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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....

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