# How can parameters be modeled differently if they share hyperparameters?

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

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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