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.
This paper explores policy learning from observational data, focusing on a nonlinear welfare criterion in a binary treatment setting. The nonlinear criterion is inspired by scenarios where policymakers prioritize specific population segments. We model this criterion using a utility function that encompasses potential outcomes and intermediate parameters, with the latter capturing higher moments of the outcome distributions. When formulated in the context of observational data, both the intermediate parameters and the welfare criterion depend on the propensity score, which we estimate using machine-learning techniques. To address bias in machine learning estimates, we introduce a novel reweighting-based debiasing approach that offers a promising alternative to traditional orthogonality-based methods. To tackle the complexities of infinite-dimensional policy spaces, we employ sieve approximations and $K$-fold cross-validation for model selection, thereby fully automating the policy-learning process. Despite these complexities, we demonstrate that both the welfare regret and the average welfare regret of our proposed policy learning method satisfy an oracle inequality, thereby providing theoretical guarantees on the performance of the estimated policy relative to the best possible policy. This finding extends the existing results from linear to nonlinear welfare criteria, from finite-dimensional to infinite-dimensional policy spaces, and from a known propensity score to a machine-learned one.