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I was reading some papers related to Bayesian model selection.

The Bayesian information criterion (BIC) model selection criterion is given by $$\text{BIC}=-2\log f({\bf y}|\hat{\theta})+p\log n$$ where $p$ is the dimension of the model parameter $\theta$, and $n$ is the sample size.

In non-i.i.d settings, like mixed model for clustered data or autoregression model for longitudinal data, the sample size $n$ in BIC criterion is not well defined.

There are some papers I was reading talking about using the inverse of the information matrix (or similar method) to get the effective sample size $n_e$, however, in practice or in some programming package that implements BIC, the sample size is still used either as the total sample size $n$, or the number of subjects $N$.

I was wondering what is the difficulties in implementation of BIC? If the effective sample size could be well defined as in those papers, why we do not use it widely?

If you by chance know some resource about this topics I would like to read them.

Thanks in advance!

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