Equal likelihoods preserve model odds ($P(D\mid H_1)=P(D\mid H_2)\Rightarrow O_{12}(D)=O_{12}$)

For two models with finite positive prior odds, a sequence of data with identical conditional likelihoods under both models has Bayes factor one. The posterior odds therefore equal the prior odds. This requires equality of the probability model for all the data considered, not merely that both models fit a selected data set.