Controlling Bias in Adaptive Data Analysis Using Information Theory
Modern data is messy and high-dimensional, and it is often not clear a priori what are the right questions to ask. Instead, the analyst typically needs to use the data to search for interesting analyses to perform and hypotheses to test. This is an adaptive process, where the choice of analysis to be performed next depends on the results of the previous analyses on the same data. It's widely recognized that this process, even if well-intentioned, can lead to biases and false discoveries, contributing to the crisis of reproducibility in science. But while adaptivity renders standard statistical theory invalid, folklore and experience suggest that not all types of adaptive analysis are equally at risk for false discoveries. In this paper, we propose a general information-theoretic framework to quantify and provably bound the bias and other statistics of an arbitrary adaptive analysis process. We prove that our mutual information based bound is tight in natural models, and then use it to give rigorous insights into when commonly used procedures do or do not lead to substantially biased estimation. We first consider several popular feature selection protocols, like rank selection or variance-based selection. We then consider the practice of adding random noise to the observations or to the reported statistics, which is advocated by related ideas from differential privacy and blinded data analysis. We discuss the connections between these techniques and our framework, and supplement our results with illustrative simulations.