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B1282
Title: Convergence properties of data augmentation algorithms for high-dimensional robit regression Authors:  Sourav Mukherjee - University of Florida (United States)
Kshitij Khare - University of Florida (United States)
Saptarshi Chakraborty - State University of New York at Buffalo (United States) [presenting]
Abstract: The logistic and probit link functions are common choices for binary regression problems. However, they are not robust to the presence of outliers. The robit link function, defined as the inverse CDF of the Student's t-distribution, provides a robust alternative to them. A multivariate normal prior for the regression coefficients is customary for Bayesian inference in robit regression models. The resulting posterior density is intractable, and a Data Augmentation (DA) Markov chain is used to generate approximate samples from the desired posterior. Establishing geometric ergodicity for this Markov chain is important as it provides theoretical guarantees for asymptotic validity of MCMC standard errors for desired posterior expectations/quantiles. Previous work established the geometric ergodicity of this robit DA chain, assuming restrictions on the sample size $n$, the number of predictors $p$, and design matrix $X$. We show that the robit DA Markov chain is trace-class for arbitrary choices of $n, p$, and $X$, and the prior mean and variance parameters. The trace-class property implies geometric ergodicity. Moreover, it allows us to conclude that the sandwich robit chain obtained by inserting an inexpensive extra step in between the two steps of the DA chain is strictly better than the robit DA chain in an appropriate sense, and enables the use of methods to estimate the spectral gap of trace class DA chains.