Terminology: "hierarchical", "multi-level", "mixed effect", "partially pooled" models - is it all the same?

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Dear Bayesians,

I have a question about terminology that has been puzzling me.

From a purely structural perspective, a “hierarchical model” could be described as one where a parameter itself is modeled as a function of other parameter(s)—a “turtles all the way down” structure :turtle: :turtle: :turtle:. This definition doesn’t inherently imply the presence of group effects or clusters.

However, I’ve noticed that terms like “mixed effects models” or “partially pooled models” are often used interchangeably with “hierarchical” or “multi-level” models. While these models do require a hierarchical structure, the reverse doesn’t necessarily hold true. In other words, just because a parameter is modeled as a function of others, it doesn’t mean the model must involve shared priors across groups or clusters.

Am I misunderstanding something about the terminology here, or is this just a case of conventions differing across contexts?

Looking forward to your insights!

Best regards,
Matthias

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Andrew Gelman has a post about this. I use the term “hierarchical” all the time and I use it loosely. But once I need to convey precise information about what my model is doing, I stop trying to use English language to describe it and switch to code or math. The same is true for things like “random effects”. The model names are supposed to be useful shortcuts, but they just cause confusion.

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Great link, thanks!

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I also asked a similar question here and it generated some discussion that might be useful (or not!)

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