[Paper Review] Admissible anytime-valid sequential inference must rely on nonnegative martingales

Aaditya Ramdas, Johannes Ruf+2 more|arXiv (Cornell University)|Sep 7, 2020

Statistical Methods and Inference20 references30 citations

TL;DR

The paper proves that admissible constructions for confidence sequences, p-processes, e-processes, and sequential tests in anytime-valid sequential inference must rely on nonnegative martingales, including extensions with max-martingales and Snell envelopes for composite settings.

ABSTRACT

Confidence sequences, anytime p-values (called p-processes in this paper), and e-processes all enable sequential inference for composite and nonparametric classes of distributions at arbitrary stopping times. Examining the literature, one finds that at the heart of all these (quite different) approaches has been the identification of nonnegative (super)martingales. Thus, informally, nonnegative (super)martingales are known to be sufficient for \emph{anytime-valid} sequential inference, even in composite and nonparametric settings. Our central contribution is to show that nonnegative martingales are also universal -- after appropriately defining \emph{admissibility}, we show that all admissible constructions of confidence sequences, p-processes, or e-processes must necessarily utilize nonnegative martingales. Our proofs utilize several modern mathematical tools for composite testing and estimation problems: max-martingales, Snell envelopes, transfinite induction, and new Doob-Lévy martingales make appearances in previously unencountered ways. Informally, if one wishes to perform anytime-valid sequential inference, then any existing approach can be recovered or dominated using nonnegative martingales. We provide several nontrivial examples, with special focus on testing symmetry, where our new constructions render past methods inadmissible. We also prove the subGaussian supermartingale to be admissible.

Motivation & Objective

• Motivate and formalize anytime-valid sequential inference tools (p-processes, confidence sequences, e-processes, sequential tests).
• Characterize admissibility and show nonnegative martingales are necessary for all admissible constructions in composite and point-null settings.
• Develop reductions and tools (max-martingales, Snell envelopes) to establish necessity and sufficiency results for admissibility.
• Provide examples (Gaussian, subGaussian, symmetry) illustrating admissibility and inadmissibility of prior methods.
• Extend admissibility analysis to estimation tasks via confidence sequences and composite hypotheses.

Proposed method

• Define and relate p-processes, e-processes, confidence sequences, and Robbins-style sequential tests in a unified time-uniform framework.
• Use nonnegative (super)martingales, max-martingales, and Ville’s inequality to derive admissibility criteria.
• Establish necessary and sufficient conditions for admissibility in point-null and composite-null settings via Doob–Lévy max-martingales and Snell envelope decompositions.
• Demonstrate reductions from composite to point-null cases to extend admissibility results to estimation (confidence sequences).
• Provide illustrative exponential-family examples (Gaussian, subGaussian, symmetry) to show admissible constructions can dominate previous methods.

Experimental results

Research questions

• RQ1 What are the necessary and sufficient conditions for admissible p-processes under composite nulls?
• RQ2 Are nonnegative martingales necessary for all admissible anytime-valid sequential inference tools (p-processes, e-processes, confidence sequences, sequential tests)?
• RQ3 How can max-martingales and Snell envelopes be used to establish admissibility results in composite settings?
• RQ4 Can reductions from composite to point-null cases yield general admissibility conclusions for confidence sequences and e-processes?
• RQ5 Do standard constructions (e.g., subGaussian martingales) achieve admissibility, and where do previous methods become inadmissible (e.g., testing symmetry)?

Key findings

• Nonnegative martingales are universally necessary for admissible anytime-valid sequential inference across p-processes, e-processes, confidence sequences, and sequential tests.
• Max-martingales and Doob–Lévy constructions, together with Snell envelopes, provide the framework to prove necessary and sufficient admissibility conditions in composite settings.
• Admissible inference can be reduced from composite to point-null cases, enabling general admissibility results for confidence sequences and estimation.
• Robbins’ subGaussian martingale is shown to be admissible, while some symmetry-based tests are shown to be inadmissible against new admissible constructions.
• The paper presents new Doob-Lévy max-martingale tools and anti-concentration bounds to deepen the admissibility analysis.