Question: Suppose the following problem: I have $n$ models, $M\_k$, each with parameters $\mathbf{\theta}\_k$ for a data set $D$. There where previous observations of a subset of the parameters which are common to every model $M\_k$ (i.e., I have well defined priors for a subset of the parameters $\theta\_k$), so I performed an MCMC algorithm in order to obtain the posterior distribution of each model using that prior information, i.e., I have $p(\theta\_k|D,M\_k)$, and have to decide which of those models is the 'correct' one.

I was thinking in defining what do I mean by 'the correct' one, and came up with the idea that I have to decide which of the posterior distributions is closer to the 'real' posterior distribution that generated the data (which may or may not be in my set of posterior distributions). I was thinking of using bayes factors, but I keep thinking that I need something like the AIC which, instead of using the likelihood and the corresponding MLE estimates, uses the posterior distributions and the corresponding maximum-a-posteriori estimates. My idea is to obtain an unbiased (or nearly unbiased) estimator of the KL divergence between the real posterior and my posteriors (understanding that the AIC is an estimator of the KL divergence between the 'real' likelihood and the likelihood of my models).

Is there something like this in the statistical literature? I'm just kind of crazy of thinking the problem like this?

Answer: 
Strictly speaking, the question **"to decide which of those models is the 'correct' one"** makes no sense in a Bayesian analysis. In the Bayesian framework, what you do is to compare the models *with respect to each other*. Bayesian inference always gives you a *relative* comparison of competing models. There is a lot of information on chapter 7 of O'Hagan and Forster nice [book](http://rads.stackoverflow.com/amzn/click/0470685697). And yes, this kind of analysis will rely on the full posteriors.
Nestor: You seem to be misinterpreting BIC and DIC. They are based on a Bayesian approach. The fact that you observe the likelihood in their expressions is due to an approximation.

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> The BIC was developed by Gideon E. Schwarz, who gave a Bayesian argument for adopting it.
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> It [DIC] is particularly useful in Bayesian model selection problems where the posterior distributions of the models have been obtained by Markov chain Monte Carlo (MCMC) simulation.
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Log-Predictive scores ARE purely Bayesian and they are NOT plug-in estimators. They are complementary to Bayes factors since evaluate the predictive performance of a model.

There also seem to be a contradiction in "It is particularly useful in Bayesian model selection problems where the posterior distributions of the models have been obtained by Markov chain Monte Carlo (MCMC) simulation." and "And, if you read carefully my question, I AM searching for something like an information criterion which includes the posterior distribution (and therefore, the variability of the parameters on each model)."

Anyway, good luck ...