Tree Normals

Why

We name multivariate normal densities which factor according to a tree. These densities require tabulating fewer numbers.

Definition

A tree multivariate normal (or tree normal, tree gaussian) density is a multivariate normal density which factors according to a tree.

Fewer numbers

A multivariate normal on $\R ^n$ requires tabulating $n$ numbers for the mean and $n(n+1)/2$ numbers for the covariance matrix. If it factors according to a tree, we need only $n$ numbers to specify the covariance (one for the one-variable marginal, and $n-1$ for the covariance of the two-variable conditionals).