Making uncertainty measurable when AI predicts whole distributions

A model might predict next month's distribution of house prices rather than a single average. That raises a second question: which alternative distributions remain plausible? Trung-Khang Tran, Tuan A. Hoang, Viet-Hoang Tran and Tan M. Nguyen address this in Conformal Prediction for Distribution-to-Distribution Regression. Their study is available as an arXiv preprint submitted on November 4, 2026. The manuscript carries a NeurIPS 2026 conference footer; the supplied landing page does not establish journal publication.

The task involves learning from paired distributions, each observed through finite samples. Both the input and the output are therefore more complicated than ordinary feature vectors and numerical labels. The authors' contribution is an uncertainty-quantification layer around an existing distribution-regression model, not a replacement for that predictor.

Why a valid set can still be hard to use

Split conformal prediction separates model training from calibration. After fitting a predictor, it measures prediction errors on held-out examples and chooses a threshold corresponding to a requested coverage level. Under exchangeability—roughly, calibration and test examples being statistically interchangeable—the resulting sets have a finite-sample marginal coverage guarantee.

Marginal coverage is an average-over-examples statement, not a promise for every particular input. A further difficulty arises when the outputs are distributions: directly defining a set of acceptable density functions leaves an infinite-dimensional object. Its volume may be ill-defined or computationally intractable, and there is no canonical way to sample uniformly from it. The authors argue that formal coverage alone does not make such a set practically interpretable.

Turning curves into coordinates

Their solution represents output densities using an orthonormal basis: a collection of component functions whose weighted sum describes a curve. Experiments primarily use Fourier bases. Keeping only finitely many components turns each output into a coefficient vector estimated from its observed samples.

The construction assumes smoothness expressed through a Sobolev ellipsoid condition, which limits weighted high-frequency coefficients while allowing different regularity across dimensions. Calibration scores measure weighted distances between predicted and observed coefficients. A calibrated threshold then defines an ellipsoid around the model's prediction in coefficient space.

This region has a closed-form volume and supports uniform sampling. Mapping sampled vectors back through the basis produces candidate output functions. Here, interpretability means accessible geometry and sampling, rather than an explanation of the predictor's internal reasoning.

An important boundary is that the formal coverage theorem concerns the finite-sample, truncated approximation of the output density, not exact inclusion of an unrestricted latent density. The formulation also explicitly allows functions that need not be nonnegative or integrate to one. Consequently, the ellipsoid should not automatically be read as containing only valid probability densities.

Adapting uncertainty to the input

The adaptive variant calibrates separately for each test input using its nearest neighboring input distributions. This allows uncertainty regions to change size instead of sharing one global threshold.

Its conditional-coverage result is asymptotic. It requires regular conditional score distributions, consistent estimation of the local score distribution, and a neighborhood whose size grows while its fraction of the calibration set shrinks. It is not an unconditional finite-data guarantee of the requested coverage for every individual input.

What the experiments show

Tests include mixtures of truncated Gaussians and Beta distributions, three- and six-dimensional dark-matter simulation data, and house-price distributions. The framework wraps nonparametric distribution regression, optimal transport regression and Wasserstein regression; only the nonparametric model is used for multidimensional experiments.

For the one-dimensional synthetic benchmarks, the default setup uses 1,000 training, 1,000 calibration and 1,000 test distribution pairs, with 1,000 observations per distribution. The adaptive method generally produces comparable or smaller uncertainty regions than global calibration. To avoid numerical instability in volume calculations, the authors mainly report the calibration threshold as a size proxy; for fixed geometry, a smaller threshold means a smaller region.

Reported grouped coverage deviations reach six percentage points on the Beta-mixture and house-price tasks. These checks sort test cases by prediction-set size and assess coverage within equally populated bins. They support adaptation across uncertainty levels, but do not directly establish coverage at every fixed input.

A mortality experiment with only 36 distribution-valued observations illustrates the data constraint: splitting scarce observations among training, calibration and testing produces noisier results. Local calibration also adds nearest-neighbor search, increasing inference costs with calibration-set size.

The central advance is making distribution-valued uncertainty geometrically usable. Its practical limits remain substantial: truncation trades fine detail against dimension, local calibration becomes harder in high-dimensional distribution spaces, and the framework does not directly handle covariate shift or other failures of exchangeability. Smaller, sampleable regions are useful, but only alongside a clear account of what they cover and under which assumptions.