## Mathematics Information and Applications Seminar: Dr Elina Robeva

- Date: 01/17/2019
- Time: 12:30

University of British Columbia

Maximum likelihood 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 allows a path forward. In this talk I will discuss non-parametric density estimation under total positivity (i.e. log-supermodularity). 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.

Location: ESB 4133