Essays on Probabilistic Belief Revision and Updating

Abstract

This dissertation studies probabilistic belief revision and updating. In the first chapter, we propose an axiomatic framework for belief revision when new information is of the form \event A is more likely than event B." Our decision maker need not have beliefs about the joint distribution of the signal she will receive and the payoff-relevant states. With the pseudo-Bayesian updating rule that we propose, the decision maker behaves as if she selects her posterior by minimizing Kullback-Leibler divergence (or, maximizing relative entropy) subject to the constraint that A is more likely than B. The two axioms that yield the representation are exchangeability and symmetry. Exchangeability is the requirement that the order in which the information arrives does not matter whenever the dierent pieces of information neither reinforce nor contradict each other. Symmetry requires that the decision maker be neutral when receiving two directly opposite signals. We show that pseudo-Bayesian agents are susceptible to recency bias and honest persuasion. We also show that the beliefs of pseudo-Bayesian agents communicating within a network will converge but that they may disagree in the limit even if the network is strongly connected.

In the second chapter, we focus on belief updating. We provide a framework for analyzing a range of well-documented non-Bayesian updating behaviors including base rate neglect, conjunction fallacy and disjunction fallacy. Our model links the concept of similarity in theoretical psychology with belief updating. We follow Kahneman and Tversky (1974) and assume that when attempting to respond to the question "How likely is A given B?", people mistakenly respond to the question "How representative is A of B (i.e. how similar are A and B)?" With a similarity-based updating rule the conditional probability of AUC given B might be less than the conditional probability of A given B if B and C have empty intersection;, simply because the pair of events AUC and B differ more from each other. Our axioms yield a Cobb-Douglas weighted geometric mean of P(AjB) and P(BjA) as the behavioral conditional probability of A given B, where P is the correct subjective probability and P(.|.) is the Bayesian conditional of P. That is, we provide a model of behavioral decision makers who confuse these two conditional probabilities but have correct unconditional beliefs. This combination of correct priors and incorrect updating occurs often since in many experiments subjects are explicitly given the relevant prior probabilities.

In the third chapter we present the tools that we developed through the course of writing the second chapter. In particular, we extend the Anscombe-Aumman theorem of subjective probability to allow for general mixture operations. Applying our theorem, we characterize quasi-linear means with a simple condition that resembles the classic independence axiom. We show that within the framework introduced in the second chapter, in addition to our Cobb-Douglas similarity index, the condition also enables us to recover Tversky's similarity index, which is a weighted harmonic mean of P(AjB) and P(BjA).