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# Professor Esteban Tabak, Courant Institute of Mathematical Sciences – Mathematics Colloquium

## September 26, 2019 @ 4:00 pm - 5:00 pm

**Title:** Conditional probability estimation and factor discovery through optimal transport

**Abstract:** Conditional probability estimation provides data-based answers to critical questions, such as the expected response of specific patients to different medical treatments, weather forecasts and the effect of political measures on the economy. In the complex systems behind these examples, the outcome x of a process depends on many and diverse factors z. In addition, x is probabilistic in nature due in part to our ignorance of other relevant factors and to the chaotic nature of the underlying dynamics.

This talk will describe a general procedure for the estimation and simulation of the conditional probability density P(x|z) underlying a sample set {x_i, z_i}. The methodology relies on a data-driven formulation of the Wasserstein barycenter problem, posed as an adversarial problem in terms of two flows: one that carries each sample point x_i to a corresponding sample y_i of the barycenter, and another that deforms a test function so as to enforce the independence of the variables {y_i} and {z_i}.

Time allowing, we will discuss an extension of this methodology to factor discovery: the determination of latent variables w that, combined with the z, help explain the largest possible amount of variability in x, generalizing classical ideas such as clustering and principal component analysis.