You can find a discussion on this type of clustering in McLachlan &
Basford (1988). It's called Classification Maximum Likelihood and there
are some problems with it. Generally in mixture models it is not
advisable to treat the component assignments as parameters.
Murray Jorgensen
Torvik, Vetle Ingvald wrote:
I have what appears to be a fundamental clustering problem but I have not
been able to find much relevant literature on it: Given an n-by-n matrix
of probabilities P, find the maximum likelihood clustering solution
represented by an n-by-n binary matrix X where Xij = 1 if elements i and j
are assigned to the same cluster, and 0 otherwise. That is,
maximize product{ [Pij*Xij + (1-Pij)*(1-Xij)], over all pairs (i,j): i < j}
subject to Xij + Xik + Xjk != 2 for all triples (i,j,k): i<j<k.
P satisfies symmetry (i.e., Pij = Pji) but may have many transitivity
violations of the form Pij + Pik – 1 > Pjk.
Do you know of someone who has studied this problem? I am particularly
interested in fast algorithms that solve it optimally (or near-optimally).
If not, do you know of any closely related problems (beyond traditional
clustering) that could shed some light on this? If not, maybe you can
point me to a person who is likely to know?
Vetle Torvik
PS. We have developed a procedure that seems to work well when applied to
author name disambiguation (Torvik et al., A probabilistic similarity
metric for Medline records: a model for author name disambiguation. JASIST
2005). This paper describes a model to estimate P. To find a clustering
solution X, we first correct all transitivity violations by weighted least
squares, and then perform agglomerative clustering (iteratively merge the
pair of clusters that give the greatest instantaneous increase in
likelihood until the likelihood hits max).
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Dr Murray Jorgensen http://www.stats.waikato.ac.nz/Staff/maj.html
Department of Statistics, University of Waikato, Hamilton, New Zealand
Email: [EMAIL PROTECTED] Fax 7 838 4155
Phone +64 7 838 4773 wk Home +64 7 825 0441 Mobile 021 1395 862
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