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On the hardness of group-conditional distribution-free predictive inference, an application to prediction with missing covariates.
Abstract
Predictive uncertainty quantification is crucial in decision-making problems.In this talk, we will focus on distribution-free uncertainty quantification by considering predictive intervals for the target Y enjoying validity (i.e. nominal coverage) with no assumptions on the underlying data generating process nor the sample size. After introducing the framework, we will detail the nuance between marginal validity and conditional---on the test point---validity. We will review the existing (impossibility) results on conditional validity. This will lead us to our main question: how can we relax the goal of conditional validity to make it achievable? We will present new hardness results, that characterize the limits of group conditional coverage (e.g., achieving nominal coverage not only on average but also among women on the one hand, and among men on the other hand), a weaker goal extensively used in the literature in place of the impossible perfect conditional validity.
Finally, we will dive into applying these results in the context of prediction with missing values. There, one wants to obtain not only marginally valid intervals despite missing values, but also intervals that achieve the nominal coverage regardless of which values are missing at test time. We provide an algorithm reaching this goal by constructing informative predictive intervals in light of our hardness results.
Based on a joint work with J. Josse, Y. Romano & A. Dieuleveut
