Randomized Transport Plans via Hierarchical Fully Probabilistic Design

TL;DR

This paper introduces a hierarchical probabilistic framework for designing uncertain, randomized mass transport plans, extending classical optimal transport by incorporating Bayesian uncertainty and enabling diverse, fair market matching strategies.

Contribution

It develops HFPD-OT, a Bayesian approach to uncertain transport plans, providing methods for sampling, contracts, and uncertainty quantification, advancing beyond deterministic OT.

Findings

Enables generation of randomized transport plan samples

Provides measures of uncertainty in plans and contracts

Supports more diverse and fair market matching

Abstract

An optimal randomized strategy for design of balanced, normalized mass transport plans is developed. It replaces -- but specializes to -- the deterministic, regularized optimal transport (OT) strategy, which yields only a certainty-equivalent plan. The incompletely specified -- and therefore uncertain -- transport plan is acknowledged to be a random process. Therefore, hierarchical fully probabilistic design (HFPD) is adopted, yielding an optimal hyperprior supported on the set of possible transport plans, and consistent with prior mean constraints on the marginals of the uncertain plan. This Bayesian resetting of the design problem for transport plans -- which we call HFPD-OT -- confers new opportunities. These include (i) a strategy for the generation of a random sample of joint transport plans; (ii) randomized marginal contracts for individual source-target pairs; and (iii) consistent measures of uncertainty in the plan and its contracts. An application in fair market matching is outlined, in which HFPD-OT enables the recruitment of a more diverse subset of contracts -- than is possible in classical OT -- into the delivery of an expected plan.