# Posterior mean of $\mu$ in Bayesian Hierarchical model (Poisson-Gamma)

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?