The sparse additive model (SpAM) offers a trade-off between interpretability and flexibility, and hence is a powerful model for high-dimensional research. This paper focuses on the variable selection, i.e., the univariate function selection problem in SpAM. We establish the minimax separation rates from both the perspectives of sparse multiple testing (FDR + FNR control) and support recovery (wrong recovery probability control). We further study how adaptation to unknown smoothness affects the minimax separation rate, and propose an adaptive selection procedure. Finally, we discuss the difference between estimation and selection in SpAM: Procedures achieving optimal function estimation may fail to achieve optimal univariate function selection.

Large-scale hypothesis testing supports probability claims about individual hypotheses, as in empirical Bayes methods for estimating local false discovery rates. We study how such claims can be interpreted as approximately calibrated forecasts of the null hypothesis, yielding interpretable error probabilities even under model misspecification. Our approach draws conceptual inspiration from probabilistic forecasting but addresses a different challenge: unlike forecasting, where labels are eventually observed, in multiple testing the ground truth is never revealed, so calibration must be assessed stochastically and established indirectly. We address this challenge by constructing a set of pseudo-labels, derived from the spacings of ordered $p$-values, which have the local false discovery rate as their regression target. Our construction unlocks existing tools for assessing and performing post-hoc calibration in multiple testing. Notably, we find on a large-scale empirical survey of published psychology and neuroscience literature that the $q$-value, a popular error measure based on the false discovery rate, can be severely miscalibrated.

E-values have recently emerged as a versatile alternative to p-values for statistical inference (Ramdas and Wang, 2025). They offer several advantages: they remain valid under optional stopping (Grünwald et al., 2024a), combine easily under arbitrary dependence, and exist for irregular problems where no other inferential method is known (Wasserman et al., 2020), among others. Beyond being useful, they have also proven necessary in various problems, such as multiple testing (Wang and Ramdas, 2022; Fischer and Ramdas, 2024; Xu et al., 2025), statistical contract theory (Bates et al., 2022), and post-hoc inference (Grünwald, 2024). Formally, an e-value is a nonnegative test statistic whose expected value is at most one under the null hypothesis. Ideally, analysts want e-values that are large under the alternative--that is, e-values with high power.

Real-time decision-making gets more attention in the big data era.

In bandit multiple hypothesis testing, each arm corresponds to a different null hypothesis that we wish to test, and the goal is to design adaptive algorithms that correctly identify large set of interesting arms (true discoveries), while only mistakenly identifying a few uninteresting ones (false discoveries). One common metric in non-bandit multiple testing is the false discovery rate (FDR). We propose a unified, modular framework for bandit FDR control that emphasizes the decoupling of exploration and summarization of evidence. We utilize the powerful martingale-based concept of e-processes to ensure FDR control for arbitrary composite nulls, exploration rules and stopping times in generic problem settings. In particular, valid FDR control holds even if the reward distributions of the arms could be dependent, multiple arms may be queried simultaneously, and multiple (cooperating or competing) agents may be querying arms, covering combinatorial semi-bandit type settings as well. Prior work has considered in great detail the setting where each arm's reward distribution is independent and sub-Gaussian, and a single arm is queried at each step. Our framework recovers matching sample complexity guarantees in this special case, and performs comparably or better in practice. For other settings, sample complexities will depend on the finer details of the problem (composite nulls being tested, exploration algorithm, data dependence structure, stopping rule) and we do not explore these; our contribution is to show that the FDR guarantee is clean and entirely agnostic to these details.

Neural Information Processing Systems http://nips.cc/

We propose an adaptive sampling approach for multiple testing which aims to maximize statistical power while ensuring anytime false discovery control.