Posterior predictive checks for hyperprior distributions of multilevel models

This is a methodological question.

Suppose we have a multilevel model

x_i \sim F(\alpha), y_i \sim G(x_i, \beta), \text{for}\ i = 1, \dots, N

where F and G are some distributions, and only y_i is observable.

Assume however that the cross-sectional distribution of x_i is of interest, and it is important go get it right. How would one do PPC in this context? Eg is comparing x_i to simulated values using estimated \alpha s appropriate?