Chapter 7 of Jim Albert's book considers the case of using a hierarchical model, to estimate heart-transplant mortality rates ($\lambda_i$) from 94 hospitals, each with it's own exposure (# of operations, $e_i$), with:

$$ y_i \sim Poisson(e_i\lambda_i) \\ \lambda_i \sim Gamma(\alpha, \frac{\alpha}{\mu}) \\ $$

Under this model, we get that the posterior probability of $\lambda_i$ is $Gamma(y_i + \alpha, e_i + \frac{\alpha}{\mu})$. It is also shown that the posterior mean of $\lambda_i$ is equal to $\mathbb E(\lambda_i | y_i,\alpha,\mu) = (1-B_i)\frac{y_i}{e_i} + B_i \mu$, which demonstrates Shrinkage.

In addition, after assuming the following prior distribution for the hyperparameters $\mu$ and $\alpha$:

$$ g(\mu) \propto\frac{1}{\mu} \\ h(\alpha) = \frac{z_0}{(\alpha+z_0)^2} $$

We get also a Marginal Posterior density of the two. Which allows us to sample values for these hyperparameters, and then use these samples to sample plausible $\lambda_i$ values and in turn also $y_i$.

Now at some point he states that $\mathbb E(\lambda_i | data)$ can be approximated by $(1-\mathbb E(B_i|data))\frac{y_i}{e_i} + \mathbb E(B_i|data) \frac{\sum y_i}{\sum e_i}$.

Now I think that he meant that we replace the parameters with our simulated averages of these parameters. This is what he does with $\mathbb E(B_i|data)$ anyway. But I wonder why he replaced $\mu$ with $\frac{\sum y_i}{\sum e_i}$ ?

My hunch is that the posterior mean of $\mu$ is equal to $\frac{\sum y_i}{\sum e_i}$. At least they are very close numerically from the simulated samples. But I wonder if there is a way to show this analytically?


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