Theory question: 2-stage hierarchical modelling

#1

Hello,
I’m using a 2-stage hierachical model for model fitting. I think, I already implemented it correctly, since the results are very satisfying. Nevertheless, I have a theory question.

At first, a 2-stage hierarchical model is defined as:
P(A,B|C) \approx P(C|A)\ P(A|B)\ P(B),
with P(C|A) the likelihood, the prior P(A|B) and the hyperprior P(B).

In my Gaussian likelihood, I have a (shared) variance parameter \sigma_l, to be identical for each experiment. For this variance \sigma_l, I specified a weakly informative prior, such as the Gaussian f(\sigma_l|0.5,0.2).
Now, my question is the following: Is this prior part of P(C|A), P(A|B), or P(B)?
I’m a little bit confused, since P(B) is the prior for all hyperparameters, but the mentioned variance is not a hyperparameter. So I guess, it might be assigned to be part of P(A|B)?

I would be really happy, if someone could help me with my theory problem.

Regards