A Tight Convex Upper Bound on the Likelihood of a Finite Mixture
This provides a method for evaluating solution quality in mixture model estimation, which is incremental as it builds on existing discrete constraint approaches to offer theoretical guarantees.
The paper tackles the problem of assessing proximity to the global maximum of the likelihood in finite mixture models, which is non-convex and prone to local optima, by proposing a tight convex upper bound computed via convex optimization under a discrete parameter constraint, enabling quality evaluation of solutions like EM with dataset-specific bounds.
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