  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.
