The conditional-mean barrier: From deterministic regression to conditional distribution learning

• URL: http://arxiv.org/abs/2605.28076v1
• Date: Wed, 27 May 2026 07:31:42 GMT
• Title: The conditional-mean barrier: From deterministic regression to conditional distribution learning
• Abstract summary: This tutorial develops a self-contained module centered on the conditional-mean barrier.<n>Adding latent randomness to a squared-loss predictor collapses it back to the conditional mean.<n>We briefly organize common distributional objectives, including negative log-likelihood, moment and observable matching, variational objectives, adversarial divergences, and score matching.
• Score: 6.160113667162026
• License: http://creativecommons.org/licenses/by-nc-nd/4.0/
• Abstract: Many problems in computational science and engineering become one-to-many after coarse graining, partial observation, or inverse reconstruction: a resolved state may not determine a unique subgrid forcing, a structural descriptor may not determine a unique effective response, and a low-resolution observation may correspond to many plausible high-resolution fields. In such settings, deterministic surrogates may learn a well-defined mathematical object while still missing application-relevant uncertainty. This tutorial develops a self-contained module centered on the conditional-mean barrier: the point at which a squared-loss predictor has reached the conditional mean and the remaining error is irreducible aleatoric variance. We give two diagnostics for locating this barrier, residual-feature orthogonality and the coefficient of determination against its explained-variance ceiling, and prove that adding latent randomness to a squared-loss predictor collapses it back to the conditional mean. Crossing the barrier therefore requires a loss that scores distributions rather than point predictions. We briefly organize common distributional objectives, including negative log-likelihood, moment and observable matching, variational objectives, adversarial divergences, and score matching, by the feature of the conditional law each targets. The emphasis is the boundary itself and a finite-data procedure for recognizing it, rather than a survey of methods beyond it. CPU-based demonstrations on a two-branch law and a two-scale Lorenz-96 closure problem show how the diagnostics distinguish deterministic underfitting from residual distributional variability.

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