Flow-Based Conformal Predictive Distributions
Trevor Harris
TL;DR
The paper tackles the challenge of turning distribution-free conformal prediction sets into actionable predictive distributions in high-dimensional, structured output spaces. It introduces nonconformity flows, a training-free dynamical system driven by the score gradient, which globally converges to conformal boundaries, enabling efficient boundary sampling. By reconformalizing boundary samples, the method yields risk-controlling prediction bands, and by mixing across confidence levels, it constructs conformal predictive distributions (CPDs) that are calibrated and sampleable without likelihood or transport models. Empirical results across PDE inverse problems, precipitation downscaling, climate model debiasing, and hurricane trajectory forecasting demonstrate fast convergence, adaptive uncertainty representation, and competitive performance against established UQ baselines, with targeted sampling capabilities for rare events. This approach operationalizes conformal prediction for complex domains while preserving finite-sample guarantees and scalability.
Abstract
Conformal prediction provides a distribution-free framework for uncertainty quantification via prediction sets with exact finite-sample coverage. In low dimensions these sets are easy to interpret, but in high-dimensional or structured output spaces they are difficult to represent and use, which can limit their ability to integrate with downstream tasks such as sampling and probabilistic forecasting. We show that any differentiable nonconformity score induces a deterministic flow on the output space whose trajectories converge to the boundary of the corresponding conformal prediction set. This leads to a computationally efficient, training-free method for sampling conformal boundaries in arbitrary dimensions. Boundary samples can be reconformalized to form pointwise prediction sets with controlled risk, and mixing across confidence levels yields conformal predictive distributions whose quantile regions coincide exactly with conformal prediction sets. We evaluate the approach on PDE inverse problems, precipitation downscaling, climate model debiasing, and hurricane trajectory forecasting.