  We prove a theorem justifying the regularity conditions which are needed for
Path Sampling in Factor Models. We then show that the remaining ingredient,
namely, MCMC for calculating the integrand at each point in the path, may be
seriously flawed, leading to wrong estimates of Bayes factors. We provide a new
method of Path Sampling (with Small Change) that works much better than
standard Path Sampling in the sense of estimating the Bayes factor better and
choosing the correct model more often. When the more complex factor model is
true, PS-SC is substantially more accurate. New MCMC diagnostics is provided
for these problems in support of our conclusions and recommendations. Some of
our ideas for diagnostics and improvement in computation through small changes
should apply to other methods of computation of the Bayes factor for model
selection.
