Beyond Black Box Densities: Parameter Learning for the Deviated Components

TL;DR

This paper introduces a method for parameter learning in deviated mixture models, where the true density deviates from a known estimate, providing convergence rates for maximum likelihood estimates under Wasserstein metric.

Contribution

It proposes a novel distinguishability concept and establishes convergence rates for MLE of mixture parameters in deviated mixture models.

Findings

Convergence rates for MLE of mixture parameters are established.

Simulation studies validate the theoretical results.

Abstract

As we collect additional samples from a data population for which a known density function estimate may have been previously obtained by a black box method, the increased complexity of the data set may result in the true density being deviated from the known estimate by a mixture distribution. To model this phenomenon, we consider the \emph{deviating mixture model} $(1-\lambda^{*})h_0 + \lambda^{*} (\sum_{i = 1}^{k} p_{i}^{*} f(x|\theta_{i}^{*}))$, where $h_0$ is a known density function, while the deviated proportion $\lambda^{*}$ and latent mixing measure $G_{*} = \sum_{i = 1}^{k} p_{i}^{*} \delta_{\theta_i^{*}}$ associated with the mixture distribution are unknown. Via a novel notion of distinguishability between the known density $h_{0}$ and the deviated mixture distribution, we establish rates of convergence for the maximum likelihood estimates of $\lambda^{*}$ and $G^{*}$ under Wasserstein metric. Simulation studies are carried out to illustrate the theory.