Conformal Prediction for Distribution-to-Distribution Regression

Trung-Khang Tran; Tuan A. Hoang; Viet-Hoang Tran; Tan M. Nguyen

cs.LGNov 4, 2026

Conformal Prediction for Distribution-to-Distribution Regression

Authors: Trung-Khang Tran, Tuan A. Hoang, Viet-Hoang Tran, Tan M. Nguyen

Organizations: School of Computing, National University of Singapore · Center for AI Research, VinUniversity · Department of Mathematics, National University of Singapore

Abstract

Conformal Prediction (CP) provides a model-agnostic framework for constructing finite-sample prediction sets with coverage guarantees and has become a tool for reliable decision-making in high-risk settings. Recent advances have extended Split Conformal Prediction to increasingly complex learning problems, including multilabel, high-dimensional, and functional outputs. In this work, we study CP for Distribution-to-Distribution Regression, a setting of growing interest across many application domains. We show that a naive application of Split CP yields prediction sets that, while marginally valid, are non-interpretable, e.g., lacking a well-defined volume or a tractable sampling procedure, and of limited practical value. To overcome this, we propose a new conformal framework that operates in an orthogonal basis space, producing interpretable prediction sets while preserving coverage guarantees. We further develop an adaptive variant with asymptotic conditional coverage under mild assumptions. Empirical results on synthetic and real-world datasets validate the effectiveness of the proposed approach. The code is publicly available at https://github.com/tuanhacs/conformal_d2d.

Figures & tables

Panel of Figure 1 — Figure 1: Visualization of distributions uniformly sampled from the proposed conformal prediction sets (blue), compared with the true density (red). Panel (a) presents results on synthetic datasets (Truncated Gaussians and Mixture of Betas), while panel (b) shows results on real-world datasets (House Price and Mortality).

Figure 2: Results on synthetic datasets using the NDR baseline. Rows correspond to Truncated Gaussian, Mixture of Betas, and 3D Dark Matter datasets (top to bottom). Columns correspond to tasks (I) – (III) (left to right): efficiency-coverage trade-off between adaptive and non-adaptive CP, comparison of empirical coverage with the theoretical Beta-Binomial distribution, and empirical conditional coverage across different input groups, respectively. Dashed horizontal lines in the rightmost column indicate the nominal coverage level.

Figure 3: House Price experiments using the NDR base model. The left panel (task (I) ) compares the efficiency-coverage trade-off between adaptive and non-adaptive CP, while the right panel (task (III) ) reports empirical conditional coverage across different input groups, with the dashed line indicating the nominal coverage level.

Appendix figures & tables11 assets

Supplementary material from the paper’s appendix.

Appendix

$\mathbb{R}^{k}$	$k$ -dimensional Euclidean space
$\\|\cdot\\|_{2}$	Euclidean norm
$\text{Ball}_{\\|\cdot\\|}(R)$	a ball of radius $R$ , induced by $\\|\cdot\\|$
$\Gamma(\cdot)$	Gamma function
$\mathbf{1}\{\cdot\}$	Indicator function
$\text{Beta}(\cdot,\cdot)$	Beta distribution
$\text{BetaBinomial}(\cdot,\cdot,\cdot)$	Beta-Binomial distribution

Appendix

Table 4

\mathcal{E}(X_{0})\leftarrow\{a\in\mathbb{R}^{|T_{R}|}:g(W^{1/2}(\theta(\hat{f}(X_{0}))-a))\leq\hat{q}\}.

Appendix

Algorithm 1 Conformal Prediction for Distribution-to-Distribution Regression

\mathcal{E}(X_{0})\leftarrow\{a\in\mathbb{R}^{|T_{R}|}:g(W^{1/2}(\theta(\hat{f}(X_{0}))-a))\leq\hat{q}_{X_{0}}\}.

Appendix

Algorithm 2 Adaptive Conformal Prediction for Distribution-to-Distribution Regression

Figure 4: Experiments on Truncated Gaussian (first row) and Mixture of Betas (second row) for task (I) (first column), (II) (second column), and (III) (third column) using the OTR base model.

Figure 5: Experiments on Truncated Gaussian (first row) and Mixture of Betas (second row) for task (I) (first column), (II) (second column), and (III) (third column) using the WR base model.

Figure 6: Experiments on Dark Matter 6D for task (I) (first column), (II) (second column), and (III) (third column) using the NDR base model.

Figure 7: Experiments on House Price Data for task (I) (first row), and task (III) (second row) using the OTR base model (first column) and the WR base model (second column).

Figure 8: Ablation on the truncation parameter for Truncated Gaussian (first row) and Mixture of Betas (second row) using the NDR (first column), OTR (second column), and WR (third column) base models.

Figure 9: Ablation on Inference Time for Truncated Gaussian (first column) and Mixture of Betas (second column) using the NDR base models.

Varied Parameter	Truncated Gaussians	Mixture of Betas
Legendre Basis	$0.90\pm 0.013$	$0.90\pm 0.014$
Wavelet Basis	$0.90\pm 0.013$	$0.90\pm 0.013$
$\nu=1$	$0.90\pm 0.014$	$0.90\pm 0.014$
$\nu=1.5$	$0.90\pm 0.013$	$0.90\pm 0.013$
$\nu=2$	$0.90\pm 0.013$	$0.90\pm 0.013$
$\sigma=0.5$	$0.90\pm 0.013$	$0.90\pm 0.013$

Appendix

Table 1: Ablation on Hyperparameter Sensitivity results.

Figure 10: Ablation in the low-data regime on the Mortality dataset for task (I) (first row) and task (III) (third row) using the NDR (first column), OTR (second column), and WR (third column) base models.

Conformal Prediction

Conformal prediction wraps predictive models with calibrated sets or intervals that provide statistical coverage guarantees.

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CRPS-Optimal Binning for Univariate Conformal Regression

Paolo Toccaceli

Centre for Reliable Machine Learning, Royal Holloway, University of London, Egham, Surrey TW20 0EX, UK

Conditional Distribution Estimation Regression Prediction Intervals Conformal Prediction

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We propose a method for non-parametric conditional distribution estimation based on partitioning covariate-sorted observations into contiguous bins and using the within-bin empirical CDF as the predictive distribution. Bin boundaries are chosen to minimise the total leave-one-out Continuous Ranked Probability Score (LOO-CRPS), which admits a closed-form cost function with

O(n^2 \log n)

precomputation and

O(n^2)

storage; the globally optimal

K

-partition is recovered by a dynamic programme in

O(n^2 K)

time. Minimisation of within-sample LOO-CRPS turns out to be inappropriate for selecting

K

as it results in in-sample optimism. We instead select

K

K

-fold cross-validation of test CRPS, which yields a U-shaped criterion with a well-defined minimum. Having selected

K^*

and fitted the full-data partition, we form two complementary predictive objects: the Venn prediction band and a conformal prediction set based on CRPS as the nonconformity score, which carries a finite-sample marginal coverage guarantee at any prescribed level

\varepsilon

. The conformal prediction is transductive and data-efficient, as all observations are used for both partitioning and p-value calculation, with no need to reserve a hold-out set. On real benchmarks against split-conformal competitors (Gaussian split conformal, CQR, CQR-QRF, and conformalized isotonic distributional regression), the method produces substantially narrower prediction intervals while maintaining near-nominal coverage.

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