Graphical Representations of Consensus Belief
This addresses a foundational problem in multi-agent systems and probabilistic graphical models, with incremental contributions to belief aggregation methods.
The paper tackles the problem of representing the consensus belief of a group of agents in graphical models, proving that under mild assumptions, no method can maintain a common graph topology, but shows that using the logarithmic opinion pool preserves Markov independencies, enabling a consensus Markov network with an algorithm of comparable time complexity to exact Bayesian inference.
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.