  Optimal transport and Wasserstein distances are flourishing in many
scientific fields as a means for comparing and connecting random structures.
Here we pioneer the use of an optimal transport distance between L\'{e}vy
measures to solve a statistical problem. Dependent Bayesian nonparametric
models provide flexible inference on distinct, yet related, groups of
observations. Each component of a vector of random measures models a group of
exchangeable observations, while their dependence regulates the borrowing of
information across groups. We derive the first statistical index of dependence
in $[0,1]$ for (completely) random measures that accounts for their whole
infinite-dimensional distribution, which is assumed to be equal across
different groups. This is accomplished by using the geometric properties of the
Wasserstein distance to solve a max-min problem at the level of the underlying
L\'{e}vy measures. The Wasserstein index of dependence sheds light on the
models' deep structure and has desirable properties: (i) it is $0$ if and only
if the random measures are independent; (ii) it is $1$ if and only if the
random measures are completely dependent; (iii) it simultaneously quantifies
the dependence of $d \ge 2$ random measures, avoiding the need for pairwise
comparisons; (iv) it can be evaluated numerically. Moreover, the index allows
for informed prior specifications and fair model comparisons for Bayesian
nonparametric models.
