= Solution
Model the count relative to exposure, for example
$$
y_i\mid b_i\sim\operatorname{Poisson}(E_i e^{x_i^T\beta+b_i}),\qquad b_i\mid\tau\sim N(0,\tau^2).
$$
Here $E_i$ is a measured exposure such as patient-days, $x_i$ contains hospital predictors, and $b_i$ is a <random effect>. This <Poisson regression> separates hospital size from rate variation and supplies <partial pooling>. Use proper, scale-appropriate priors on coefficients and variation; add regional or temporal effects if the data demand them. A gamma mixing law with an estimated shape is another way to allow more flexible <overdispersion>. Compare the fitted model's counts and dispersion through <posterior predictive checks>.
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