## uniform correlation mixtures

Kai Zhang and my friends from Wharton, Larry Brown, Ed George and Linda Zhao arXived last week a neat mathematical foray into the properties of a marginal bivariate Gaussian density once the correlation ρ is integrated out. While the univariate marginals remain Gaussian (unsurprising, since these marginals do not depend on ρ in the first place), the joint density has the surprising property of being

[1-Φ(max{|x|,|y|})]/2

which turns an infinitely regular density into a density that is not even differentiable everywhere. And which is constant on squares rather than circles or ellipses. This is somewhat paradoxical in that the intuition (at least my intuition!) is that integration increases regularity… I also like the characterisation of the distributions factorising through the infinite norm as scale mixtures of the infinite norm equivalent of normal distributions. The paper proposes several threads for some extensions of this most surprising result. Other come to mind:

1. What happens when the Jeffreys prior is used in place of the uniform? Or Haldane‘s prior?
2. Given the mixture representation of t distributions, is there an equivalent for t distributions?
3. Is there any connection with the equally surprising resolution of the Drton conjecture by Natesh Pillai and Xiao-Li Meng?
4. In the Khintchine representation, correlated normal variates are created by multiplying a single χ²(3) variate by a vector of uniforms on (-1,1). What are the resulting variates for other degrees of freedomk in the χ²(k) variate?
5. I also wonder at a connection between this Khintchine representation and the Box-Müller algorithm, as in this earlier X validated question that I turned into an exam problem.

### One Response to “uniform correlation mixtures”

1. Integrating out a 1D Gaussian with a standard deviation that has a prior denisty such that pi(\sigma=0) = c \in (0,\infty) leads to a density with a logarithmic spike at zero, so it’s not surprising that marginalising out a variance parameter makes the density rougher.

I imagine it would be even worse with a Jeffreys prior or something else that had spikes at the end points.