Graphical Representations of Consensus Belief

TL;DR

This paper investigates how group consensus beliefs can be represented using graphical models, revealing limitations of belief combination methods and proposing the LogOP as a solution for maintaining independencies.

Contribution

It proves that common belief combination methods cannot preserve the structure of individual graphical models, but shows LogOP can maintain Markov independencies and offers an efficient algorithm.

Findings

Belief combination methods often fail to preserve graphical structures.

Logarithmic opinion pool (LogOP) maintains Markov independencies.

An efficient algorithm for computing LogOP is proposed.

Abstract

Graphical models based on conditional independence support concise encodings of the subjective belief of a single agent. A natural question is whether the consensus belief of a group of agents can be represented with equal parsimony. We prove, under relatively mild assumptions, that even if everyone agrees on a common graph topology, no method of combining beliefs can maintain that structure. Even weaker conditions rule out local aggregation within conditional probability tables. On a more positive note, we show that if probabilities are combined with the logarithmic opinion pool (LogOP), then commonly held Markov independencies are maintained. This suggests a straightforward procedure for constructing a consensus Markov network. We describe an algorithm for computing the LogOP with time complexity comparable to that of exact Bayesian inference.