We consider the task of aggregating beliefs of several experts. We assume that these beliefs are represented as probability distributions. We argue that the evaluation of any aggregation technique depends on the semantic context of this task. We propose a framework, in which we assume that nature generates samples from a `true' distribution and different experts form their beliefs based on the subsets of the data they have a chance to observe. Naturally, the optimal aggregate distribution would be the one learned from the combined sample sets. Such a formulation leads to a natural way to measure the accuracy of the aggregation mechanism. We show that the well-known aggregation operator LinOP is ideally suited for that task. We propose a LinOP-based learning algorithm, inspired by the techniques developed for Bayesian learning, which aggregates the experts' distributions represented as Bayesian networks. We show experimentally that this algorithm performs well in practice. 1

Ensemble learning algorithms combine the results of several classifiers to yield an aggregate classification. We present a normative evaluation of combination methods, applying and extending existing axiomatizations from Social Choice theory and Statistics. For the case of multiple classes, we show that several seemingly innocuous and desirable properties are mutually satisfied only by a dictatorship. A weaker set of properties admit only the weighted average combination rule. For the case of binary classification, we give axiomatic justifications for majority vote and for weighted majority. We also show that, even when all component algorithms report that an attribute is probabilistically independent of the classification, common ensemble algorithms often destroy this independence information.

The securities market is the fundamental theoretical framework in economics and finance for resource allocation under uncertainty. Securities serve both to reallocate risk and to disseminate probabilistic information. Complete securities markets---which contain one security for every possible state of nature---support Pareto optimal allocations of risk. Complete markets suffer from the same exponential dependence on the number of underlying events as do joint probability distributions. We examine whether markets can be structured and "compacted" in the same manner as Bayesian network representations of joint distributions. We show that, if all agents' risk-neutral independencies agree with the independencies encoded in the market structure, then the market is operationally complete: risk is still Pareto optimally allocated, yet the number of securities can be exponentially smaller. For collections of agents of a certain type, agreement on Markov independencies is su...

In this paper we analyze the problem of opinion pooling. We introduce a divergence minimization framework to solve the problem of standard opinion pooling. Our results show that various existing pooling mechanisms like LinOp and LogOp are an special case of this framework. This framework is then extended to address the problem of generalized opinion pooling. We show that this framework does satisfies various desiderata and we give an EM algorithm for solving this problem. Finally we present some results on synthetic and real world data and the results obtained are encouraging.

We present ongoing research on large-scale decision models in which there are many invested individuals. We apply our unique Bayesian belief aggregation approach to decision problems, taking into consideration the beliefs and utilities of each individual. Instead of averaging all beliefs to form a single consensus, our aggregation approach allows divergence in beliefs and utilities to emerge. In decision models this divergence has implications for game theory – enabling the competitive aspects in an apparent cooperative situation to emerge. Current approaches to belief aggregation assume cooperative situations by forming one consensus from diverse beliefs. However, many decision problems have individuals and groups with opposing goals, therefore

We present a new approach for combining the beliefs of many individuals using graphical models. Existing Bayesian belief aggregation methods break several theoretical assumptions for Bayesian reasoning. More practically, existing opinion pool functions that compute a single value to represent the belief of all contributors do not represent reality well, especially in cases where there are many diverse opinions. Divergence is a natural result of combining opinions from individuals with different beliefs, backgrounds and experiences. Instead of forming a single consensus value that will average out this diversity, we find clusters of agreement for each probability distribution and propagate the cluster means throughout the network during inference. We utilize a social network that tracks the agreement between individuals and the normalized graph cut algorithm to find emerging groups of consensus in the agreement network. We leverage the agreement that occurs across multiple belief estimates to help reduce the complexity that may arise as the means are propagated throughout a belief network. By monitoring agreement over time we may also expose the variety of backgrounds that will help explain divergence in belief. This paper discusses the approach, background and our motives for ongoing research.

e probabilities [3,4]. The modeler seeks to describe the joint distribution across a set of S binary, uncertain events, Z = fA 1 ; A 2 ; : : : ; AS g. Let\Omega = f! 1 ; ! 2 ; : : : ; ! 2 Sg be the set of all 2 S possible joint outcomes of the events, also called the atomic states. The expert has a subjective probability distribution Pr over\Omega and utility u(y) for y dollars, which may in general depend on the state. Let CI(A j ; W;X) be shorthand for the conditional independence Pr(A j jWX) = Pr(A j jW ). Risk-neutral probabilities are defined as Pr RN (!) /<F12.24

The use of domain knowledge in a learner can greatly improve the models it produces.

(extended abstract)

We consider a multi-agent system where each agent is equipped with a Bayesian network, and present an open framework for the agents to compromise on a possible consensus network. The framework builds on formal argumentation, and unlike previous solutions on graphical consensus belief, it is sufficiently general to allow for a wide range of compromises to be identified.