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I have recently been trying to understand the X-means method for deciding on K, using BIC. However I have become stuck on one particular equation in the original paper. On the 4th page, when discussing BIC calculations, the authors write:

Focusing just on the set $D_n$ of points which belong to centroid $n$ and plugging in the maximum likelihood

What do they mean by 'centroid $n$'?

My initial intuition was one of the clusters in the model. However if this is true, why does likelihood equation for $D_n$ include K? Because we have subset to a single cluster, this would always be 1. So this cannot be the correct interpretation.

The full equation is:

$l_{hat}(D_n) = -R_n / 2 * log(2 \pi) - R_n M / 2* log(\sigma_{hat}^2) - R_n - K / 2 + R_nlogR_n - R_n log R $

Which is the last equation on the first column of page 4.

There are two other questions around BIC calculations in X-Means, but I don't think they explain this particular point:

X-Means Calculation of BIC

X-mean algorithm BIC calculation question

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