Bayesian Learning Without Recall

This work is concerned with the problem of social learning. A network of agents attempt to learn some unknown state of the world which is drawn by nature from a finite set. The sources of information available to the agents are their own private observations as well as the beliefs of their neighboring agents in the social structure in which they interact. The rational approach to the problem of social learning is for each agent to refine her opinion, by using the Bayes' rule to incorporate her neighbors' beliefs and own private signals over time. However, repeated applications of the Bayes' rule in social networks can become computationally intractable, partly due to the fact that each agent needs to use her local data that is increasing over time and make inferences about the global network structure. The inherent complexity of the Bayesian approach has lead to the consideration of behavioral, non-Bayesian updates, where each agent instead of processing all new information in the optimum Bayesian way uses simple rules such as linear or convex combinations and forms updated beliefs. In this paper, it is shown that by replicating the rule that maps the agents' common prior to their Bayesian posterior at the initial time-step for all future steps, one can derive a memoryless Bayesian update that has a log-linear format and can serve as a theoretical justification for some of the non-Bayesian rules suggested in the literature. Convergence and learning under the derived rules are also investigated, and it is shown that while two communicating agents always learn the true state, the beliefs may fail to converge in the general network setting. The proposed approach has the advantage that while preserving some features of the Bayesian inference, they are made tractable.