Validation techniques for hierarchical model

Question

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.

Answer 1

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.

Answer 2

Perhaps posterior predictive checking (i.e., comparing your data to predictions generated by the model). For instance along these lines:

http://scholar.harvard.edu/brucewestern/files/smr.pdf