How to set up a posterior predictive test quantities (Bayesian context) to check for independent Poisson distributions?

Suppose we are given data $y_j \sim \text{Poi}(\lambda)$ and assume $y_j$ are iid. We can assume the prior distribution for $\theta$ follows $\text{Gamma}(\alpha, \beta)$,

The posterior distribution of $\theta | y$ is also Gamma while the posterior predictive distribution is Negative Binomial.

My question is that after deriving everything above, how would you go about setting up a posterior predictive test quantities to check for independent Poisson distributions?

Notice that test quantity is a function of parameters and data in a Bayesian context. It is similar to a test statistic in classical statistic.

• The question as it stands is pretty broad. Do you have any anticipated forms of potential dependence? – Glen_b Feb 17 '13 at 1:28
• we assumed y_i are iid. And we are just asked to find a test quantity to test for this independence assumption – user1769197 Feb 17 '13 at 1:46
• If the y's are actually iid there's nothing to do. You are checking because you anticipate the possibility that they aren't. But there's an infinity of infinity of ways that they can be dependent. Given that your original question about checking is predicated on the possibility of dependence, please look again at my question in the previous comment. – Glen_b Feb 17 '13 at 3:03