  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 models a set of conditional quantiles of one variable as a sparse
function of all others, and hence offers a much richer, more expressive class
of conditional distribution estimates. We establish that, under suitable
regularity conditions, the MQGM identifies 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 alternating 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 data.
