# How to use BIC (Bayesian Information Criterion) if the data are not identically distributed but rely on other covariate?

I'm now constructing some models and would like to compare and select the models. I read the wikipedia and some slides about BIC, then I found the ML(maximum likelihood) part in BIC seems to be based on the i.i.d data, like:

$$l_n(\hat \theta) = \sum_{i=1}^{n} \log f(x_i|\hat \theta)$$

where $$\theta$$ is the parameter unchanged through all data points $$x_i$$. Now, in my model, I have another factor or covariate called time $$t$$, the parameter changes over the time $$t$$. For example, suppose $$A$$ and $$B$$ are two positive definite matrix, at time $$t$$, $$x_t \sim N(0, (A + Bt)^{-1})$$. In this case, when I sample $$n$$ data uniformly through $$t \in [t_a, t_b]$$, could I write the ML part in BIC like the following?

$$l_n(\hat A, \hat B) = \sum_{i=1}^{n} \log f(x_{t_i} | \hat \theta_t)$$, where $$\theta_t = A + Bt$$

Do I need to make other adjustment on BIC in this case? I saw a lot of extension and variant on google scholar, but get drown into these papers....