Convergence Rates for Gaussian Mixtures of Experts

We provide a theoretical treatment of over-specified Gaussian mixtures of experts with covariate-free gating networks. We establish the convergence rates of the maximum likelihood estimation (MLE) for these models. Our proof technique is based on a novel notion of \emph{algebraic independence} of the expert functions. Drawing on optimal transport theory, we establish a connection between the algebraic independence and a certain class of partial differential equations (PDEs). Exploiting this connection allows us to derive convergence rates and minimax lower bounds for parameter estimation.

Posterior Distribution for the Number of Clusters in Dirichlet Process Mixture Models

Dirichlet process mixture models (DPMM) play a central role in Bayesian nonparametrics, with applications throughout statistics and machine learning. DPMMs are generally used in clustering problems where the number of clusters is not known in advance, and the posterior distribution is treated as providing inference for this number. Recently, however, it has been shown that the DPMM is inconsistent in inferring the true number of components in certain cases. This is an asymptotic result, and it would be desirable to understand whether it holds with finite samples, and to more fully understand the full posterior. In this work, we provide a rigorous study for the posterior distribution of the number of clusters in DPMM under different prior distributions on the parameters and constraints on the distributions of the data. We provide novel lower bounds on the ratios of probabilities between $s+1$ clusters and $s$ clusters when the prior distributions on parameters are chosen to be Gaussian or uniform distributions.