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CLAPS: Posterior-Aware Conformal Intervals via Last-Layer Laplace

Dongseok Kim, Hyoungsun Choi, Mohamed Jismy Aashik Rasool, Gisung Oh

TL;DR

CLAPS addresses the challenge of producing reliable, narrow prediction intervals in regression by tying a posterior-aware scoring rule to split-conformal calibration. It pairs a Last-Layer Laplace Approximation with a Gaussian predictive distribution to define a two-sided centrality score, aligning the conformal nonconformity with the full predictive distribution rather than raw residuals. Theoretical results guarantee marginal coverage and reveal HPD-like efficiency under correct specification, while a posterior-diagnostics suite guides method choice and helps interpret interval behavior. Empirically, CLAPS matches target coverage and achieves competitive or superior width and MAE on small-to-medium tabular datasets, with diagnostic tools clarifying when posterior information is most informative. The work offers a practical, low-overhead upgrade to residual-based conformal baselines and introduces diagnostics to assist practitioners in uncertainty decomposition and method selection.

Abstract

We present CLAPS, a posterior-aware conformal regression method that pairs a Last-Layer Laplace Approximation with split-conformal calibration. From the resulting Gaussian posterior, CLAPS defines a simple two-sided posterior CDF score that aligns the conformity metric with the full predictive shape, not just a point estimate. This alignment yields narrower prediction intervals at the same target coverage, especially on small to medium tabular datasets where data are scarce and uncertainty modeling matters. We also provide a lightweight diagnostic suite that separates aleatoric and epistemic components and visualizes posterior behavior, helping practitioners understand why intervals shrink when they do. Across multiple benchmarks using the same MLP backbone, CLAPS consistently attains nominal coverage with improved efficiency and minimal overhead, offering a clear, practical upgrade to residual-based conformal baselines.

CLAPS: Posterior-Aware Conformal Intervals via Last-Layer Laplace

TL;DR

CLAPS addresses the challenge of producing reliable, narrow prediction intervals in regression by tying a posterior-aware scoring rule to split-conformal calibration. It pairs a Last-Layer Laplace Approximation with a Gaussian predictive distribution to define a two-sided centrality score, aligning the conformal nonconformity with the full predictive distribution rather than raw residuals. Theoretical results guarantee marginal coverage and reveal HPD-like efficiency under correct specification, while a posterior-diagnostics suite guides method choice and helps interpret interval behavior. Empirically, CLAPS matches target coverage and achieves competitive or superior width and MAE on small-to-medium tabular datasets, with diagnostic tools clarifying when posterior information is most informative. The work offers a practical, low-overhead upgrade to residual-based conformal baselines and introduces diagnostics to assist practitioners in uncertainty decomposition and method selection.

Abstract

We present CLAPS, a posterior-aware conformal regression method that pairs a Last-Layer Laplace Approximation with split-conformal calibration. From the resulting Gaussian posterior, CLAPS defines a simple two-sided posterior CDF score that aligns the conformity metric with the full predictive shape, not just a point estimate. This alignment yields narrower prediction intervals at the same target coverage, especially on small to medium tabular datasets where data are scarce and uncertainty modeling matters. We also provide a lightweight diagnostic suite that separates aleatoric and epistemic components and visualizes posterior behavior, helping practitioners understand why intervals shrink when they do. Across multiple benchmarks using the same MLP backbone, CLAPS consistently attains nominal coverage with improved efficiency and minimal overhead, offering a clear, practical upgrade to residual-based conformal baselines.

Paper Structure

This paper contains 57 sections, 7 theorems, 50 equations, 2 figures, 7 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Contributions.
  3. Related Work
  4. Conformal for Regression: from residual to distribution-shaped scores.
  5. Heteroscedastic Adaptation and Quantile-Based Methods.
  6. Posterior-Aware Scores and Last-Layer Bayesianization.
  7. Diagnostics and Method Selection.
  8. Complementary Lines: Localization, Weighting, and Bootstrap Ensembles.
  9. Method
  10. Problem Setup
  11. Last-Layer Laplace Approximation (LLLA)
  12. Two-Sided Posterior CDF Score
  13. Split-Conformal Calibration
  14. CLAPS Procedure
  15. Complexity and Deployment
  16. ...and 42 more sections

Key Result

Proposition 4.4

Under Assumption 1, the CLAPS interval satisfies $\Pr\{\,y\in\mathcal{C}(x)\,\}\ge 1-\alpha$ marginally over the calibration set and the test point.

Figures (2)

  • 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.
  • 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), $\mathrm{tr}(\Sigma)$, and $\hat{\sigma}^2$ 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 different absolute sample sizes.

Theorems & Definitions (20)

  • Definition 4.1: LLLA posterior and predictive
  • Definition 4.2: Two-sided posterior CDF score
  • Proposition 4.4: Finite-sample marginal coverage
  • proof : Proof sketch
  • Lemma 4.5: Monotone transform of absolute $z$
  • Corollary 4.6: Equivalence to central/HPD sets
  • Theorem 4.7: Oracle efficiency under correct specification
  • proof : Proof sketch
  • Remark 4.8: Misspecification and efficiency
  • Theorem 4.9: Posterior contraction and regime change
  • ...and 10 more