Generalized mixtures of finite mixtures and telescoping sampling

TL;DR

This paper introduces a generalized class of mixtures of finite mixtures with a hyperparameter-dependent prior, and proposes a novel telescoping sampler for efficient Bayesian inference without reversible jump MCMC.

Contribution

It extends MFMs by allowing hyperparameters to depend on component count and develops a telescoping sampler for flexible, efficient inference.

Findings

The generalized MFM model can be regarded as a Bayesian non-parametric mixture outside Gibbs-type priors.

The telescoping sampler enables inference for arbitrary component distributions without reversible jump MCMC.

Demonstrated the method's effectiveness on multiple data sets.

Abstract

Within a Bayesian framework, a comprehensive investigation of mixtures of finite mixtures (MFMs), i.e., finite mixtures with a prior on the number of components, is performed. This model class has applications in model-based clustering as well as for semi-parametric density estimation and requires suitable prior specifications and inference methods to exploit its full potential. We contribute by considering a generalized class of MFMs where the hyperparameter $\gamma_K$ of a symmetric Dirichlet prior on the weight distribution depends on the number of components. We show that this model class may be regarded as a Bayesian non-parametric mixture outside the class of Gibbs-type priors. We emphasize the distinction between the number of components $K$ of a mixture and the number of clusters $K_+$, i.e., the number of filled components given the data. In the MFM model, $K_+$ is a random variable and its prior depends on the prior on $K$ and on the hyperparameter $\gamma_K$. We employ a flexible prior distribution for the number of components $K$ and derive the corresponding prior on the number of clusters $K_+$ for generalized MFMs. For posterior inference, we propose the novel telescoping sampler which allows Bayesian inference for mixtures with arbitrary component distributions without resorting to reversible jump Markov chain Monte Carlo (MCMC) methods. The telescoping sampler explicitly samples the number of components, but otherwise requires only the usual MCMC steps of a finite mixture model. The ease of its application using different component distributions is demonstrated on several data sets.