Interpretation of posterior distribution for Gelman's Rat Example

Question

Introduction

Chapter 5 of Bayesian Data Analysis 3rd Edition uses an example of rat endometrial stromal polyps to illustrate the concept of hierarchical regression.

In particular, Gelman and coauthors go on to compute the posterior of the following model of 71 binomial observations:

$$y_i \sim \operatorname{Bin}(\theta_i; n_i)$$ $$\theta_i \sim \operatorname{Beta}(\alpha, \beta)$$ $$p(\alpha,\beta) \propto \alpha \beta (\alpha + \beta)^{-5/2}$$

Gelman writes something I find a little confusing on page 112 when describing how one may simulate from the posterior distribution:

For each $j = 1, \dots ,71$, sample $\theta_j$ from its conditional posterior distribution $\theta_j \vert \alpha, \beta, y \sim \operatorname{Beta}(\alpha+ y_j, \beta+n_j - y_j)$.

Question

Once I obtain posterior estimates for alpha and beta, can I not interpret the thetas as coming from a beta distribution with those shape parameters?

What is the difference in interpretation between $\theta_j \vert \alpha, \beta, y \sim \operatorname{Beta}(\alpha+ y_j, \beta+n_j - y_j)$ and a beta distirnution parameterized with the posterior estimates for alpha and beta?

Answer 1

Intuitively, the point of a hierarchical model is that we want the $(\theta_j)$ to be similar (they share a prior distribution), but not equal; this is explained in BDA section 5.5 (pp. 119-121) for example. In particular, we want the data $y_j$ to inform on $\theta_j$. The marginal posterior distribution of $\theta_j$ will end up somewhere between the distribution we would have got with a model estimating each effect separately, and the distribution we would have got with a pooled estimate.

If you were to follow your suggestion, then you would not allow each $\theta_j$ to be specifically informed by $y_j$, and so you would essentially be using a pooled estimate.