Accurate Estimation of Mutual Information in High Dimensional Data

Abstract: Mutual information (MI) is a measure of statistical dependencies between two variables, widely used in data analysis. Thus, accurate methods for estimating MI from empirical data are crucial. Such estimation is a hard problem, and there are provably no estimators that are universally good for finite datasets. Common estimators struggle with high-dimensional data, which is a staple of modern experiments. Recently, promising machine learning-based MI estimation methods have emerged. Yet it remains unclear if and when they produce accurate results, depending on dataset sizes, statistical structure of the data, and hyperparameters of the estimators, such as the embedding dimensionality or the duration of training. There are also no accepted tests to signal when the estimators are inaccurate. Here, we systematically explore these gaps. We propose and validate a protocol for MI estimation that includes explicit checks ensuring reliability and statistical consistency. Contrary to accepted wisdom, we demonstrate that reliable MI estimation is achievable even with severely undersampled, high-dimensional datasets, provided these data admit accurate low-dimensional representations. These findings broaden the potential use of machine learning-based MI estimation methods in real-world data analysis and provide new insights into when and why modern high-dimensional, self-supervised algorithms perform effectively.