Fixed-Horizon Self-Normalized Inference for Adaptive Experiments via Martingale AIPW/DML with Logged Propensities

TL;DR

This paper develops a method for valid statistical inference in adaptive experiments by using self-normalized martingale techniques, ensuring accurate confidence intervals even when variances are unpredictable.

Contribution

It introduces a martingale-based inference approach for adaptive experiments with logged treatment probabilities, allowing for valid inference without fixed variance assumptions.

Findings

Studentized statistics are asymptotically normal under the proposed method.

Simulations show the method outperforms traditional fixed-variance Wald tests.

The approach remains valid even with unpredictable variance regimes.

Abstract

Adaptive randomized experiments update treatment probabilities as data accrue, but still require an end-of-study interval for the average treatment effect (ATE) at a prespecified horizon. Under adaptive assignment, propensities can keep changing, so the predictable quadratic variation of AIPW/DML score increments may remain random. When no deterministic variance limit exists, Wald statistics normalized by a single long-run variance target can be conditionally miscalibrated given the realized variance regime. We assume no interference, sequential randomization, i.i.d. arrivals, and executed overlap on a prespecified scored set, and we require two auditable pipeline conditions: the platform logs the executed randomization probability for each unit, and the nuisance regressions used to score unit $t$ are constructed predictably from past data only. These conditions make the centered AIPW/DML scores an exact martingale difference sequence. Using self-normalized martingale limit theory, we show that the Studentized statistic, with variance estimated by realized quadratic variation, is asymptotically N(0,1) at the prespecified horizon, even without variance stabilization. Simulations validate the theory and highlight when standard fixed-variance Wald reporting fails.