown to be complete and separable with two additional useful properties. First, set-theoretic convergent (monotone) sequences of information converge, so martingale convergence is modelled by this metric. Second, the set of finite partitions of the state space is dense, so any information can be finitely approximated. The topology derived from this metric is more tractible than the one used by Allen (1982). A generalization of the topology is provided to handle special cases such as information consisting of state plus noise.