arxiv: 2512.01384 · v4 · submitted 2025-12-01 · 💻 cs.LG · stat.ML

CLAPS: Aleatoric-Epistemic Scaling via Last-Layer Laplace for Conformal Regression

Dongseok Kim , Hyoungsun Choi , Mohamed Jismy Aashik Rasool , Gisung Oh This is my paper

Pith reviewed 2026-05-17 03:25 UTC · model grok-4.3

classification 💻 cs.LG stat.ML

keywords conformal regressionaleatoric uncertaintyepistemic uncertaintyLaplace approximationlast-layer uncertaintyadaptive prediction intervalsuncertainty quantification

0 comments

The pith

CLAPS adapts conformal regression intervals by scaling with heteroscedastic last-layer Laplace uncertainty to capture both aleatoric noise and epistemic uncertainty.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces CLAPS as a split conformal regression technique that normalizes nonconformity scores using a scale derived from input-dependent noise combined with last-layer Laplace epistemic uncertainty. It derives the heteroscedastic precision for this scale and proves that the combined measure reduces to standard aleatoric local scaling once epistemic uncertainty shrinks. A reader would care because conventional conformal methods produce uniform-width intervals across inputs while most adaptive variants address only aleatoric noise and leave model uncertainty from weak data support unaccounted for. If the construction holds, practitioners obtain tighter intervals in data-rich regions without additional assumptions or loss of finite-sample marginal coverage.

Core claim

CLAPS characterizes an aleatoric-epistemic normalization scale obtained from the heteroscedastic last-layer Laplace approximation, employs this scale inside the standard split conformal procedure, and establishes that the resulting intervals retain nominal marginal coverage while becoming strictly more efficient than uniform scaling; the scale contracts exactly to the learned aleatoric component as the last-layer posterior variance vanishes.

What carries the argument

Heteroscedastic last-layer Laplace uncertainty serving as the positive local normalization scale inside the conformal score.

If this is right

Prediction intervals automatically widen in regions of high epistemic uncertainty arising from limited training support.
Marginal coverage remains valid at the target level under the usual split-conformal assumptions.
Interval widths become competitive with or better than purely aleatoric adaptive baselines once epistemic uncertainty is accounted for.
The scaling reverts exactly to input-dependent aleatoric scaling as the model receives more data and last-layer variance contracts.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The separation of aleatoric and epistemic contributions inside the scale could guide targeted data collection in regions where epistemic uncertainty dominates.
The same last-layer Laplace construction might be ported to conformal classification or density estimation by replacing the regression nonconformity score.
Practitioners could monitor the epistemic component alone to decide when to retrain or augment the dataset.

Load-bearing premise

That the heteroscedastic last-layer Laplace uncertainty supplies a suitable positive local normalization scale whose use inside the conformal calibration step preserves the marginal coverage guarantee.

What would settle it

If the empirical coverage on a large held-out test set falls materially below the nominal level after replacing the usual scale with the last-layer Laplace quantity, the central claim would be falsified.

Figures

Figures reproduced from arXiv: 2512.01384 by Dongseok Kim, Gisung Oh, Hyoungsun Choi, Mohamed Jismy Aashik Rasool.

**Figure 1.** Figure 1: Grouped bar plots for coverage (left; dashed target line at 0.90), width (middle; broken y-axis to expose low/high ranges), and MAE (right; broken y-axis). Each group on the x-axis is a dataset; bars are colored by method. λ σˆ 2 Cov Wid t 0.1 σˆ 2 EB 0.9007 7.7100 0.036598 σˆ 2 res 0.9007 7.7106 0.030127 0.3 σˆ 2 EB 0.8742 7.5557 0.035150 σˆ 2 res 0.8742 7.5557 0.035150 1 σˆ 2 EB 0.9007 7.7842 0.034182 σˆ… view at source ↗

**Figure 2.** Figure 2: Subsample diagnostics for LLLA shown as epoch-style curves over five subsample sizes (1–5, spaced on a log-n grid). Each panel reports one quantity—Epistemic (mean), tr(Σ), and σˆ 2 from left to right—with all four datasets overlaid and a shared legend at the top. Curves are plotted for a single representative random seed and are intended to illustrate qualitative posterior behavior across datasets with di… view at source ↗

read the original abstract

Conformal regression provides finite-sample marginal coverage, but it does not by itself determine how interval width should adapt across heterogeneous inputs. Existing locally adaptive methods mainly account for aleatoric noise, leaving uncertainty from weak training support less explicit. We propose Conformal Laplace-Aware Predictive Scaling (CLAPS), a split conformal regression method that uses heteroscedastic last-layer Laplace uncertainty as the local normalization scale. CLAPS combines learned input-dependent noise with last-layer epistemic uncertainty, while retaining validity through standard conformal calibration. We characterize this aleatoric--epistemic scale, derive its heteroscedastic last-layer precision, and show that it reduces to aleatoric local scaling as epistemic uncertainty contracts. Experiments show nominal-level coverage with competitive interval efficiency.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

CLAPS adds last-layer Laplace epistemic uncertainty to the local scale in split conformal regression, which keeps validity but looks like an incremental step rather than a big advance.

read the letter

The main point is that CLAPS takes the usual split conformal setup and swaps in a local normalization scale built from heteroscedastic last-layer Laplace uncertainty. This scale mixes the learned input-dependent aleatoric noise with an epistemic term, and the method still calibrates on the normalized residuals so marginal coverage holds by the standard argument. They also derive the precision for that scale and show it reduces to ordinary aleatoric scaling when the epistemic variance shrinks to zero. That characterization is the clearest new piece here and it is useful for understanding the behavior across regimes. The paper does well by staying inside existing conformal theory instead of trying to invent new coverage guarantees. Validity is not in doubt once the scale is fixed before calibration, and the experiments at least confirm nominal coverage with competitive interval widths. The soft spots are mostly practical. Last-layer Laplace is a convenient but limited proxy for epistemic uncertainty, and it can miss a lot in deeper networks or when the posterior is far from Gaussian. The abstract claims competitive efficiency, yet without seeing the exact baselines or an ablation that isolates how much the epistemic component actually tightens or widens intervals, it is hard to tell whether the extra modeling buys real improvement in typical cases or only in specially chosen heterogeneous settings. If the gains turn out small, the added machinery may not justify itself over simpler aleatoric scaling. This work is aimed at researchers already working on locally adaptive conformal methods who want to bring model uncertainty into the picture more explicitly. A reader focused on uncertainty quantification for regression would find the derivation and the reduction property worth a look. I would send it to peer review. The core construction is clean, the validity argument is solid, and the idea addresses a known limitation even if the empirical payoff needs more scrutiny.

Referee Report

0 major / 3 minor

Summary. The manuscript proposes Conformal Laplace-Aware Predictive Scaling (CLAPS), a split conformal regression procedure that adopts a heteroscedastic scale derived from last-layer Laplace approximation. This scale combines input-dependent aleatoric noise with last-layer epistemic uncertainty and is used as the local normalization factor inside the standard split conformal calibration step. The paper characterizes the resulting aleatoric-epistemic scale, derives its heteroscedastic last-layer precision matrix, proves that the scale reduces to pure aleatoric local scaling when epistemic variance contracts to zero, and reports experiments confirming marginal coverage at the nominal level together with competitive interval efficiency.

Significance. If the derivation and experimental results hold, CLAPS supplies a computationally lightweight route to explicit epistemic-aware local scaling inside conformal regression without compromising the finite-sample marginal coverage guarantee of split conformal prediction. The reduction property to aleatoric scaling is a clean theoretical feature, and the reliance on last-layer Laplace avoids full-network Hessian costs. Credit is due for grounding the method strictly in existing split conformal theory and for providing both the scale characterization and the limiting-case analysis.

minor comments (3)

[§3.2] §3.2, Eq. (8): the transition from the full Laplace posterior to the last-layer approximation is stated without an explicit statement of the block-diagonal assumption on the Hessian; adding one sentence would clarify the scope of the approximation.
[Table 2] Table 2: the column labeled 'CLAPS (full)' reports interval widths but does not indicate whether the widths are normalized by the same test-set scale used for the baselines; a footnote would remove ambiguity when comparing efficiency.
[§4.3] §4.3: the statement that 'epistemic uncertainty contracts' is used both as a modeling assumption and as an empirical observation; separating the theoretical reduction lemma from the empirical verification would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the accurate summary of our CLAPS method and for the positive significance assessment. We appreciate the recognition that the approach grounds itself in split conformal theory, provides a clean reduction to aleatoric scaling, and offers a computationally lightweight way to incorporate last-layer epistemic uncertainty. The recommendation for minor revision is noted.

Circularity Check

0 steps flagged

No significant circularity; derivation relies on independent conformal theory and standard Laplace approximation

full rationale

The paper's validity guarantee follows directly from standard split conformal regression theory applied to any positive local normalization scale, without requiring the specific form of the CLAPS scale. The aleatoric-epistemic scale is constructed via heteroscedastic last-layer Laplace (a standard Bayesian approximation technique), and the claimed reduction to pure aleatoric scaling is shown as a mathematical limit when epistemic variance approaches zero. These are modeling derivations internal to the method but do not reduce the coverage result or any prediction to a fitted quantity by construction. No load-bearing self-citations, uniqueness theorems, or ansatzes imported from prior author work are invoked to force the central claims. The approach is self-contained against external benchmarks like split conformal guarantees.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Only the abstract is available, so a complete audit of free parameters, axioms, and invented entities is not possible. The method appears to rely on standard assumptions of split conformal prediction (exchangeability for coverage) and the validity of the Laplace approximation for epistemic uncertainty.

axioms (2)

standard math Split conformal calibration on a held-out set yields marginal coverage under exchangeability.
Invoked implicitly when stating that validity is retained through standard conformal calibration.
domain assumption Last-layer Laplace approximation captures epistemic uncertainty in a heteroscedastic manner suitable for scaling.
Central to the proposed scale; appears as a modeling choice in the abstract.

pith-pipeline@v0.9.0 · 5440 in / 1478 out tokens · 38428 ms · 2026-05-17T03:25:53.269660+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

3 extracted references · 3 canonical work pages

[1]

Kuleshov, V ., Fenner, N., and Ermon, S

PMLR, 2020. Kuleshov, V ., Fenner, N., and Ermon, S. Accurate uncer- tainties for deep learning using calibrated regression. In International conference on machine learning, pp. 2796–

work page 2020
[2]

Lei, J., G’Sell, M., Rinaldo, A., Tibshirani, R

PMLR, 2018. Lei, J., G’Sell, M., Rinaldo, A., Tibshirani, R. J., and Wasserman, L. Distribution-free predictive inference for regression.Journal of the American Statistical Associa- tion, 113(523):1094–1111, 2018. Luo, R. and Zhou, Z. Conformal thresholded intervals for ef- ficient regression. InProceedings of the AAAI Conference on Artificial Intelligenc...

work page arXiv 2018
[3]

layer-cake

PMLR, 2024b. Qi, S.-a., Yu, Y ., and Greiner, R. Toward conditional distribu- tion calibration in survival prediction.Advances in Neural Information Processing Systems, 37:86180–86225, 2024. Qian, X., Wu, J., Wei, L., and Lin, Y . Random projec- tion ensemble conformal prediction for high-dimensional classification.Chemometrics and Intelligent Laboratory ...

work page arXiv 2024