Hypothesis testing is a key component in Statistics. In practice, it is common for a practitioner to test several hypotheses, the so-called simultaneous hypothesis testing problem. Unfortunately, simultaneous hypothesis tests can be logically incoherent: for instance, if hypothesis H1 implies hypothesis H2, a procedure that rejects H2 should also reject H1, a property not always met by multiple test procedures. Indeed, previous results show that standard two-way hypothesis tests cannot be logically coherent. Three-way tests allow more nuanced data-based decisions. This paper studies whether Bayesian simultaneous three-way hypothesis tests can be logically coherent. Two types of results are obtained. First, under the standard error-wise constant loss, only for a limited set of models a Bayes simultaneous test can be logically coherent. Second, if more general loss functions are used, then it is possible to obtain Bayes simultaneous tests that are always logically coherent. An explicit example of such a loss function is provided. These results provide guidelines on how to build a logically coherent posterior probability three-way hypothesis test or more general Bayesian three-way hypothesis tests.

Conformal methods create prediction bands that control average coverage assuming solely i.i.d. data. Although the literature has mostly focused on prediction intervals, more general regions can often better represent uncertainty. For instance, a bimodal target is better represented by the union of two intervals. Such prediction regions are obtained by CD-split, which combines the split method and a data-driven partition of the feature space which scales to high dimensions. CD-split however contains many tuning parameters, and their role is not clear. In this paper, we provide new insights on CD-split by exploring its theoretical properties. In particular, we show that CD-split converges asymptotically to the oracle highest predictive density set and satisfies local and asymptotic conditional validity. We also present simulations that show how to tune CD-split. Finally, we introduce HPD-split, a variation of CD-split that requires less tuning, and show that it shares the same theoretical guarantees as CD-split. In a wide variety of our simulations, CD-split and HPD-split have better conditional coverage and yield smaller prediction regions than other methods.

Conformal methods create prediction bands that control average coverage assuming solely i.i.d. data. Besides average coverage, one might also desire to control conditional coverage, that is, coverage for every new testing point. However, without strong assumptions, conditional coverage is unachievable. Given this limitation, the literature has focused on methods with asymptotical conditional coverage. In order to obtain this property, these methods require strong conditions on the dependence between the target variable and the features. We introduce two conformal methods based on conditional density estimators that do not depend on this type of assumption to obtain asymptotic conditional coverage: Dist-split and CD-split. While Dist-split asymptotically obtains optimal intervals, which are easier to interpret than general regions, CD-split obtains optimal size regions, which are smaller than intervals. CD-split also obtains local coverage by creating prediction bands locally on a partition of the features space. This partition is data-driven and scales to high-dimensional settings. In a wide variety of simulated scenarios, our methods have a better control of conditional coverage and have smaller length than previously proposed methods.

The quantification problem consists of determining the prevalence of a given label in a target population. However, one often has access to the labels in a sample from the training population but not in the target population. A common assumption in this situation is that of prior probability shift, that is, once the labels are known, the distribution of the features is the same in the training and target populations. In this paper, we derive a new lower bound for the risk of the quantification problem under the prior shift assumption. Complementing this lower bound, we present a new approximately minimax class of estimators, ratio estimators, which generalize several previous proposals in the literature. Using a weaker version of the prior shift assumption, which can be tested, we show that ratio estimators can be used to build confidence intervals for the quantification problem. We also extend the ratio estimator so that it can: (i) incorporate labels from the target population, when they are available and (ii) estimate how the prevalence of positive labels varies according to a function of certain covariates.