When do we use a hierarchical model structure in Bayesian Analysis?

Question

I am having trouble understanding when it is advantageous or when it is rational to use a hierarchical model set up in Bayesian Analysis. Basically what kinda of data do I have or what kind of analysis do I want to perform that warrants using a hierarchical model set up.

Here's an example of where the author uses a hierarchical set up:

This is my understanding of the set up of his model. There are s prior distributions for θ, each having the same given K value. The ω for each distribution of θ may not be the same, but come from the same distribution for ω. If we are interested in the overall success rate of the drug, we want to look at expected value of ω, which is the mode of all the distributions of θ. (Thus if K is the same for all distributions of θ and ω comes from the same distribution, then all the prior distributions of θ are the same as well).

My question is, why do we need such a complicated model? Can we not just have only one distribution for θ, where instead of θ following a beta distribution, we have it follow a normal distribution. Depending on what we end up for parameters mu and SD, we get almost the same information as before for the overall effect of the drug? Wouldn't (mu, SD) provide almost the same information as (ω, K) for the effectiveness and variability among patients for the drug? Except our model is simpler.

Thanks!

Answer 1

Beta distribution is more flexible than normal (here) in the sense that it can be asymmetric (around mean/median) and multimodal. For the same or more flexibility, you need a mixture of gaussians. Also, modelling a parameter that represents some kind of probability, i.e. $\in [0,1]$ via normal distribution is ill-defined since its support is the whole real axis. A better approach would be using a truncated normal.

Answer 2

Wouldn't (mu, SD) provide almost the same information as (ω, K) for the effectiveness and variability among patients for the drug? Except our model is simpler.

This sounds fine if the variance of the true distribution for the $\theta$ is small and far from the boundaries of 0/1. But what happens when the true $\theta$ is close to 0 or 1? Is the normal still a good model? No, because the normal is supported on the entire real line. The use of the beta is more appropriate because it is has the same support as the phenomenon we are studying.

Answer 3

The motivation for hierarchical models include if you want to model more granular data, if you have access to subsets of data, or if you want to model different levels of data. A hierarchical model allows you to jointly estimate parameters for all subsets and categories of data. This may be better than estimating each subset’s parameters individually as it may reduce some noise.

Couldn't delete my question so I'll answer it myself.