An information theoretic approach to statistical dependence: copula information

We discuss the connection between information and copula theories by showing that a copula can be employed to decompose the information content of a multivariate distribution into marginal and dependence components, with the latter quantified by the mutual information. We define the information excess as a measure of deviation from a maximum entropy distribution. The idea of marginal invariant dependence measures is also discussed and used to show that empirical linear correlation underestimates the amplitude of the actual correlation in the case of non-Gaussian marginals. The mutual information is shown to provide an upper bound for the asymptotic empirical log-likelihood of a copula. An analytical expression for the information excess of T-copulas is provided, allowing for simple model identification within this family. We illustrate the framework in a financial data set.Comment: to appear in Europhysics Letter

The mutual information between graphs

The estimation of mutual information between graphs has been an elusive problem until the formulation of graph matching in terms of manifold alignment. Then, graphs are mapped to multi-dimensional sets of points through structure preserving embeddings. Point-wise alignment algorithms can be exploited in this context to re-cast graph matching in terms of point matching. Methods based on bypass entropy estimation must be deployed to render the estimation of mutual information computationally tractable. In this paper the novel contribution is to show how manifold alignment can be combined with copula-based entropy estimators to efficiently estimate the mutual information between graphs. We compare the empirical copula with an Archimedean copula (the independent one) in terms of retrieval/recall after graph comparison. Our experiments show that mutual information built in both choices improves significantly state-of-the art divergences.Funding. F. Escolano, M.A. Lozano: Project TIN2012-32839 (Spanish Gov.). M. Curado: BES-2013-064482 (Spanish Gov.). E. R. Hancock: Royal Society Wolfson Research Merit Award

Non–Parametric Estimation of Mutual Information through the Entropy of the Linkage

A new, non–parametric and binless estimator for the mutual information of a d–dimensional random vector is proposed. First of all, an equation that links the mutual information to the entropy of a suitable random vector with uniformly distributed components is deduced. When d = 2 this equation reduces to the well known connection between mutual information and entropy of the copula function associated to the original random variables. Hence, the problem of estimating the mutual information of the original random vector is reduced to the estimation of the entropy of a random vector obtained through a multidimensional transformation. The estimator we propose is a two–step method: first estimate the transformation and obtain the transformed sample, then estimate its entropy. The properties of the new estimator are discussed through simulation examples and its performances are compared to those of the best estimators in the literature. The precision of the estimator converges to values of the same order of magnitude of the best estimator tested. However, the new estimator is unbiased even for larger dimensions and smaller sample sizes, while the other tested estimators show a bias in these cases