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Bayesian $\chi^2$ discrepancy

We define the $\chi^2$ discrepancy as $$T(y,\theta)=\sum_i\frac{(y_i-E(y_i|\theta))^2}{\mathrm{var}(y_i|\theta)}$$ (definition from Gelman). How do we apply this if $\theta$ is a random variable, and ...
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Validation of Bayesian hierarchical model [duplicate]

Possible Duplicate: Validation techniques for hierarchical model I already posted this to Math.SE, but I realized this would probably be a better place to post to. I have a hierarchical ...