Pooling Sequential Evidence Across Hypotheses: Rate-Optimal Multiple Testing at a Fixed Horizon

Abstract

We study sequential testing of a fixed family of hypotheses when observations are costly and a sampling horizon is specified in advance. The challenge is to pool evidence for earlier decisions when the number and identities of false hypotheses are unknown, while controlling the probability of any false rejection at level $α$. Existing merges attain the pooled growth rate at a single number of false hypotheses: averaging when one is false, multiplying when all are. Under an independent-stream model with common simple null and alternative distributions, we test each intersection with a prior-weighted mixture of products of marginal likelihood ratios. Closed testing combines these elementary-symmetric-polynomial mixtures to identify individual false hypotheses. Design-specific boundary calibration gives finite-horizon family-wise error control, with exact finite-state guarantees or a confidence qualification for Monte Carlo calibration. The prior-matched mixture uniquely maximizes expected log evidence at each horizon. Mixtures assigning positive weight to every nonempty subset of streams attain log-growth rate $lD$ when $l$ streams follow the alternative. Here $D$ is the mean log likelihood ratio per alternative observation, and a round supplies one observation per stream. This rate attains the first-order intersection-delay lower bound as $α\downarrow0$ at fixed dimension, configuration, weights, and a long enough horizon. Power for an individual hypothesis cannot exceed the best single-stream power at a given deadline, but closure removes the multiplicity penalty when all are false. Gaussian, basket-trial, language-model and advertising studies illustrate both. The primary basket boundaries are 40-52% below $1/α$. Across 41 simulated configurations, the median reduction in capped mean patient outcomes relative to prespecified interim-look Bonferroni tests is 31%.