Hierarchical Models

In Bayesian hierarchical models, we start by imposing a prior that is a function of different parameters. We'll introduce a new variable γ, which is called the hyper-prior; The prior of α depends on β, which depends on γ.

A typical example of this in medical research is population of hospitals, providers within hospitals, and patients within providers. Any datasets with such a structure are called hierarchical.

We are interested in making inference of specific units. In doing so observations may be collected over time on the same individuals, so repeated measures must be accounted for.

Bayesian Ranking

We can derive the posterior distribution of each ranking by ranking the estimates at each iteration of the Gibbs Sampling and generate the posterior distributions of the ranks. Below the ranks are in parenthesis:

The posterior distribution of ranks gives us a measure of the uncertainty of the ordering.