Do posterior probability values from an MCMC analysis have any use? I need to estimate the Bayesian posterior of my model parameters ($\theta$) for some observed data ($D$), given a likelihood $P(\theta|D)$, and assumed priors $P(\theta)$:
$$P(\theta|D)= \frac{P(\theta|D)\; P(\theta)}{P(D)}$$
I use a MCMC algorithm which as far as I understand samples the unnormalized posterior (edit: I was wrong, the draws are taken from the full posterior).
After the MCMC is done, I can construct the probability density function for each $\theta_i$ in my set of parameters $\theta$ (from which I can obtain the necessary statistics: mean, median, confidence intervals, etc.) but I also have a rather large set of unnormalized posterior values.
As far as I understand, these values are not used at all in the analysis of the model parameters. Does this set of unnormalized posterior values have any use at all, or are they simply discarded?
 A: I have to disagree with the earlier answers that the values of the (unnormalised) posteriors at the MCMC simulations are not of any use. They actually provide a much more refined view of the posterior than an histogram, especially in multiple dimensions. One direct illustration is the construction of the HPD region: the easiest way to construct an HPD region at level $\alpha$ is to take the same percentage on the MCMC simulations with the largest [unnormalised] posterior values and to construct a convex envelope of these simulations.
A: The values obtained after MCMC is a sample from the posterior $P(\theta|D)$. You have to ensure that you draw a large enough sample so that the sample itself can be considered a population. Now assuming that the sample is indeed the population, you can do anything with it that you do with the probability density. For example, you can find out the credible-region by finding $2.5^{th}$ and $97.5^{th}$ percentile or you can compute the mean of $\theta|D$.
Now, you can use this sample from posterior to estimate the probability density $P(\theta|D)$ which is usefull in constructing High Posterior Density (HPD) region. I cannot think of any other application where you will need probability values themselves. It is redundant to have both the population and its probability distribution because they provide the same information.
For your doubt regarding the normalization, you can refer to the article by Chib and Greenberg: http://www.tandfonline.com/doi/pdf/10.1080/00031305.1995.10476177. This article helped me understand the MCMC very well.
A: The draws are from the normalized posterior density. The reason why we write the posterior as proportional to the unnormalized posterior density is that the normalizing constant does not matter and drops out of the computations. Thus, the draws from an MCMC sampler are from the normalized posterior density.
