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
Figure 1: Simulation I: Non-normality of OLS and homoskedastic BOLS
Figure 2: ε -Greedy
Figure 3: Bernoulli Thompson Sampling
Appendix figures & tables1 asset
Supplementary material from the paper’s appendix.
Appendix
Non-adaptive
Adaptive
Homoskedastic case: σ1,t2=σ0,t2=σ2
Realized share π^t
π^t=πt , constant across batches
πt depends on Ht−1 , π^t random
Variance vt
vt=ntπt(1−πt)σ2
vt=ntπ^t(1−π^t)σ2
Weight ωt=Swt
∑s=1Tnsnt
∑s=1Tnsπ^s(1−π^s)ntπ^t(1−π^t)
Heteroskedastic case: σ1,t2=σ0,t2
Realized share π^t
π^t=πt , constant across batches
πt depends on Ht−1 , π^t random
Appendix
Table 1: Weight structure under homoskedasticity and heteroskedasticity for two-armed experiments
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.
João Victor Ferreira Alves, Eduardo Rocha Laurentino, Gustavo de Oliveira Kanno +1
Instituto de Ciência e Tecnologia Itaú (ICTi), Brasil
Adaptive experimentation enables efficient estimation of causal effects, but existing methods are not designed for survival data with censoring, where event times are only partially observed (e.g., overall survival in cancer trials but with dropout). In this paper, we develop a novel framework for adaptive experimentation to estimate causal effects under right censoring. For this, we derive the semiparametric efficiency bound for the average survival effect curve as a function of the treatment allocation policy and thereby obtain a closed-form efficiency-optimal allocation policy. The policy generalizes classical Neyman allocation to survival settings by prioritizing patient strata where both event and censoring dynamics induce high uncertainty. Building on this, we propose the Adaptive Survival Estimator (ASE), an adaptive framework that learns the allocation policy and estimates the average survival effect curve sequentially. Our framework has three main benefits: (i) it accommodates arbitrary machine learning models for nuisance estimation; (ii) it is guided by a closed-form efficiency-optimal allocation policy; and (iii) it admits strong theoretical guarantees, including asymptotic normality via a martingale central limit theorem. We demonstrate our framework across various numerical experiments to show consistent efficiency gains over uniform randomization and censoring-agnostic baselines.
Yuxin Wang, Dennis Frauen, Jonas Schweisthal +3
LMU Munich Munich Center of Machine Learning (MCML)
Adaptive experiments for average treatment effects (ATE) require randomized allocations balancing valid inference with statistical efficiency. The oracle design is a covariate-dependent Neyman rule governed by unknown arm-conditional outcome variances. We investigate whether this sequential variance-estimation and allocation process can be amortized via in-context learning. We introduce Bayesian in-context experimenters: transformer policies trained to imitate a Bayesian posterior Neyman teacher. The teacher updates nonparametric beliefs over potential outcomes using experimental history to assign posterior Neyman treatment probabilities. This design converges to the oracle rule, supporting efficient ATE inference. Transformers constructively implement this mapping through attention-based sufficient statistics and projected gradient descent, imitating Bayesian updating for Gaussian-series priors. To address unknown outcome smoothness, we combine smoothness-indexed experimenters using a mixture-of-experts transformer. The gate acts as a hierarchical posterior over smoothness classes, concentrating on near-oracle experts. By bounding the complexity of the transformer class, we prove this amortized policy can be learned via empirical risk minimization using supervised pretraining. Experiments confirm accurate teacher imitation, adaptive allocation, and improved ATE precision over baselines.
Jiachun Li, David Simchi-Levi
Laboratory for Information and Decision Systems, MIT