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.
Contextual bandit algorithms have transformed modern experimentation by enabling real-time adaptation for personalized treatment. Yet these advantages create challenges for statistical inference due to adaptivity. We study inference with contextual-bandit data without assuming a well-specified outcome model. In this setting, we show a previously overlooked issue: standard algorithms such as LinUCB may fail to stabilize under misspecified working models, leading to non-Gaussian estimator behavior and invalid inference. This issue is practically important, as misspecified working models -- such as approximations of complex dynamical systems -- are often employed by online agents in real-world adaptive experiments to balance reward, computational tractability, and robustness. We develop an inverse-probability-weighted Z-estimation framework for a broad class of marginal moment targets, including projection parameters, structural parameters with noisy contexts, and off-policy values. We identify a stability condition tailored to this framework, scaled inverse-propensity convergence, under which the IPW-Z estimator is consistent and asymptotically normal with a consistent sandwich variance estimator. We further establish sufficient conditions for scaled inverse-propensity convergence for several policy classes, including multi-armed bandit algorithms and smooth contextual allocation policies. Simulations and a HeartSteps V1 real-data-calibrated application show reliable coverage and competitive performance across multiple targets. Overall, our results highlight the importance of stability-aware adaptive design for valid post-experiment inference.