Bayesian Model Specification

I've been giving this some thought lately. Given several models (N) describing the same phenomenon with observable x, each with drastically different parameter estimates and predictors, i.e. with probability densities given by

f_n(\mathbf{x}|\theta_n), n = 1, ... , N,

for this specific set of models, how do you decide which one would be more apt given a set of data?

The bayesian model specification framework suggests the following; given data x, we have

\pi(\theta_n, n | \mathbf{x}) \propto \pi(\mathbf{x} | \theta_n, n) \pi(\theta_n| n) \pi(n).

Substitute the appropriate model density to complete the proportionality,

\pi(\mathbf{x} | \theta_n, n) = f_n(\mathbf{x} | \theta_n).

We can specify suitable priors for

\pi(\theta_n| n) & \pi(n).

Finally the probability density of the "correct" model n is given by the marginal density

\pi(n | \mathbf{x}) = \int \pi(\theta_n, n | \mathbf{x}) d\theta_n

where we integrate out the parameter estimates.

The actual difficulty of this problem involves generating the posterior density. Unlike ordinary simulation methods, the values and the dimensions of the parameter vector depend entirely on the chosen model, n. In order to simulate the posterior density, a practitioner would likely have to use the Reversible Jump Markov Chain Monte Carlo method, which while difficult, is tractable in its implementation.

What's very interesting is the latitude one can have with the posterior. If we were discussing forecast models, it is easy to include an estimator y in the density,

\pi(\mathbf{y}, \theta_n, n | \mathbf{x}) \propto \pi(\mathbf{y}, \mathbf{x}| \theta_n, n) \pi(\theta_n| n) \pi(n).

This way, we're effectively producing a mixed model density "estimator" of y, given x.