Information Geometry of Bivariate Families

From the study by Arwini [13], we provide information geometry, including the α-geometry, of several important families of bivariate probability density functions. They have marginal density functions that are gamma density functions, exponential density functions and Gaussian density functions. These are used for applications in the sequel, when we have two random variables that have non-zero covariance—such as will arise for a coupled pair of random processes.

The multivariate Gaussian is well-known and its information geometry has been reported before [183, 189]; our recent work has contributed the bivariate Gaussian α-geometry. Surprisingly, it is very difficult to construct a bivariate exponential distribution, or for that matter a bivariate Poisson distribution that has tractable information geometry. However we have calculated the case of the Freund bivariate mixture exponential distribution [89]. The only bivariate gamma distribution for which we have found the information geometry tractable is the McKay case [146] which is restricted to positive covariance, and we begin with this.