Efficient Estimation of Mutual Information for Strongly Dependent Variables

We demonstrate that a popular class of nonparametric mutual information (MI) estimators based on k-nearest-neighbor graphs requires number of samples that scales exponentially with the true MI. Consequently, accurate estimation of MI between two strongly dependent variables is possible only for prohibitively large sample size. This important yet overlooked shortcoming of the existing estimators is due to their implicit reliance on local uniformity of the underlying joint distribution. We introduce a new estimator that is robust to local non-uniformity, works well with limited data, and is able to capture relationship strengths over many orders of magnitude. We demonstrate the superior performance of the proposed estimator on both synthetic and real-world data.Comment: 13 pages, to appear in International Conference on Artificial Intelligence and Statistics (AISTATS) 201

Quantifying dependencies for sensitivity analysis with multivariate input sample data

We present a novel method for quantifying dependencies in multivariate datasets, based on estimating the R\'{e}nyi entropy by minimum spanning trees (MSTs). The length of the MSTs can be used to order pairs of variables from strongly to weakly dependent, making it a useful tool for sensitivity analysis with dependent input variables. It is well-suited for cases where the input distribution is unknown and only a sample of the inputs is available. We introduce an estimator to quantify dependency based on the MST length, and investigate its properties with several numerical examples. To reduce the computational cost of constructing the exact MST for large datasets, we explore methods to compute approximations to the exact MST, and find the multilevel approach introduced recently by Zhong et al. (2015) to be the most accurate. We apply our proposed method to an artificial testcase based on the Ishigami function, as well as to a real-world testcase involving sediment transport in the North Sea. The results are consistent with prior knowledge and heuristic understanding, as well as with variance-based analysis using Sobol indices in the case where these indices can be computed

Advances in Feature Selection with Mutual Information

The selection of features that are relevant for a prediction or classification problem is an important problem in many domains involving high-dimensional data. Selecting features helps fighting the curse of dimensionality, improving the performances of prediction or classification methods, and interpreting the application. In a nonlinear context, the mutual information is widely used as relevance criterion for features and sets of features. Nevertheless, it suffers from at least three major limitations: mutual information estimators depend on smoothing parameters, there is no theoretically justified stopping criterion in the feature selection greedy procedure, and the estimation itself suffers from the curse of dimensionality. This chapter shows how to deal with these problems. The two first ones are addressed by using resampling techniques that provide a statistical basis to select the estimator parameters and to stop the search procedure. The third one is addressed by modifying the mutual information criterion into a measure of how features are complementary (and not only informative) for the problem at hand