My research focuses on studying models that depict complex dependencies between random variables. Such models include directed graphical models with hidden variables, discrete mixture models (which give rise to low rank tensors), and models that impose strong positive dependence called total positivity. In this talk I will give a brief overview of my work in these areas, and will particularly focus on the problem of density estimation under total positivity.
Nonparametric density estimation is a challenging statistical problem -- in general the maximum likelihood estimate (MLE) does not even exist! Introducing shape constraints such as total positivity allows a path forward. Though they possess very special structure, totally positive random variables are quite common in real world data and exhibit appealing mathematical properties. Given i.i.d. samples from a totally positive distribution, we prove that the MLE exists with probability one if there are at least 3 samples. We characterize the domain of the MLE, and give algorithms to compute it. If the observations are 2-dimensional or binary, we show that the logarithm of the MLE is a piecewise linear function and can be computed via a certain convex program. Finally, I will discuss statistical guarantees for the convergence of the MLE, and will conclude with a variety of further research directions.