Scientists and engineers are often interested in learning the number of subpopulations (or components) present in a data set. A common suggestion is to use a finite mixture model (FMM) with a prior on the number of components. Past work has shown the resulting FMM component-count posterior is consistent; that is, the posterior concentrates on the true generating number of components. But existing results crucially depend on the assumption that the component likelihoods are perfectly specified. In practice, this assumption is unrealistic, and empirical evidence suggests that the FMM posterior on the number of components is sensitive to the likelihood choice. In this paper, we add rigor to data-analysis folk wisdom by proving that under even the slightest model misspecification, the FMM component-count posterior diverges: the posterior probability of any particular finite number of latent components converges to 0 in the limit of infinite data. We illustrate practical consequences of our theory on simulated and real data sets.

Type

Publication

Proceedings of the 38th International Conference on Machine Learning (ICML), to appear

Preliminary versions appeared in *NeurIPS 2019 Workshop on Machine Learning with Guarantees*
and also the *Symposium on Advances in Approximate Bayesian Inference 2017*,
co-located with NeurIPS 2017.

```
@article{cai2020finite,
title={Finite mixture models do not reliably learn the number of components},
author={Cai, Diana and Campbell, Trevor and Broderick, Tamara},
journal={arXiv preprint arXiv:2007.04470},
year={2020}
}
```