Aggregating probabilistic beliefs: Market mechanisms and graphical representations.

Abstract

A long-standing question in statistics is how best to aggregate the probabilistic beliefs of multiple agents. Related is the practical question of how to represent the combined beliefs efficiently. This dissertation reports contributions on both fronts. First, I formulate and analyze a securities market mechanism for aggregating beliefs. Equilibrium prices in the market are interpreted as consensus beliefs. Under homogeneity conditions regarding agents' utilities, the market mechanism corresponds with standard aggregation functions, and the market's outward behavior is indistinguishable from that of an individual. I also explore extensions to the model in which agents learn from prices and the market as a whole adapts over time. In certain circumstances, price fluctuations can be viewed as the Bayesian updates of a rational individual. Second, I investigate the use of graphical models, and in particular Bayesian networks, for representing aggregate beliefs. I derive two impossibility theorems which contradict widely held intuitions about how Bayesian networks should be combined. The so-called logarithmic opinion pool is shown to admit relatively concise encodings. I describe the nature of graphical structures consistent with this pooling function, and give algorithms for computing the logarithmic and linear opinion pools with, in some cases, exponential speedups over standard methods. Finally, I apply and extend the graphical modeling results to the market framework, deriving sufficient conditions for compact markets to be operationally complete. Such markets still induce a complete consensus distribution and support Pareto optimal allocations of risk, but with exponentially fewer securities than required for traditional completeness.