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 \emph{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).

翻译：我们建议对两个 Odinal 矢量 $xx 和 $y 之间的相互信息进行新的估计。 我们的方法是 $1 left(y; x\right) = I\ left(u_y; u_xright) = I\ left( lift( y_ y; u_x\right) = I droft( lift( i_ y; u_ x_right) $1 美元和 $u_x$ 美元是 emph{ copulla- unifyle 双面表示, 因为它仅通过某些非参数的 美元和 $x 来生成分配数据, (i. 在概率整体变异形中) 估计 $( left( u_ y_ y\right), 美元到 left( ru_xright) $left(y) $rental(y, y, u_xright) modeal_right) modeal-modeal-mode mode, modeal modeal_ dismodeal_ dismodeal.