2020/06/26 by Anya Katsevich, Afonso S. Bandeira, Katsevich, Anya +1 · 2 citations
Computer Science · Mathematics · Biochemistry, Genetics and Molecular Biology · #Bayesian Methods and Mixture Models #Statistical Methods and Bayesian Inference #Gene expression and cancer classification
paper · pdf · doi:10.48550/arxiv.2006.15202
We derive an asymptotic expansion for the log likelihood of Gaussian mixture models (GMMs) with equal covariance matrices in the low signal-to-noise regime. The expansion reveals an intimate connection between two types of algorithms for parameter estimation: the method of moments and likelihood optimizing algorithms such as Expectation-Maximization (EM). We show that likelihood optimization in the low SNR regime reduces to a sequence of least squares optimization problems that match the moments of the estimate to the ground truth moments one by one. This connection is a stepping stone toward the analysis of EM and maximum likelihood estimation in a wide range of models. A motivating application for the study of low SNR mixture models is cryo-electron microscopy data, which can be modeled as a GMM with algebraic constraints imposed on the mixture centers. We discuss the application of our expansion to algebraically constrained GMMs, among other example models of interest.