## Gauss to Laplace transmutation!

**W**hen browsing X validated the other day [translate by procrastinating!], I came upon the strange property that the marginal distribution of a zero mean normal variate with exponential variance is a Laplace distribution. I first thought there was a mistake since we usually take an inverse Gamma on the variance parameter, not a Gamma. But then the marginal is a t distribution. The result is curious and can be expressed in a variety of ways:

– the product of a χ^{2}_{1} and of a χ_{2} is a χ^{2}_{2};

– the determinant of a 2×2 normal matrix is a Laplace variate;

– a difference of exponentials is Laplace…

The OP was asking for a direct proof of the result and I eventually sorted it out by a series of changes of variables, although there exists a much more elegant and general proof by Mike West, then at the University of Warwick, based on characteristic functions (or Fourier transforms). It reminded me that continuous, unimodal [at zero] and symmetric densities were necessary scale mixtures [a wee misnomer] of Gaussians. Mike proves in this paper that exponential power densities [including both the Normal and the Laplace cases] correspond to the variances having an inverse positive stable distribution with half the power. And this is a straightforward consequence of the exponential power density being proportional to the Fourier transform of a stable distribution and of a Fubini inversion. (Incidentally, the processing times of Biometrika were not that impressive at the time, with a 2-page paper submitted in Dec. 1984 published in Sept. 1987!)

This is a very nice and general derivation, but I still miss the intuition as to why it happens that way. But then, I know nothing, and even less about products of random variates!

October 30, 2015 at 5:57 am

Here is a simple proof of the Gauss-Laplace transmutation by using representation and moment generating functions:

http://arxiv.org/abs/1510.08765

October 15, 2015 at 10:35 pm

Christian, voir aussi http://arxiv.org/pdf/1408.5297.pdf pour une densité prédictive obtenue comme la différence de deux lois exponentielles. Éric

October 14, 2015 at 12:59 am

Thanks Christian for reminding us that property of the Laplace as a mixture of a centered normal and an exponential variance. Is not that property which is used in the Bayesian Lasso as shown in the 2008 JASA paper by Park & Casella and based on a more general theorem by

Andrews & Mallows (1974)?

October 14, 2015 at 9:37 am

Merci, Jean-Louis!, I had forgotten about this connection with George’s Bayesian Lasso.