Ah! I was thinking (obviously) of a totally different question. I’m pretty sure that was more my fault than yours :p

]]>Dan: my writing skills were never that great but they must have been dropping even further: the question is about the genuine number of modes in a given mixture density, so there is no data, no uncertainty, no prior, no statistician involved! This is pure calculus, in a way, except that there is no formula that tells how many modes result from a certain collection of parameters in a k-component mixture. And hence the need to resort to numerical analysis or simulation…

]]>Dear Alan: Thank you for pointing out your paper and this great Mexican stamp dataset! I am afraid I was not clear enough in this post: the question is not to estimate the number of modes based on a sample of observations, but to determine the number of modes of a given Gaussian mixture with all parameters known and set. This is thus a purely mathematical problem.

]]>I like the idea of committing to an interpretation (clusters WILL BE SEPARATE) and using a prior to enforce that, rather than just let these modes form. Or, to put it differently, I don’t know how to interpret the individual components of a mixture model without some sort of “separating” condition. (The sum is always interpretable as a “non-parameteric” model for the density, but I would suspect that the sum has fewer modes as all roads lead to Rome)

]]>“Philatelic Mixtures and Multimodal Densities,” JASA, 83 (1988), 941-953.

The data set used is fascinating and has a great story attached to it.

The analysis can also be found in Section 4.5.3 in my book,

“Modern Multivariate Statistical Techniques: Regression, Classification, and Manifold Learning,” Springer, 2013.

I’m surprised that it was ignored in the paper.

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