A Tight Convex Upper Bound on the Likelihood of a Finite Mixture

TL;DR

This paper introduces a convex optimization-based method to compute a tight upper bound on the likelihood of finite mixture models, helping to evaluate how close a given solution is to the global maximum.

Contribution

It proposes a novel convex optimization approach to bound the likelihood of finite mixture models within a discrete parameter set, enabling assessment of solution optimality.

Findings

The bound is computationally feasible using convex optimization.

The bound is tight under certain conditions.

The method can evaluate the quality of EM solutions.

Abstract

The likelihood function of a finite mixture model is a non-convex function with multiple local maxima and commonly used iterative algorithms such as EM will converge to different solutions depending on initial conditions. In this paper we ask: is it possible to assess how far we are from the global maximum of the likelihood? Since the likelihood of a finite mixture model can grow unboundedly by centering a Gaussian on a single datapoint and shrinking the covariance, we constrain the problem by assuming that the parameters of the individual models are members of a large discrete set (e.g. estimating a mixture of two Gaussians where the means and variances of both Gaussians are members of a set of a million possible means and variances). For this setting we show that a simple upper bound on the likelihood can be computed using convex optimization and we analyze conditions under which the bound is guaranteed to be tight. This bound can then be used to assess the quality of solutions found by EM (where the final result is projected on the discrete set) or any other mixture estimation algorithm. For any dataset our method allows us to find a finite mixture model together with a dataset-specific bound on how far the likelihood of this mixture is from the global optimum of the likelihood