Estimating Conditional Mutual Information for Discrete-Continuous Mixtures using Multi-Dimensional Adaptive Histograms

TL;DR

This paper introduces a novel method for estimating conditional mutual information in mixed discrete-continuous data using adaptive histograms, enabling consistent estimation and improved benchmarking performance.

Contribution

It presents a new approach to estimate CMI for mixed data by leveraging adaptive histograms, extending existing methods to handle discrete-continuous mixtures.

Findings

The proposed estimator is consistent for discrete-continuous mixtures.

It outperforms state-of-the-art CMI estimators in benchmarks.

Effective in causal discovery tasks.

Abstract

Estimating conditional mutual information (CMI) is an essential yet challenging step in many machine learning and data mining tasks. Estimating CMI from data that contains both discrete and continuous variables, or even discrete-continuous mixture variables, is a particularly hard problem. In this paper, we show that CMI for such mixture variables, defined based on the Radon-Nikodym derivate, can be written as a sum of entropies, just like CMI for purely discrete or continuous data. Further, we show that CMI can be consistently estimated for discrete-continuous mixture variables by learning an adaptive histogram model. In practice, we estimate such a model by iteratively discretizing the continuous data points in the mixture variables. To evaluate the performance of our estimator, we benchmark it against state-of-the-art CMI estimators as well as evaluate it in a causal discovery setting.