A hierarchical model captures grouped data with multiple levels of variation, organized in a hierarchy. The datapoint is observation within group . We assume that there are groups, each with observations.

The key is that we want to allow variation between groups. Extending the 🛎️ Normal Model, we have

where the mean comes from

This accounts for variation within each group with and variation between groups with .

Known Variance#

First, let's assume is known. Our remaining parameters are

The prior for is already , so we just need priors for and . We'll use a non-informative prior and slightly different , which is less informative than . Note that this is different than the simple Normal model!

With these priors, our joint posterior is

Breaking this down, the group means' conditional posterior is

where is the mean of group .

The conditional posterior for global mean is

Finally, the marginal posterior for is non-standard and simplifies to

which requires 🧱 Grid Sampling.

Interpretation#

The group means are a compromise between the group's data mean and global prior mean . The between-group variance is important here because if , the prior mean is ignored and we have

On the other hand, with , there is no variation between groups, so .

Sampling#

To sample from this hierarchical model, we perform the steps below:

1. Select grid of values for .
2. Calculate for each grid value.
3. Sample grid value with probability .
4. Sample .
5. For each group , sample .
6. If we want posterior predictions for group , sample .
7. If we want a new group, sample and .