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.
Hi dreycenfoiles,
so if I understand you correctly, you’re currently estimating the A_i and B_i as constants, but you think they might be functions of some covariates x_i that you have, right? I think making A_i and B_i functions of x_i makes sense. The only potential issue I see (and perhaps this is what is bothering you) is that currently, you are allowing A_i and B_i to have a separate value for each group, but if you impose your function
A_i \approx A_m x_i + A_b
you force the values to lie on this line. What I might try in your position would be to model
A_i \sim \mathcal{N}(A_m x_i + A_b, \sigma_A^2)
i.e. draw the value A_i from a hyperprior, and infer \sigma_A^2 from the data. If your covariates x_i are very informative and the linear relationship is good, \sigma_A^2 will be very small, but if it’s not very good, \sigma_A^2 will be large and you’ll be close to your original model. In that case, maybe you could experiment with more non-linear functions – a Gaussian Process or a spline, maybe.
I hope I understood you correctly and this is helpful, let me know! 