Finite mixture models are typically inconsistent for the number of components

A generative model is, of necessity, a vast simplification of the deeply complex real-world phenomena that govern any observed data set. It is only via this simplification that we can arrive at a tractable data analysis and discover meaningful and actionable patterns in data. In this sense, typically any model of a real-world data set is misspecified, and misspecification is unavoidable. But while misspecification in the form of simplification is powerful, it can also be potentially dangerous. In particular, certain kinds of misspecification can lead to fundamentally inaccurate or misleading inferences. For instance, recent work by Miller and Harrison (2013, 2014) serves as a cautionary tale about mixture modeling. In particular, mixture models are often matched with a nonparametric Bayesian prior by practitioners in order to discover the number of clusters in a set of data. Miller and Harrison (2013, 2014) demonstrate that such models are “severely” inconsistent for the number of clusters; that is, the probability of the correct number of clusters being recovered decreases to zero as the amount of data increases. An implication is that finite mixture models would be a more appropriate modeling choice. But empirical work by Miller and Dunson (2015) suggests otherwise. We here aim to demonstrate theoretically that even finite mixture models with an unknown number of clusters generally exhibit severe inconsistency, just as the nonparametric Bayesian models do. We discuss the implications for practical modeling and inference in mixture models.