What is hierarchical prior in Bayesian statistics?

Question

What are hierarchical priors?

How do they differ from the general concept of priors?

Answer 1

A regular Bayesian model has the form $p(\theta |y) \propto p(\theta)p(y|\theta)$. Essentially the posterior is proportional to the product of the likelihood and the prior. Hierarchical models put priors on the prior (called a hyperprior) $p(\theta |y) \propto p(y|\theta)p(\theta |\lambda)p(\lambda)$. We can do this as often as we want.

See Gelman's "Bayesian Data Analysis" for a good explanation.

Answer 2

When you have a hierarchical Bayesian model (also called multilevel model), you get priors for the priors and they are called hierarchical priors.

Consider for example:

$z = \beta_0+\beta_1{y}+\epsilon, \\ \epsilon \mathtt{\sim} N(0,σ)\\ \beta_0\mathtt{\sim} N(\alpha_0,σ_0), \beta_1\mathtt{\sim} N(\alpha_1,σ_1), \beta_2\mathtt{\sim} N(\alpha_2,σ_2)\\ \alpha_0\mathtt{\sim} inverse-\gamma(\alpha_{01},\theta_0)\\ $

In this case, you can say that, $inverse$-$\gamma$ is a hyperprior.

EDIT: This was very useful to me when I learned about Hierarchical Bayesian Modeling. For an in depth explanation and detail, you may refer to Gelman's Data Analysis Using Regression and Multilevel/Hierarchical Models.