Inference for Batched Adaptive Experiments

Jan Kemper; Davud Rostam-Afschar

econ.EMDec 10, 2025

Inference for Batched Adaptive Experiments

Authors: Jan Kemper, Davud Rostam-Afschar

Organizations: University of Mannheim, ZEW · University of Mannheim, IZA, GLO, NeSt

Abstract

The advantages of adaptive experiments have led to their rapid adoption in economics, other fields, and among practitioners. However, adaptive experiments pose challenges for causal inference. This note suggests a BOLS (batched ordinary least squares) test statistic for inference on treatment effects in adaptive experiments. The statistic provides a precision-equalizing aggregation of per-period treatment-control differences under heteroskedasticity. The combined test statistic is a linear combination of heteroskedastic per-period z-statistics and can be used to construct asymptotically valid confidence intervals. We provide simulation results comparing rejection rates in the typical case with few treatment periods and few (or many) observations per batch.

Figures & tables

Panel of Figure 1 — Figure 1: Simulation I: Non-normality of OLS and homoskedastic BOLS

Panel of Figure 2 — Figure 2: $\varepsilon$ -Greedy

Panel of Figure 3 — Figure 3: Bernoulli Thompson Sampling

Appendix figures & tables1 asset

Supplementary material from the paper’s appendix.

Appendix

Homoskedastic case: $\sigma_{1,t}^{2}=\sigma_{0,t}^{2}=\sigma^{2}$
	Non-adaptive	Adaptive
Realized share $\hat{\pi}_{t}$	$\hat{\pi}_{t}=\pi_{t}$ , constant across batches	$\pi_{t}$ depends on $\mathcal{H}_{t-1}$ , $\hat{\pi}_{t}$ random
Variance $v_{t}$	$\displaystyle v_{t}=\frac{\sigma^{2}}{n_{t}\,\pi_{t}(1-\pi_{t})}$	$\displaystyle v_{t}=\frac{\sigma^{2}}{n_{t}\,\hat{\pi}_{t}(1-\hat{\pi}_{t})}$
Weight $\displaystyle\omega_{t}=\frac{w_{t}}{S}$	$\displaystyle\frac{\sqrt{n_{t}}}{\sum_{s=1}^{T}\sqrt{n_{s}}}$	$\displaystyle\frac{\sqrt{n_{t}\,\hat{\pi}_{t}(1-\hat{\pi}_{t})}}{\sum_{s=1}^{T}\sqrt{n_{s}\,\hat{\pi}_{s}(1-\hat{\pi}_{s})}}$
Heteroskedastic case: $\sigma_{1,t}^{2}\neq\sigma_{0,t}^{2}$
Realized share $\hat{\pi}_{t}$	$\hat{\pi}_{t}=\pi_{t}$ , constant across batches	$\pi_{t}$ depends on $\mathcal{H}_{t-1}$ , $\hat{\pi}_{t}$ random

Appendix

Table 1: Weight structure under homoskedasticity and heteroskedasticity for two-armed experiments

Causal Effect Estimation

Causal effect estimation measures how outcomes would change under interventions, using experiments or observational data and explicit assumptions.

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Offline Policy Evaluation as a decision support tool for designing Adaptive Experiments

João Victor Ferreira Alves, Eduardo Rocha Laurentino, Gustavo de Oliveira Kanno, Thiago Costa Rizuti da Rocha

Instituto de Ciência e Tecnologia Itaú (ICTi), Brasil

Contextual Bandits A/b Testing Off-Policy Evaluation

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We investigate how historical data from fixed randomized experiments (A/B tests) can be used to inform the deployment of adaptive experiments based on contextual bandits. Given data collected under a static allocation, our goal is to assess which adaptive policies, if any, would have outperformed the original design and under what conditions. To this end, we combine off-policy evaluation (OPE) with a controlled warm-start simulation. From logged A/B test data exhibiting heterogeneous treatment effects, we estimate nuisance components and use doubly robust estimators to rank a portfolio of pre-specified adaptive and non-adaptive policies. When ground truth is available, we then deploy the same offline-trained policies in a simulator that reuses the exact data-generating reward probabilities, providing a safe, ground-truth-anchored environment to study the offline-to-online transition under warm starting. Using synthetic randomized controlled trials with known heterogeneity structures and an oracle policy, our results indicate that adaptive, context-aware policies improve upon fixed allocations when meaningful heterogeneity is present, while providing little benefit in its absence. We reinforce our findings on standard open benchmarks (Hillstrom, Criteo Uplift, and LaLonde), reinterpreted through a policy-value and regret perspective. Overall, our results provide a practical methodology for deciding when adaptive experimentation is worth deploying and how to select among competing adaptive policies using existing A/B test data.

Abstract

Causal Effect Estimation

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