How can parameters be modeled differently if they share hyperparameters?

by mnmn   Last Updated October 19, 2019 20:19 PM - source

In one popular example of multilevel Bayesian models (2007 Gelman et. al paper), radon exposure in a household is modeled as a function of the county and whether the house has a basement.

In this explanation of the paper, the author writes:

...we may assume that while $\alpha$s and $\beta$s are different for each county as in the unpooled case, the coefficients all share similarity. We can model this by assuming that each individual coefficient comes from a common group distribution:

$\alpha_c \sim N( \mu_{\alpha},\sigma_{\alpha}^2)$

$\beta_c \sim N( \mu_{\beta},\sigma_{\beta}^2)$

We thus assume the intercepts $\alpha$ and slopes $\beta$ to come from a normal distribution centered around their respective group mean $\mu$ with a certain standard deviation $\sigma^2$, the values (or rather posteriors) of which we also estimate.

Let's focus on the intercepts ($\alpha$) right now. My question is: If each county's $\alpha_c$ shares exact the same distribution (namely $N( \mu_{\alpha},\sigma_{\alpha}^2)$) how can the various $\alpha_c$'s ever be different from each other?

In other words, if $\alpha_{Aitkin} \sim N( \mu_{\alpha},\sigma_{\alpha}^2)$, $\alpha_{Anoka} \sim N( \mu_{\alpha},\sigma_{\alpha}^2)$, and $\alpha_{County X} \sim N( \mu_{\alpha},\sigma_{\alpha}^2)$, etc. etc., why should we ever suspect that the posterior distribution for $\alpha_{Aitkin}$ look any different from $\alpha_{Anoka}$ or any other county's $\alpha$? We specified in our model that each distribution was identical!

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