Regression with n$\to$1 by Expert Knowledge Elicitation

We consider regression under the "extremely small $n$ large $p$" condition, where the number of samples $n$ is so small compared to the dimensionality $p$ that predictors cannot be estimated without prior knowledge. This setup occurs in personalized medicine, for instance, when predicting treatment outcomes for an individual patient based on noisy high-dimensional genomics data. A remaining source of information is expert knowledge, which has received relatively little attention in recent years. We formulate the inference problem of asking expert feedback on features on a budget, propose an elicitation strategy for a simple "small $n$" setting, and derive conditions under which the elicitation strategy is optimal. Experiments on simulated experts, both on synthetic and genomics data, demonstrate that the proposed strategy can drastically improve prediction accuracy.