Difference between Bayesian Hierarchical Model and a Bayesian regression model?

Question

Are Bayesian Hierarchical models and Bayesian regression models the same in books, papers?

Answer 1

In general, they are not the same: One can have a single-level Bayesian regression model without any notion of hierarchy. What you might call the book on hierarchal models devotes ~100 pages to single-level models. Here's an excerpt from the introduction that helps make the hierarchal distinction clear:

The two key parts of a multilevel model are varying coefficients, and a model for those varying coefficients (which can itself include group-level predictors). Classical regression can sometimes accommodate varying coefficients by using indicator variables. The feature that distinguishes multilevel models from classical regression is the modeling of the variation between groups.

Contrariwise, an analysis—regression or otherwise—is distinguished as Bayesian by its reliance on the posterior for inference.