Toward a Market Model for Bayesian

We present a methodology for representing probabilistic relationships in a general-equilibrium economic model. Speciically, we deene a precise mapping from a Bayesian network with binary nodes to a market price system where consumers and producers trade in uncertain propositions. We demonstrate the correspondence between the equilibrium prices of goods in this economy and the probabilities represented by the Bayesian network. A computational market model such as this may provide a useful framework for investigations of belief aggregation, distributed probabilistic inference, resource allocation under uncertainty, and other problems of decentralized uncertainty. A principled market model for Bayesian inference would potentially address a variety of important and interesting problems of distributed uncertain reasoning. Although the particular contributions of our work to date do not deliver on these general problems, we present the big picture at the outset as underlying motivation for our speciic developments. There are several reasons one might want to build markets for probabilistic reasoning. Researchers in uncertain reasoning are likely to be acquainted with the rst two aims enumerated in sections below. The third may be more familiar to those with some background in economics. The approach we follow in this work owes much to both economic theory (especially general equilibrium under uncertainty) and uncertain reasoning technology (especially Bayesian networks). 1.1 AGGREGATING BELIEFS Given several agents with incompatible beliefs, how can we aggregate their individual beliefs into a characterization of the group's beliefs? This is a classical question in uncertain reasoning, one that has eluded deenitive answers despite the research attention it has attracted (Genest and Zidek 1986). Although we doubt that deenitive solutions are forthcoming from any quarter, we point out that market mechanisms of various sorts are widely used in uncertain contexts, and their function as aggregators of belief are well-recognized. For example, the price of a stock represents the \market evaluation" of the expected present value of future dividends, and odds in a horse race aggregate the bettors' beliefs about the winning horse's identity. Despite their commonality and well-developed underlying theory, there appear to have been few attempts in the uncertain reasoning community at principled application of market ideas for belief aggregation. Specifically , to our knowledge, nobody has proposed a comprehensive market architecture for belief aggregation. 1 1.2 DISTRIBUTED PROBABILISTIC INFERENCE AND DECISION UNDER UNCERTAINTY Belief aggregation is an instance of the more general problem of coordinating belief and decision among a collection of agents. …