Calibrated Multivariate Distributional Regression with Pre-Rank Regularization

The goal of probabilistic prediction is to issue predictive distributions that are as informative as possible, subject to being calibrated. Despite substantial progress in the univariate setting, achieving multivariate calibration remains challenging. Recent work has introduced pre-rank functions, scalar projections of multivariate forecasts and observations, as flexible diagnostics for assessing specific aspects of multivariate calibration, but their use has largely been limited to post-hoc evaluation. We propose a regularization-based calibration method that enforces multivariate calibration during training of multivariate distributional regression models using pre-rank functions. We further introduce a novel PCA-based pre-rank that projects predictions onto principal directions of the predictive distribution. Through simulation studies and experiments on 18 real-world multi-output regression datasets, we show that the proposed approach substantially improves multivariate pre-rank calibration without compromising predictive accuracy, and that the PCA pre-rank reveals dependence-structure misspecifications that are not detected by existing pre-ranks.

💡 Research Summary

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The paper tackles the problem of achieving calibrated multivariate probabilistic predictions, a task that is considerably more difficult than its univariate counterpart because one must ensure not only that each marginal distribution is well‑calibrated but also that the joint dependence structure is correctly represented. Existing approaches largely rely on post‑hoc diagnostics such as multivariate rank histograms or on extending the univariate probability integral transform (PIT) in ways that are either non‑differentiable or computationally prohibitive for training.

The authors introduce a general framework that enforces multivariate calibration during training by augmenting the usual proper scoring‑rule loss (e.g., CRPS, log‑likelihood) with a differentiable regularization term derived from a pre‑rank function. A pre‑rank is any scalar mapping ρ(x, y) from a forecast–observation pair to a real number. For a given ρ, the projected PIT Zρ = F̂T|X(T|X) is defined, where T = ρ(x, y) and T̂ = ρ(x, Ŷ) with Ŷ ∼ F̂Y|X. If the model is perfectly calibrated with respect to ρ, Zρ follows a uniform distribution on