  We commonly encounter the problem of identifying an optimally weight adjusted
version of the empirical distribution of observed data, adhering to predefined
constraints on the weights. Such constraints often manifest as restrictions on
the moments, tail behaviour, shapes, number of modes, etc., of the resulting
weight adjusted empirical distribution. In this article, we substantially
enhance the flexibility of such methodology by introducing a nonparametrically
imbued distributional constraints on the weights, and developing a general
framework leveraging the maximum entropy principle and tools from optimal
transport. The key idea is to ensure that the maximum entropy weight adjusted
empirical distribution of the observed data is close to a pre-specified
probability distribution in terms of the optimal transport metric while
allowing for subtle departures. The versatility of the framework is
demonstrated in the context of three disparate applications where data
re-weighting is warranted to satisfy side constraints on the optimization
problem at the heart of the statistical task: namely, portfolio allocation,
semi-parametric inference for complex surveys, and ensuring algorithmic
fairness in machine learning algorithms.
