The Multiple Quantile Graphical Model

Alnur Ali, Carnegie Mellon University

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We introduce the Multiple Quantile Graphical Model (MQGM), which extends the neighborhood selection approach of Meinshausen and Buhlmann for learning sparse graphical models. The latter is defined by the basic subproblem of modeling the conditional mean of one variable as a sparse function of all others. Our approach jointly models a set of conditional quantiles of one variable as a sparse function of all others and thus offers a much richer, more expressive class of conditional distribution estimates than standard neighborhood regression. We show that, under suitable regularity conditions, the MQGM recovers the exact conditional independencies with probability tending to one as the problem size grows, even outside of the usual homoskedastic Gaussian data model. We develop an efficient algorithm for fitting the MQGM using the alternative direction method of multipliers. We also describe a strategy for sampling from the joint distribution that underlies the MQGM estimate. Lastly, we present detailed experiments that demonstrate the flexibility and effectiveness of the MQGM in modeling hetereoskedastic non-Gaussian synthetic data, as well as real flu epidemic and sustainable energy examples.

Abstract Author(s): A. Ali, Z. Kolter, R. Tibshirani