Hybrid mutual information between directional and linear variables

Abstract

Measuring the mutual dependence between two linear variables has been studied

at length in Rényi (1959a,b) and Lloyd (1962), among many others. Mutual in-
formation (Shannon 1949, Cover and Thomas 2012) between two linear variables

is a general measure that determines the similarity between the joint distribu-
tion and the product of their marginal distributions. For directional statistics,

the circular mutual information was recently proposed in Leguey et al. (2016).
This is suitable when the underlying paired distributions follow bivariate wrapped
Cauchy distributions (Kato and Pewsey 2015), whose marginals and conditionals
belong to the univariate wrapped Cauchy family.
Here we go one step further by presenting the hybrid mutual information, which
allows to express in a closed form the mutual information measure between a
circular-linear or a linear-circular pair of variables regardless of the marginal
distribution of each variable.