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I'm now trying to implement the algorithm presented in https://www.stat.washington.edu/raftery/Research/PDF/fraley2005.pdf.

The algorithm is the following one: enter image description here

First I get a mixture model for classification in step 1, call it M. Then using the method in step 2 I choose the set Q and assign to a new class. Here we get a new model call M'. By doing step 3, we get a new BIC called $BIC_{M'}$, by compare to $BIC_{M}$, we can choose to keep this model M' or go to fit a less parsimonious model starting with the current classification.

My question is, what does is mean for "fit a less parsimonious model starting with the current classification"? What should I manipulate in this step??

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I'm not familiar with the algorithm but, in general, "fit a less parsimonious model starting with the current classification" means try to simplify the model by removing something, in your example, from model M'.

The information criteria, like $BIC$ usually decrease with the number of variables if the fit does not improve enough (in the sense that the residuals will be similar). Similarly, $BIC$ is expected to increase if we remove "superfluous" items from the model, i.e, the removal of some set of items (that I called "superfluous") does not worsen the fit (again, in the sense that the residuals will be similar).

(I've written this as an answer because it was too long for a comment.)

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