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For groupings G1 and G2, the Bayes factor is given by: B=P(N|G1)P(N|G2)(4) where P(N∣Gi) is the marginal likelihood ∫01⋯∫01P(cij,πij,i,j∈1:S|Gi)P(N|cij,πij,i,j∈1:g,Gi)dc11…dcggdπ11…dπgg(5) which can be analytically integrated to give: ∏i=1g∏j=1gKij!Zij!(Lij-Kij)!(1+Lij)(1+Lij+Zij)! (6) Because there are many possible groupings to choose from, we compared the marginal likelihoods of the groupings when searching for the best grouping, rather than explicitly calculating B for each pair.
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