# Validation techniques for hierarchical model

I have a hierarchical model that I need to validate. My model is as follows: we have a collection of $\lambda_i$ that we draw from $Gamma(\alpha,\beta)$. Then, we draw our data point $y_i$ from $Poisson(\lambda_i)$. I get a distribution of $\alpha,\beta,\{\lambda_i\}$ via a Gibbs sampler combined with a Metropolis step. This part is fine. My question is how do I validate such a model? I have my set of data, each one corresponding to one particular $\lambda_i$. I'm not sure what statistical tests/ other steps I should take to check are.

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Cross posting is discouraged. You should probably close the question on Math.SE, especially if you haven't yet received an answer and you feel that this is the better place. –  Jeromy Anglim Apr 23 '12 at 6:59
I don't have enough reputation to close it. –  Andrew Apr 23 '12 at 7:14
okay. no problems, one idea is to press "flag" and write a comment to the moderators indicating that you are new to the site and ultimately decided to post to stats.SE. –  Jeromy Anglim Apr 23 '12 at 7:18
Done. You wouldn't happen to have any ideas about my problem, would you :) ? –  Andrew Apr 23 '12 at 7:23
I am not aware of a general Bayesian model validation technique. It is more common to conduct model comparison via Bayes factor, Scoring rules such as the log-predictive scores, and etcetera. –  user10525 Apr 23 '12 at 7:30

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Posterior predictive checks, outlined in Gelman et al (1996), are an obvious starting point. Given how simple the model is, it probably makes sense to use graphical checks. Plot the histogram of $(y_1,...,y_n)$ against histograms of several posterior predictive replications $(y_1^{rep},...,y_n^{rep})$. If you spot a feature that doesn't fit, you can formalize things by defining an appropriate discrepancy statistic and computing the posterior predictive p-value of your model against that statistic.