Finite mixture models are widely used in the life sciences for data analysis. Yet, the calibration of these models to data is still challenging as the optimization problems are often ill-posed. This holds for censored and uncensored data, and is caused by symmetries and other types of non-identifiabilities. Here, we discuss the problem of parameter estimation and model selection for finite mixture models from a theoretical perspective. We provide a review of the existing literature and illustrate the ill-posedness of the calibration problem for mixtures of uniform distributions and mixtures of normal distributions. Furthermore, we assess the effect of interval censoring on this estimation problem. Interestingly, we find that a proper treatment of censoring can facilitate the estimation of the number of mixture components compared to inference from uncensored data, which is an at first glance surprising result. The aim of the manuscript is to raise awareness of challenges in the calibration of finite mixture models and to provide an overview about available techniques.
Keywords: finite mixture model; interval censoring; model selection; unbounded likelihood.