# Duke Probability Seminar

### A seminar for the probability community at Duke, both in and outside of the Mathematics Department.

#### More information on the Duke Probability community can be found on the Probability: Theory and Applications Page

To joint the Duke probability mailing list visit: subscribe to probability-seminar.

Upcoming Seminars:
• Thursday, August 23, 2018, 3:15pm, 235 Physics, Probability Seminar
Conjugate gradient-accelerated Gibbs sampler for "large n and large p" sparse Bayesian logistic regression
Aki Nishimura

In a modern observational study based on healthcare databases, the number of observations is typically in the order of 10^5 ~ 10^6 and that of the predictors in the order of 10^4 ~ 10^5. Despite the large sample size, the data rarely provide enough information to reliably estimate such a large number of parameters. Sparse regression provides a potential solution to this problem. There is a rich literature on desirable theoretical properties of the Bayesian approach based on shrinkage prior. On the other hand, the development of scalable methods for the required posterior computation has largely been limited to the p >> n case. While shrinkage priors are designed to make the posterior amenable to Gibbs sampling, a major computational bottleneck still arises from the need to sample from a high-dimensional Gaussian distribution at each iteration. The closed form expression for the precision matrix $\Phi$ is available, but computing and factorizing such a large matrix is computationally expensive nonetheless. In this article, we present a novel algorithm to speed up this bottleneck based on the following observation: we can cheaply generate a random vector $b$ such that the solution of a linear system $\Phi \beta = b$ has the desired Gaussian distribution. An accurate solution of the linear system can then be found by the conjugate gradient algorithm with only a small number of the matrix-vector multiplications by $\Phi$, without ever explicitly inverting $\Phi$. We apply our algorithm to a large-scale observational study with n = 72,489 and p = 22,175, designed to assess the relative risk of intracranial hemorrhage from two alternative blood anti-coagulants. Our algorithm demonstrates an order of magnitude speed-up in the posterior computation.

• Thursday, September 13, 2018, 3:15pm, 119 Physics, Probability Seminar
TBA

• Thursday, September 20, 2018, 3:15pm, 119 Physics, Probability Seminar
Stable coexistence of savannah and forest in a spatial model
Carla Staver

The goal of this talk is to further a joint project involving Carla Staver, Simon Levin, Rick Durrett, and Ruibo Ma. The puzzle is: why can savannah and forest display stable coexistence when this is not possible in a spatially homogeneous system.

• Thursday, September 27, 2018, 3:15pm, 119 Physics, Probability Seminar
TBA
Yu-ting Chen (U of Tennessee, Knoxviller)

• Thursday, October 4, 2018, 3:15pm, 119 Physics, Probability Seminar
TBA
Erik Slivken (Paris Diderot)

• Thursday, November 8, 2018, 3:15pm, 119 Physics, Probability Seminar
title
Pascal Maillard (CRM (Montréal) and Université Paris-Sud)

abstract

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