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?