Model selection and Monte Carlo integration

Current state:

I have a dataset. I have a model I am fitting to this dataset. I am doing this via Metropolis-Hastings MCMC.

A simple description of the model: There are K continuous distributions. Let us say they are distributions over a single dimension. They overlap. Each of the K distributions has sets of labeled data from this dimension. Gaussians weren't appropriate so we're not using that, if that's an important point to make.

I am trying to find a good way to compare this model to a baseline. There is the area under the ROC, but I want to explore all of my model comparison options.

In looking at Bayes Factor, you can use a ratio of the integrated posteriors.

My question is this:

When doing MCMC to learn parameters and using a posterior ratio as your acceptance ratio, are you effectively approximating the integral mentioned in the Bayes Factor?

Sorry if I left out any important information!