Hierarchical vs. graphical models

An email from a reader:

I’m studying your book  The Bayesian Choice (2007). I know that you mentioned graphics models in the Notes of Section 10. But I’m still confused on what is Hierarchical Bayesian Models and Graphical Models. It seems to me that Hierarchical models are just special cases of graphical models. The Bayesian network that corresponds to a graphical model is a DAG in general and the Bayesian network that corresponds to a hierarchical model is just a chain of directed edges.

Would you please take a few seconds to point me some references or discussions so that I can understand their differences better?

To which I can only reply that hierarchical models are indeed special cases of graphical models for which the edges have some kind of causal interpretation and where some conditional independence relations are imposed by the hierarchy, which is not always the case for graphical models. As probabilistic objects and as distributions, both structures belong to the same family. Inference on those objects may be different though, in that [presence or absence of] edges may be examined individually in graphical models, less so in hierarchical models where they come [and go] in batches respecting the hierarchy (see e.g. random effect models).

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