  We propose a novel estimator of the mutual information between two ordinal
vectors $x$ and $y$. Our approach is inductive (as opposed to deductive) in
that it depends on the data generating distribution solely through some
nonparametric properties revealing associations in the data, and does not
require having enough data to fully characterize the true joint distributions
$P_{x, y}$. Specifically, our approach consists of (i) noting that $I\left(y;
x\right) = I\left(u_y; u_x\right)$ where $u_y$ and $u_x$ are the copula-uniform
dual representations of $y$ and $x$ (i.e. their images under the probability
integral transform), and (ii) estimating the copula entropies
$h\left(u_y\right)$, $h\left(u_x\right)$ and $h\left(u_y, u_x\right)$ by
solving a maximum-entropy problem over the space of copula densities under a
constraint of the type $\alpha_m = E\left[\phi_m(u_y, u_x)\right]$. We prove
that, so long as the constraint is feasible, this problem admits a unique
solution, it is in the exponential family, and it can be learned by solving a
convex optimization problem. The resulting estimator, which we denote MIND, is
marginal-invariant, always non-negative, unbounded for any sample size $n$,
consistent, has MSE rate $O(1/n)$, and is more data-efficient than competing
approaches. Beyond mutual information estimation, we illustrate that our
approach may be used to mitigate mode collapse in GANs by maximizing the
entropy of the copula of fake samples, a model we refer to as Copula Entropy
Regularized GAN (CER-GAN).
