 Solution

Model the count relative to exposure, for exampleHere $E_i$ 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.