Date of this Version
The American Statistician
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.
This is an Accepted Manuscript of an article published by Taylor & Francis in The American Statistician on 14 Apr 2014, available online: http://wwww.tandfonline.com/10.1080/00031305.2014.909741.
bivariate normal mixture, khintchine mixture, uniform prior over correlation
Zhang, K., Brown, L. D., George, E. I., & Zhao, L. (2014). Uniform Correlation Mixture of Bivariate Normal Distributions and Hypercubically-Contoured Densities That Are Marginally Normal. The American Statistician, 68 (3), 183-187. http://dx.doi.org/10.1080/00031305.2014.909741
Date Posted: 27 November 2017