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# STOR Colloquium: Elina Robeva, MIT

## February 15, 2019 @ 2:30 pm - 3:30 pm

**Elina Robeva
**

**Massachusetts Institute of Technology**

**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 and log-concave 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.