# Determining number of components in mixtures of normal distributions with common mean

This is a pretty simple question, suppose we want to fit a mixture distribution of multivariate normals with common mean

$$y_i \sim \sum_k \pi_k N(\mu, \Sigma_k)$$

What is the preferred approach for deciding the number of components in this situation?

If the means differ, even slightly, then to me it seems like the problem is significantly more intuitive since the peaks will cause multi-modal data. But if the peaks are all shared, then how would you go about approaching this?

• Separated peaks will not necessarily cause the distribution to be multi-modal. You can most easily see this in one dimension with equal variances, where the peaks must be separated by at least two sd before multimodality appears. Note, too, the distinction between a multimodal distribution and multimodal data. – whuber Nov 3 '18 at 21:47
• There is a significant amount of literature on the Bayesian approach to this problem, e.g., Richardson and Green (1997). – Xi'an Nov 9 '18 at 11:18