A note on maximum likelihood estimation for mixture models

Practitioners as well as some statistics students often blindly use standard software or algorithms to get maximum likelihood estimator (MLE) without checking the validity of existence of such an estimator. Even in simple situations where data comes from mixtures of Gaussians, global MLE does not exist. This note is intended as a teachers corner, highlighting existential issues related to MLE for mixture models, even when the components are not necessarily Gaussian.