Transformers as Bayesian In-Context Experimenters: Smoothness-Adaptive Efficient ATE Estimation
Authors: Jiachun Li, David Simchi-Levi
Abstract
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.
Bayesian online learning promises uncertainty-aware prediction on data streams, but its performance hinges on inferential choices, including learning rates, prior distributions and variational families, which are usually fixed before seeing the stream. We address this by treating Bayesian update rules as experts and aggregating the Bayesian experts according to sequential predictive losses. We prove that the resulting aggregate competes with the best expert in hindsight at an aggregation cost determined by how each expert's per-round performance is evaluated. We instantiate the framework in online conformal inference and Gaussian process regression. The conformal inference application yields a smoothed Bayesian counterpart of adaptive conformal inference with long-run randomized coverage, while the Gaussian process application gives an oracle inequality in cumulative predictive Kullback-Leibler risk and adaptation to unknown Hölder smoothness up to logarithmic factors. Experiments show that the aggregate tracks strong experts without oracle expert selection.
Amortized inference is the task of training a parametric model, such as a neural network, to approximate a distribution with a given unnormalized density where exact sampling is intractable. When sampling is implemented as a sequential decision-making process, reinforcement learning (RL) methods, such as generative flow networks, can be used to train the sampling policy. Off-policy RL training facilitates the discovery of diverse, high-reward candidates, but existing methods still face challenges in efficient exploration. We propose to use an adaptive training distribution (the \teacher) to guide the training of the primary amortized sampler (the \student). The \teacher, an auxiliary behavior model, is trained to sample high-loss regions of the \student and can generalize across unexplored modes, thereby enhancing mode coverage by providing an efficient training curriculum. We validate the effectiveness of this approach in a synthetic environment designed to present an exploration challenge, two diffusion-based sampling tasks, and four biochemical discovery tasks demonstrating its ability to improve sample efficiency and mode coverage. Source code is available at https://github.com/alstn12088/adaptive-teacher.
We theoretically justify the recent empirical finding of [Teh et al., 2025] that a transformer pretrained on synthetically generated data achieves strong performance on empirical Bayes (EB) problems. We take an indirect approach to this question: rather than analyzing the model architecture or training dynamics, we ask why a pretrained Bayes estimator, trained under a prespecified training distribution, can adapt to arbitrary test distributions. Focusing on Poisson EB problems, we identify the existence of universal priors such that training under these priors yields a near-optimal regret bound of O(n1) uniformly over all test distributions. Our analysis leverages the classical phenomenon of posterior contraction in Bayesian statistics, showing that the pretrained transformer adapts to unknown test distributions precisely through posterior contraction. This perspective also explains the phenomenon of length generalization, in which the test sequence length exceeds the training length, as the model performs Bayesian inference using a generalized posterior.