Regressing through Latent Variables

Questions version agnostic
1

Hello,

I’m still fairly new to Bayesian modeling, so I think I have a naive question. I am doing growth curve modeling and trying to determine the existence of an effect coming from input variables. I have some data from group i and observation j that I am modeling as coming from an exponential function:

y_{ij} \sim \mathcal{N}(A_ie^{B_it_j}, \sigma)

In my model right now, each A_i and B_i are separate groups, but I would like to determine if there is a relationship between the components of the exponential and the input data x.

A_i = f_1(x_i) \\ B_i = f_2(x_i)

I suspect that the relationship, if it exists, is nonlinear, but I don’t think I have enough data to fully realize it. For now, I would just like to determine if there is a simple linear effect. My question is how would I go about modeling this in PyMC? Would I include another hierarchy level in the same model? Something to the effect of

A_i = f_1(x_i) \approx A_m x_i + A_b

Like I mentioned above, I believe that this function is, in reality, nonlinear. Would including this bias the model away from the “true” value of A_i? Are there alternative ways to do this that I am not seeing?

Thank you very much for your time.