An official website of the United States government
Conditional Mutual Information (CMI) is a measure of conditional dependence between random variables X and Y, given another random variable Z. It can be used to quantify conditional dependence among variables in many data-driven inference problems such as graphical models, causal learning, feature selection and time-series analysis. While k-nearest neighbor (kNN) based estimators as well as kernel-based methods have been widely used for CMI estimation, they suffer severely from the curse of dimensionality. In this paper, we leverage advances in classifiers and generative models to design methods for CMI estimation. Specifically, we introduce an estimator for KL-Divergence based on the likelihood ratio by training a classifier to distinguish the observed joint distribution from the product distribution. We then show how to construct several CMI estimators using this basic divergence estimator by drawing ideas from conditional generative models. We demonstrate that the estimates from our proposed approaches do not degrade in performance with increasing dimension and obtain significant improvement over the widely used KSG estimator. Finally, as an application of accurate CMI estimation, we use our best estimator for conditional independence testing and achieve superior performance than the state-of-the-art tester on both simulated and real data-sets.
more »
« less
Potential conditional mutual information: Estimators and properties
The conditional mutual information I(X; Y|Z) measures the average information that X and Y contain about each other given Z. This is an important primitive in many learning problems including conditional independence testing, graphical model inference, causal strength estimation and time-series problems. In several applications, it is desirable to have a functional purely of the conditional distribution py|x, z rather than of the joint distribution pX, Y, Z. We define the potential conditional mutual information as the conditional mutual information calculated with a modified joint distribution pY|X, ZqX, Z, where qX, Z is a potential distribution, fixed airport. We develop K nearest neighbor based estimators for this functional, employing importance sampling, and a coupling trick, and prove the finite k consistency of such an estimator. We demonstrate that the estimator has excellent practical performance and show an application in dynamical system inference.
more »
« less
- Award ID(s):
- 1651236
- PAR ID:
- 10057054
- Date Published:
- Journal Name:
- Communication, Control, and Computing (Allerton), 2017 55th Annual Allerton Conference on
- Page Range / eLocation ID:
- 1228 to 1235
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
We consider the problem of non-parametric Conditional Independence testing (CI testing) for continuous random variables. Given i.i.d samples from the joint distribution f (x, y, z) of continuous random vectors X, Y and Z, we determine whether X is independent Y |Z. We approach this by converting the conditional independence test into a classification problem. This allows us to harness very powerful classifiers like gradient-boosted trees and deep neural networks. These models can handle complex probability distributions and allow us to perform significantly better compared to the prior state of the art, for high-dimensional CI testing. The main technical challenge in the classification problem is the need for samples from the conditional product distribution fCI(x,y,z) = f(x|z)f(y|z)f(z) – the joint distribution if and only if X is independent Y |Z. – when given access only to i.i.d. samples from the true joint distribution f (x, y, z). To tackle this problem we propose a novel nearest neighbor bootstrap procedure and theoretically show that our generated samples are indeed close to f^{CI} in terms of total variational distance. We then develop theoretical results regarding the generalization bounds for classification for our problem, which translate into error bounds for CI testing. We provide a novel analysis of Rademacher type classification bounds in the presence of non-i.i.d near- independent samples. We empirically validate the performance of our algorithm on simulated and real datasets and show performance gains over previous methods.more » « less
-
Estimation of mutual information from observed samples is a basic primitive in machine learning, useful in several learning tasks including correlation mining, information bottleneck, Chow-Liu tree, and conditional independence testing in (causal) graphical models. While mutual information is a quantity well-defined for general probability spaces, estimators have been developed only in the special case of discrete or continuous pairs of random variables. Most of these estimators operate using the 3H -principle, i.e., by calculating the three (differential) entropies of X, Y and the pair (X,Y). However, in general mixture spaces, such individual entropies are not well defined, even though mutual information is. In this paper, we develop a novel estimator for estimating mutual information in discrete-continuous mixtures. We prove the consistency of this estimator theoretically as well as demonstrate its excellent empirical performance. This problem is relevant in a wide-array of applications, where some variables are discrete, some continuous, and others are a mixture between continuous and discrete components.more » « less
-
Estimation of information theoretic quantities such as mutual information and its conditional variant has drawn interest in recent times owing to their multifaceted applications. Newly proposed neural estimators for these quantities have overcome severe drawbacks of classical kNN-based estimators in high dimensions. In this work, we focus on conditional mutual information (CMI) estimation by utilizing its formulation as a minmax optimization problem. Such a formulation leads to a joint training procedure similar to that of generative adversarial networks. We find that our proposed estimator provides better estimates than the existing approaches on a variety of simulated datasets comprising linear and non-linear relations between variables. As an application of CMI estimation, we deploy our estimator for conditional independence (CI) testing on real data and obtain better results than state-of-the-art CI testersmore » « less
-
Information bottleneck (IB) is a technique for extracting information in one random variable X that is relevant for predicting another random variable Y. IB works by encoding X in a compressed “bottleneck” random variable M from which Y can be accurately decoded. However, finding the optimal bottleneck variable involves a difficult optimization problem, which until recently has been considered for only two limited cases: discrete X and Y with small state spaces, and continuous X and Y with a Gaussian joint distribution (in which case optimal encoding and decoding maps are linear). We propose a method for performing IB on arbitrarily-distributed discrete and/or continuous X and Y, while allowing for nonlinear encoding and decoding maps. Our approach relies on a novel non-parametric upper bound for mutual information. We describe how to implement our method using neural networks. We then show that it achieves better performance than the recently-proposed “variational IB” method on several real-world datasets.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Conference Paper:
- https://doi.org/10.1109/ALLERTON.2017.8262877
