Tackling the infinite likelihood problem when fitting mixtures of shifted asymmetric Laplace distributions

TL;DR

This paper introduces a Bayesian estimation method for mixtures of shifted asymmetric Laplace distributions, addressing the infinite likelihood problem and improving parameter recovery and classification performance over traditional EM algorithms.

Contribution

A novel Bayesian parameter estimation scheme is developed for these mixtures, overcoming the limitations of EM-based methods in handling the infinite likelihood problem.

Findings

Bayesian scheme yields better parameter estimates.

Classification performance is comparable or superior.

Method tested on real data sets.

Abstract

Mixtures of shifted asymmetric Laplace distributions were introduced as a tool for model-based clustering that allowed for the direct parameterization of skewness in addition to location and scale. Following common practices, an expectation-maximization algorithm was developed to fit these mixtures. However, adaptations to account for the `infinite likelihood problem' led to fits that gave good classification performance at the expense of parameter recovery. In this paper, we propose a more valuable solution to this problem by developing a novel Bayesian parameter estimation scheme for mixtures of shifted asymmetric Laplace distributions. Through simulation studies, we show that the proposed parameter estimation scheme gives better parameter estimates compared to the expectation-maximization based scheme. In addition, we also show that the classification performance is as good, and in some cases better, than the expectation-maximization based scheme. The performance of both schemes are also assessed using well-known real data sets.