Nonparametric density estimation is a challenging problem in theoretical statistics -- in general a maximum likelihood estimate (MLE) does not even exist! Introducing shape constraints allows a path forward.
In this talk I will first discuss non-parametric density estimation under total positivity (i.e. log-supermodularity) and log-concavity. Although they possess very special structure, totally positive random variables are quite common in real world data and have appealing mathematical properties. Given i.i.d. samples from a totally positive and log-concave distribution, we prove that the MLE exists with probability one assuming there are at least 3 samples. We characterize the domain of the MLE and if the observations are 2-dimensional, we show that the logarithm of the MLE is a tent function (i.e. a piecewise linear function) with "poles" at the observations, and we show that a certain convex program can find it.
I will finish by discussing density estimation for log-concave graphical models. As before, we show that the MLE exists and is unique with probability 1. We also characterize the domain of the MLE, and show how to find it if the graphical model corresponds to a chordal graph. I will conclude by discussing some future directions.
Speaker Biography
Dr. Robeva is an Assistant Professor with the Department of Mathematics at the University of British Columbia. From 2016 – 2019, Dr. Robeva was a Statistics Instructor and an NSF Postdoctoral Fellow in the Department of Mathematics and the Institute for Data, Systems, and Society, at the Massachusetts Institute of Technology. Dr. Robeva completed her PhD in 2016 from UC Berkeley, and won the Bernard Friedman Memorial Prize in Applied Mathematics, for her thesis.
About the Prize
The UBC-PIMS Mathematical Sciences Young Faculty Award prize was created by two founding donors, Anton Kuipers and Darrell Duffie, to recognize UBC researchers for their leading edge work in mathematics or its applications in the sciences. Dr Elina Robeva is the 2020 winner and will give her colloquium on Thursday April 21, 2021.
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