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I am trying to calculate the BIC for the X-Means algorithm as described in the paper by Pelleg and Moore (https://www.cs.cmu.edu/~dpelleg/download/xmeans.pdf).

The paper describes the calculation of the pj variable, which I do not understand completely:

The number of free parameters pj is simply the sum of K-1 class probabilities, M*K centroid coordinates, and one variance estimate.

On another post (X-mean algorithm BIC calculation question), a user described the calculated of pj as simply K-1 + M*K + 1. I am wondering if anyone has any knowledge as to whether this is correct or not?

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The two sources you cite give the same answer: $p_j$ is the number of free parameters in a model with $K$ clusters:

  • $K-1$ cluster probabilities (the $K^{th}$ being forced because they sum to $1$)
  • $MK$ centroid coordinates ($K$ centroid, each with $M$ features)
  • $1$ variance estimate, assuming common covariance (per answer to the linked question)
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