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arxiv: 2605.21197 · v1 · pith:XZRZX3BTnew · submitted 2026-05-20 · 📊 stat.ME

Laplace Approximations for Mixed-Effects and Gaussian Process Quantile Regression

Andrea Nava , Fabio Sigrist This is my paper

Pith reviewed 2026-05-21 01:47 UTC · model grok-4.3

classification 📊 stat.ME
keywords quantile regressionLaplace approximationmixed-effects modelsGaussian processesasymmetric Laplace distributionFisher informationnon-smooth likelihoods
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The pith

Laplace approximations for quantile regression use Fisher information or expected loss curvature instead of the observed Hessian.

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

The paper shows how to apply Laplace approximations to quantile regression models that use the asymmetric Laplace likelihood even though the observed Hessian vanishes almost everywhere. It does this by using the Fisher information matrix when the model is correctly specified or the curvature of the population expected loss under misspecification. This matters for readers because it allows fast computation of approximate posteriors and marginal likelihoods in mixed-effects and Gaussian process models without smoothing the likelihood function or using slower sampling methods. The authors also provide practical estimators for the curvature and prove that the resulting approximations are asymptotically valid.

Core claim

The obstacle of a vanishing observed Hessian in Laplace approximations for the asymmetric Laplace likelihood can be overcome without smoothing by using the Fisher information for correctly specified models and the population curvature of the expected loss under misspecification. This basis allows development of a Laplace approximation framework for quantile regression in mixed-effects and Gaussian process models, with practical curvature estimators such as the triangular kernel curvature estimator that are asymptotically valid.

What carries the argument

Replacement of the observed Hessian with the curvature from the Fisher information or the expected loss in the quadratic expansion for the Laplace approximation.

If this is right

  • The methods enable scalable and numerically stable posterior inference for latent Gaussian quantile regression models.
  • Approximations achieve accuracy comparable to MCMC at lower computational cost.
  • Marginal likelihoods can be estimated for model selection in these settings.
  • The framework applies to both correctly specified and misspecified models.

Where Pith is reading between the lines

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

  • This could allow Laplace methods for other non-smooth or non-differentiable loss functions in Bayesian generalized linear models.
  • Connections to robust statistics suggest similar curvature-based justifications might apply to other M-estimators.
  • Future work could test the approach in high-dimensional Gaussian process quantile regression for spatial data.

Load-bearing premise

Local quadratic behavior of the expected loss or Fisher information provides a valid curvature for the Laplace approximation despite the observed Hessian vanishing almost everywhere.

What would settle it

Compare the approximate posterior means and variances from the proposed method against those obtained from long-run MCMC in a simulated mixed-effects quantile regression dataset with known parameters.

Figures

Figures reproduced from arXiv: 2605.21197 by Andrea Nava, Fabio Sigrist.

Figure 1
Figure 1. Figure 1: A motivating example: High school math scores. A random intercept and random slope quantile regression model (τ = 0.5 and τ = 0.05) with school-level random effects fitted with our proposed Laplace approximation for the Math Score of students from 160 different high-schools in the US from the High School and Beyond (HSB) dataset. The plots show the fitted function for 10 randomly selected schools. The left… view at source ↗
Figure 2
Figure 2. Figure 2: Comparison of asymmetric Laplace Fisher curvature with the asymptotic cur￾vature. Left: Fisher curvature τ (1 − τ )/λˆ2 using the estimated asymmetric Laplace scale λˆ plotted against the correct asymptotic curvature f(µτ )/λˆ. Right: same comparison when the Fisher curvature is computed with fixed scale λ = 1, i.e. τ (1 − τ ) vs. f(µτ ). Marker shape indicates the target quantile τ ; color indicates the n… view at source ↗
Figure 3
Figure 3. Figure 3: Comparison of triangular kernel curvature (TKC) and Fisher-Laplace approx￾imations to the log-likelihood for a single-level random effects model for a correctly specified likelihood (left) and a misspecified likelihood (right). The curve labeled “true curvature” corresponds to the asymptotic (population) curvature at the data-generating value b0. [Koenker, 2005]. This connection allows us to derive asympto… view at source ↗
Figure 4
Figure 4. Figure 4: Single-Level Grouped Random Effects: variance component estimation. Estimates of the random-effects variance σ 2 u across 10 replications, shown for varying group sizes nj in the Gaussian noise setting. 3.2 Crossed Random Effects We next consider a two-factor crossed random effects model with m1 = 100 levels for the first factor and m2 = 50 levels for the second factor. For each level of the first factor, … view at source ↗
Figure 5
Figure 5. Figure 5: Accuracy of log-marginal likelihood approximations, measured by relative error with respect to adaptive Gauss–Hermite quadrature, across different sample sizes. Left: correctly specified likelihood. Right: misspecified likelihood. where Cˆ∆b( ˆbj ) is our triangular kernel curvature estimate of the curvature. We evaluate the empirical coverage, defined as the proportion of confidence intervals containing t… view at source ↗
Figure 6
Figure 6. Figure 6: Sandwich correction with TKC yields calibrated coverage under misspecification. Empirical coverage over K = 10 replications of 90% Wald-type confidence intervals for the group random effects bj , using the TKC-based sandwich standard errors (TKC–SW). Left: Gaussian noise. Right: Student’s t noise. 19 [PITH_FULL_IMAGE:figures/full_fig_p019_6.png] view at source ↗
read the original abstract

Laplace approximations are a standard tool for computationally efficient inference in latent Gaussian models, but they fail for quantile regression with the asymmetric Laplace likelihood because the observed Hessian vanishes almost everywhere. We show that this obstacle can be overcome without smoothing the likelihood: the relevant local curvature is given not by the observed Hessian, but by the Fisher information when the model is correctly specified and by the population curvature of the expected loss under misspecification. On this basis, we develop a Laplace approximation framework for quantile regression with mixed-effects and Gaussian process models. We propose practical curvature estimators, including the triangular kernel curvature (TKC) estimator, that yield approximations for posterior distributions and marginal likelihoods, and we establish their asymptotic validity. Empirically, the proposed methods are scalable and numerically stable, and for latent Gaussian models, they achieve accuracy comparable to or better than MCMC and variational competitors at substantially lower computational costs. More broadly, the framework clarifies how Laplace approximations can be justified for non-smooth generalized posteriors through local quadratic behavior of the expected loss.

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.

Referee Report

2 major / 2 minor

Summary. The manuscript develops Laplace approximations for Bayesian inference in mixed-effects and Gaussian process quantile regression models that use the asymmetric Laplace likelihood. It addresses the vanishing observed Hessian by replacing it with the Fisher information matrix under correct specification or the Hessian of the expected loss under misspecification, proposes practical curvature estimators including the triangular kernel curvature (TKC) estimator, derives approximations to the posterior and marginal likelihood, and claims asymptotic validity together with empirical performance comparable to MCMC at lower cost.

Significance. If the central claims on asymptotic validity hold, the work provides a computationally attractive route to posterior approximation and model comparison for quantile regression in latent Gaussian settings where standard Laplace methods break down. The framework's emphasis on population curvature rather than observed Hessian offers a principled way to handle non-smooth generalized posteriors and could improve scalability for hierarchical and spatial quantile models.

major comments (2)
  1. [§3] §3 (framework derivation): the substitution of Fisher information or expected-loss curvature for the observed Hessian is load-bearing for the entire approximation; the manuscript must supply a rigorous bound showing that the Laplace remainder term still vanishes at the usual rate when the mode lies near a kink of the asymmetric Laplace loss and when latent variables are integrated.
  2. [Theorem on asymptotic validity] Theorem on asymptotic validity (likely §3.3 or §5): the proof sketch invokes local quadratic behavior of the expected loss, yet it is unclear whether the argument controls the additional error introduced by the TKC estimator's tuning parameters or by the non-differentiability when the posterior mass straddles a kink; an explicit rate for the total approximation error is required.
minor comments (2)
  1. [Abstract] Abstract and §2: the description of the TKC estimator would benefit from a one-sentence definition of its kernel and bandwidth choice before the empirical comparisons.
  2. [Simulation section] Simulation section: state explicitly the rule for excluding or handling data sets in which the mode coincides with a kink point of the quantile loss.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive report. The comments highlight important points regarding the rigor of our asymptotic analysis, and we address each one below. We commit to revisions that strengthen the theoretical justification without altering the core contributions of the framework.

read point-by-point responses

  1. Referee: [§3] §3 (framework derivation): the substitution of Fisher information or expected-loss curvature for the observed Hessian is load-bearing for the entire approximation; the manuscript must supply a rigorous bound showing that the Laplace remainder term still vanishes at the usual rate when the mode lies near a kink of the asymmetric Laplace loss and when latent variables are integrated.

    Authors: We agree that an explicit rigorous bound on the remainder is necessary to confirm the approximation remains valid near kinks and after marginalization. In the revised manuscript we will add a supporting lemma and proof (placed in an appendix to §3) that establishes the Laplace remainder is o_p(1) at the standard rate. The argument proceeds by showing that the population curvature (Fisher information or expected-loss Hessian) governs the local quadratic behavior almost surely, that the set of parameter values exactly at a kink has posterior measure zero, and that the integral over latent variables in the mixed-effects and GP cases preserves the rate via a dominated-convergence argument under the Gaussian prior. We will also verify the conditions under which the mode lies sufficiently far from kinks with high probability. revision: yes

  2. Referee: [Theorem on asymptotic validity] Theorem on asymptotic validity (likely §3.3 or §5): the proof sketch invokes local quadratic behavior of the expected loss, yet it is unclear whether the argument controls the additional error introduced by the TKC estimator's tuning parameters or by the non-differentiability when the posterior mass straddles a kink; an explicit rate for the total approximation error is required.

    Authors: We accept that the current sketch leaves the total error rate and the influence of the TKC tuning parameter implicit. In revision we will strengthen the theorem (likely in §3.3) to state an explicit total approximation error of order O_p(n^{-1}) for the log-posterior and O_p(n^{-1/2}) for the marginal likelihood. The proof will include (i) a separate bound showing that the TKC estimation error is o_p(n^{-1/2}) under the bandwidth condition h_n = o(1) with n h_n^2 → ∞, and (ii) a probabilistic control demonstrating that the probability of the posterior mass straddling a kink decays exponentially fast by posterior concentration, rendering its contribution negligible. These additions will be accompanied by a short simulation study confirming the rates in finite samples. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on standard asymptotic curvature arguments independent of internal fits.

full rationale

The paper's core claim replaces the vanishing observed Hessian of the asymmetric Laplace likelihood with Fisher information (correct specification) or the Hessian of the expected loss (misspecification) to justify Laplace approximations for quantile regression. This substitution is presented as following from classical results on local quadratic behavior of the log-posterior or expected loss, without defining the curvature in terms of quantities fitted inside the paper itself. The triangular kernel curvature (TKC) estimator and other practical methods are downstream tools for implementing the framework, not load-bearing definitions that reduce predictions to inputs by construction. No self-citation chains, uniqueness theorems from prior author work, or renaming of known results appear as the justification for the central result. The derivation chain remains self-contained against external benchmarks such as standard Laplace theory and asymptotic statistics.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the premise that expected-loss curvature supplies a usable quadratic approximation for non-smooth posteriors; this is treated as a domain assumption rather than derived from first principles within the paper.

free parameters (1)
  • triangular kernel curvature (TKC) estimator tuning parameters
    Practical curvature estimators including TKC are proposed; their specific bandwidth or kernel choices function as free parameters fitted or chosen for the approximation.
axioms (1)
  • domain assumption Local quadratic behavior of the expected loss supplies valid curvature for Laplace approximation when observed Hessian vanishes
    Invoked in the abstract to overcome the vanishing-Hessian obstacle for asymmetric Laplace likelihood.

pith-pipeline@v0.9.0 · 5708 in / 1318 out tokens · 67391 ms · 2026-05-21T01:47:36.160272+00:00 · methodology

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Reference graph

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