Conditional copula representations and extremal bounds for multivariate statistical functionals

What the math gives to ML

The paper provides a constructive separation between marginal behavior and dependence through probability-integral transforms and conditional copula distributions. This suggests a neural probabilistic-output head in which each marginal predictive distribution is modeled independently while a monotone conditional copula module captures cross-output dependence, avoiding the need for a full joint density network. The same representation supports exact joint sampling by recursively inverting conditional copula CDFs and permits targeted upper-tail stress tests using the survival-copula formula. The most promising initial application is calibrated multivariate prediction or generative modeling where marginal accuracy is good but correlations and tail co-movements are poorly represented.