Uniform Correlation Mixture of Bivariate Normal Distributions and Hypercubically-contoured Densities That Are Marginally Normal

Abstract

The bivariate normal density with unit variance and correlation $ρ$ is well-known. We show that by integrating out $ρ$, the result is a function of the maximum norm. The Bayesian interpretation of this result is that if we put a uniform prior over $ρ$, then the marginal bivariate density depends only on the maximal magnitude of the variables. The square-shaped isodensity contour of this resulting marginal bivariate density can also be regarded as the equally-weighted mixture of bivariate normal distributions over all possible correlation coefficients. This density links to the Khintchine mixture method of generating random variables. We use this method to construct the higher dimensional generalizations of this distribution. We further show that for each dimension, there is a unique multivariate density that is a differentiable function of the maximum norm and is marginally normal, and the bivariate density from the integral over $ρ$ is its special case in two dimensions.