## how many modes in a normal mixture?

Posted in Books, Kids, Statistics, University life with tags , , , , , , on January 7, 2015 by xi'an

An interesting question I spotted on Cross Validated today: How to tell if a mixture of Gaussians will be multimodal? Indeed, there is no known analytical condition on the parameters of a fully specified k-component mixture for the modes to number k or less than k… Googling around, I immediately came upon this webpage by Miguel Carrera-Perpinan, who studied the issue with Chris Williams when writing his PhD in Edinburgh. And upon this paper, which not only shows that

1. unidimensional Gaussian mixtures with k components have at most k modes;
2. unidimensional non-Gaussian mixtures with k components may have more than k modes;
3. multidimensional mixtures with k components may have more than k modes.

but also provides ways of finding all the modes. Ways which seem to reduce to using EM from a wide variety of starting points (an EM algorithm set in the sampling rather than in the parameter space since all parameters are set!). Maybe starting EM from each mean would be sufficient.  I still wonder if there are better ways, from letting the variances decrease down to zero until a local mode appear, to using some sort of simulated annealing…

Edit: Following comments, let me stress this is not a statistical issue in that the parameters of the mixture are set and known and there is no observation(s) from this mixture from which to estimate the number of modes. The mathematical problem is to determine how many local maxima there are for the function

$f(x)\,:\,x \longrightarrow \sum_{i=1}^k p_i \varphi(x;\mu_i,\sigma_i)$