Definition of Hierarchy vs. Level

Here’s what I think these mean:

Definition Hierarchy
For any parameter that has a prior, we can assign a hyper prior to that prior. We call this parameter a hyperparameter, and this submodel hierarchical. (This is how I define a hierarchy).

Example

\gamma \sim normal(0, \gamma_0)
\gamma_0 \sim normal(0, 100)

Definition Level
For any ordinal covariate in a dataset, we can code an indicator for each category in the ordinal covariate. This is called a level. Furthermore, we can assign a parameter to each indicator for each category for local pooling, and a shared parameter for all categories in the ordinal covariate.

Example
Let the covariate \mathbf{x} have two categories, that are 1 and 2. We then assign a global intercept parameter, and a “level” for values of \mathbf{x} that are 1 or 2 respectively. This submodel then becomes:

\beta_0 + \beta_{1,\mathbf{x}=1}\mathbf{x} + \beta_{2,\mathbf{x}=2}\mathbf{x}

Where \beta_{1,\mathbf{x}=1} is 1 if \mathbf{x}=1 and 0 otherwise. The definition for \beta_{2,\mathbf{x}=2} is analogous.

I’m honestly not sure how to write the indicator notation clearly, but if it pops in my head I’ll come back and write it out.

The reason I’m asking is because the definitions are sometimes used interchangably. But - I think this is mostly because you can assign hierarchies and levels simultaneously in a certain submodel.

In chapter 5 of BDA3, we’re usually discussing hierarchies in a similar fashion to what I’ve defined above, but there’s no rigorous definition. Readers are expected to infer the definition, leading to ambiguity and confusion. So I guess I’m wrong, defintion, I think is here:

“It is natural to model such a problem hierarchically, with observable outcomes
modeled conditionally on certain parameters, which themselves are given a probabilistic
specification in terms of further parameters, known as hyperparameters.”

Next, in chapter 15, we’re talking about hierarchical models, for something that I’d think of as a level. Yes, a multi-level model is, in fact, according to my definition, a hierarchical model, because all the parameters for each level will have priors. But this isn’t explicitly stated.

We all the sudden start referring to something that’s previously been referred to as a hierarchy as a level. Like, in maths, one of my teachers used (x',x) for an inner product, which is usually an open interval, but it was all kosher, because she stated that at the beginning of the class.

I guess in math textbooks, all the notation is defined at the beginning of the textbook, so it’s much less ambiguous. I’m not asking because I’m confused, really. I know what I want and I know how to get it, but yeah. I won’t reply to this thread.

good question - here’s a nice summary Bob put together a while back:

https://statmodeling.stat.columbia.edu/2019/09/18/all-the-names-for-hierarchical-and-multilevel-modeling/

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