  Conformal prediction has received tremendous attention in recent years and
has offered new solutions to problems in missing data and causal inference; yet
these advances have not leveraged modern semiparametric efficiency theory for
more robust and efficient uncertainty quantification. In this paper, we
consider the problem of obtaining distribution-free prediction regions
accounting for a shift in the distribution of the covariates between the
training and test data. Under an explainable covariate shift assumption
analogous to the standard missing at random assumption, we propose three
variants of a general framework to construct well-calibrated prediction regions
for the unobserved outcome in the test sample. Our approach is based on the
efficient influence function for the quantile of the unobserved outcome in the
test population combined with an arbitrary machine learning prediction
algorithm, without compromising asymptotic coverage. Next, we extend our
approach to account for departure from the explainable covariate shift
assumption in a semiparametric sensitivity analysis for potential latent
covariate shift. In all cases, we establish that the resulting prediction sets
eventually attain nominal average coverage in large samples. This guarantee is
a consequence of the product bias form of our proposal which implies correct
coverage if either the propensity score or the conditional distribution of the
response is estimated sufficiently well. Our results also provide a framework
for construction of doubly robust prediction sets of individual treatment
effects, under both unconfoundedness and allowing for some degree of unmeasured
confounding. Finally, we discuss aggregation of prediction sets from different
machine learning algorithms for optimal prediction and illustrate the
performance of our methods in both synthetic and real data.
