Mathematical Sciences Department Research Talk- Gonzalo Contador, Universidad Técnica Federico Santa María in Santiago, Chile

Speaker: Gonzalo Contador, Universidad Técnica Federico Santa María in Santiago, Chile

Friday, February 28th 

10:00 - 11:00 am

Stratton 205

Optional zoom meeting ID: 912 7870 1077

Title: Differentiability and Approximation of Probability Functions under Gaussian Mixture Models: A Bayesian Approach

Abstract: In this talk, we study stochastically constrained optimization problems probability functions associated with Gaussian mixture models. Our primary focus is on extending the use of spherical radial decomposition for multivariate Gaussian random vectors to the context of Gaussian mixture models, which are not inherently spherical but only conditionally so. Specifically, the conditional probability distribution, given a random parameter, follows a Gaussian distribution, allowing us to apply Bayesian analysis tools to the probability function. This assumption, together with spherical radial decomposition for Gaussian random vectors, enables us to represent the probability function as an integral over the Euclidean sphere. Using this representation, we establish sufficient conditions to ensure the differentiability of the probability function and provide and integral representation of its gradient. Furthermore, leveraging the Bayesian decomposition, we approximate the probability function using random sampling over the parameter space and the Euclidean sphere. Finally, we present numerical examples that illustrate the advantages of this approach over classical approximations based on random vector sampling.

Bio: Gonzalo Contador, Assistant Professor at the Mathematics Department of Universidad Técnica Federico Santa María in Santiago, Chile. PhD in Statistics from University of Wisconsin-Madison and former post doctoral scholar at the department of Mathematical Sciences of Worcester Polytechnic Institute.