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We prove a theorem justifying the regularity conditions which are needed forPath Sampling in Factor Models. We then show that the remaining ingredient,namely, MCMC for calculating the integrand at each point in the path, may beseriously flawed, leading to wrong estimates of Bayes factors. We provide a newmethod of Path Sampling (with Small Change) that works much better thanstandard Path Sampling in the sense of estimating the Bayes factor better andchoosing the correct model more often. When the more complex factor model istrue, PS-SC is substantially more accurate. New MCMC diagnostics is providedfor these problems in support of our conclusions and recommendations. Some ofour ideas for diagnostics and improvement in computation through small changesshould apply to other methods of computation of the Bayes factor for modelselection.

Bayes Model Selection with Path Sampling: Factor Models and Other Examples