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REVIEW 3 major objections 8 minor 29 references

Scalable Bayesian Quantile Regression via EM and INLA: The EM-INLA Algorithm

T0 review · 3 major / 8 minor · reviewed 2026-08-02 · deepseek-v4-flash

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Pith's one-line read The paper proposes EM-INLA, an algorithm that makes Bayesian hierarchical quantile regression tractable for very large datasets by replacing MCMC with closed-form EM steps and INLA approximations.

desk verdict A practical EM+INLA hybrid for hierarchical quantile regression that scales to 400k observations; the EM theory is loose, but the closed-form updates and empirical validation make it worth refereeing. read the letter →

arxiv 2607.10432 v2 pith:SZIYH5O4 submitted 2026-07-11 stat.CO stat.ME

Tomas Rodriguez Taborda , Johnatan Cardona-Jiménez This is my paper
classification stat.COstat.ME MSC 62F1562J0562G08
keywords BayesianquantileregressionAsymmetricLaplaceDistributionEMalgorithmIntegratedNestedApproximationshierarchicalmodelsempiricalBayescluster-robustsandwichlarge-scaleinference
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper aims to show that Bayesian quantile regression for hierarchical data, traditionally a computationally heavy MCMC problem, can be solved by an EM algorithm whose M-step is a weighted Gaussian regression handled by INLA, with scale and variance updates in closed form. If correct, the method delivers point estimates and calibrated uncertainty for quantile-specific covariate effects in settings with hundreds of thousands of observations and thousands of random effects, where Hamiltonian Monte Carlo is impractical. The authors demonstrate accuracy comparable to HMC in simulations, speedups of 15x to over 50x, and a successful fit to 415,472 students nested in 13,801 schools and 1,116 municipalities, revealing tail-specific patterns in socioeconomic effects.

What carries the argument

The engine is the normal-exponential mixture representation of the Asymmetric Laplace Distribution: Y = mu + theta V + kappa sqrt(sigma V) Z, with V exponential and Z standard normal, which makes Y conditionally Gaussian given V. The E-step then reduces to closed-form moments of a Generalized Inverse Gaussian distribution with index 1/2, and the M-step becomes a weighted Gaussian regression on pseudo-responses, solved by INLA; the scale and random-effect variances update analytically. The cluster-robust sandwich formula is the companion device that converts the working-likelihood posterior spread into calibrated frequentist intervals.

What would settle it

On a moderate simulated dataset where the exact M-step is computable, evaluate the penalized ALD log-likelihood after every EM-INLA iteration; any non-negligible decrease from one iteration to the next would falsify the implicit monotone-ascent premise of the algorithm.

Watch

Extended reading notes

Core claim

The central claim is that the Asymmetric Laplace working likelihood, via its normal-exponential mixture, turns each EM M-step into a weighted Gaussian regression that INLA can fit, so no MCMC and no smoothing of the check loss are needed. The E-step conditional distributions of latent variables are Generalized Inverse Gaussian with closed-form moments, the scale parameter has an exact closed-form update, and each random-effect variance is updated by posterior mean plus variance from INLA. A final INLA run conditional on the converged hyperparameters gives posterior marginals, and a one-pass cluster-robust sandwich correction restores near-nominal interval coverage that the naive empirical-Ba

Load-bearing premise

The M-step replaces the exact weighted-Gaussian maximizer with INLA's approximate fit, and the paper does not prove that the objective still ascends; if that approximation error accumulates across iterations, the algorithm could converge to a point far from the true marginal maximum.

Editorial extensions

If this is right

  • EM-INLA removes the need for MCMC in hierarchical quantile regression, making quantile analyses of administrative-scale datasets feasible in minutes.
  • Simulations indicate point estimates match Hamiltonian Monte Carlo across normal, heavy-tailed, and heteroscedastic error scenarios, with 15x to 50x speedups.
  • The closed-form random-effect variance update stays strictly positive, avoiding the zero-variance boundary collapse seen in frequentist mixed-model quantile fits.
  • Naive intervals from the final INLA call undercover; the analytic cluster-robust sandwich correction brings coverage to near nominal in one extra pass, matching a cluster bootstrap benchmark.
  • On the 2023 national test data, quantile effects reveal that parental education and internet access matter most in the upper tail, while the private-school premium is largest at the bottom.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Because the method inherits INLA's prior flexibility, the same EM-INLA loop should extend to spatial, temporal, or spatio-temporal random effects, giving scalable quantile versions of areal and longitudinal models.
  • The absence of a formal convergence guarantee for the approximate M-step suggests a targeted diagnostic: monitoring the penalized ALD objective directly, not just sigma, would catch drift when INLA's approximation is crude.
  • The empirical-Bayes conditioning on hyperparameters means practitioners should treat the sandwich or bootstrap intervals as the inferential output, not the raw posterior marginals; this is a design choice, not a bug.
  • Fitting each quantile independently invites quantile crossing; combining EM-INLA with non-crossing or rearrangement methods would likely preserve its speed while restoring coherent quantile functions.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 8 minor

Summary. The paper proposes EM-INLA, an empirical-Bayes algorithm for hierarchical quantile regression based on the asymmetric Laplace (ALD) working likelihood. The ALD is written as a normal-exponential mixture, and the algorithm iterates between an E-step that computes closed-form Generalized Inverse Gaussian moments for the latent variables and an M-step that fits a weighted Gaussian regression with INLA. Closed-form updates are derived for the ALD scale and for the random-effect variances. The authors also propose an analytic cluster-robust sandwich correction for the fixed-effect intervals and compare it with a cluster bootstrap. The method is evaluated in four simulation scenarios against lqmm/qrLMM, QREM, and Hamiltonian Monte Carlo, and is applied to the 2023 Prueba Saber 11 data with 415,472 students nested in schools and municipalities. The central claims are: EM-INLA avoids MCMC, recovers parameters comparably to HMC, achieves 15x–50x speedups, produces calibrated intervals via the sandwich, and scales sublinearly with sample size.

Significance. If the central claims hold, this is a practically valuable contribution: it would bring empirical-Bayes hierarchical quantile regression to data sizes at which MCMC is infeasible, while retaining closed-form updates and a single-pass uncertainty correction. The manuscript has notable strengths: the derivations in Appendix A are explicit and the σ update reproduces the known ALD maximum-likelihood estimator in the baseline case; the simulations compare against independent external implementations (lqmm, qrLMM, QREM, HMC/Stan); the reproducibility statement indicates that code is available; and the interval-coverage problem is addressed directly, with the sandwich estimator checked against a cluster bootstrap. However, the load-bearing point that EM-INLA is a valid EM algorithm with a well-defined monotonically increasing objective is not established, and the authors themselves acknowledge that strict monotone ascent is not guaranteed. This gap must be resolved or the algorithm must be reframed as an approximate fixed-point procedure with appropriate convergence diagnostics.

major comments (3)
  1. [§3.1–3.4 and Algorithm 1, especially Eq. (10), Theorem 3.1 Eq. (14), and Prop. 3.2 Eq. (19)] The algorithm is presented as an EM algorithm, but no single objective is shown to be monotonically increased. The E-step in Eq. (10) conditions the latent variables v_i on point estimates of the location parameters, while the M-step for random-effect variances in Eq. (18)–(19) treats alpha as latent and uses posterior moments. Meanwhile, Theorem 3.1’s Q-function Eq. (14) omits the random-effect prior terms that the INLA fit in Algorithm 1, lines 4–6, actually maximizes. Thus there is an inconsistency between the Q-function used in the derivation and the criterion actually optimized by INLA. The Section 6 concession that 'strict monotone ascent ... is not mathematically guaranteed' is weaker than the actual problem: the paper does not identify a well-defined objective that EM-INLA increases. This is load-bearing for the 'EM' description and for the claim that the algorithm maximizes an e
  2. [§3.5, Eq. (20)] The stopping rule monitors only the relative change in σ. The paper states that once σ stabilizes, 'the location and variance updates reach a steady state,' but no argument is given for why flatness of σ implies convergence of β, α, and the random-effect variances. This matters because the objective-function gap above means that a flat σ trajectory could stop at a point that is not a fixed point of the full update vector. At minimum, the simulation section should report the full-vector changes over iterations, or the stopping rule should include the location parameters and variance components. Without this, the empirical convergence claims are incomplete.
  3. [§3.5, Eq. (21), and §4.5] The analytic sandwich estimator is derived from a first-order expansion of the penalized criterion, and its applicability to two-level nested random effects is asserted rather than proved. The authors note that the derivation assumes Gaussian random effects, exogeneity, and a constant residual density, and they relegate a formal asymptotic treatment to future work. The simulation coverage in Table 5 is encouraging, but it is based on one design with J2 = 10 top-level clusters and does not exercise the assumptions (e.g., non-Gaussian random effects, clustered covariates, or a larger number of top-level units). Since the application and the paper’s uncertainty-quantification claims rely on Eq. (21), the manuscript should either provide a more formal justification or clearly state the conditions under which the intervals are expected to be calibrated, with targeted simulations for those con
minor comments (8)
  1. [Section 1] The introduction describes the method as an 'exact EM algorithm,' but Section 6 states that the M-step is approximate because it uses an INLA fit. Please replace 'exact' with 'closed-form EM-type' or similar throughout, so the terminology is consistent.
  2. [Section 3.1, Eq. (10)] The conditioning notation in Eq. (10) is inconsistent with later expressions: Eq. (10) conditions on current parameter estimates, while Prop. 3.2 conditions on '˜y, β̂'. Please unify the notation and clarify that the posterior moments in Eq. (19) are INLA approximations.
  3. [Algorithm 1, step 6] Step 6 uses Var(α_j^(k) | ·) from INLA, but the variance update in Eq. (19) is derived as an EM conditional expectation. Since the INLA fit conditions on the current hyperparameters and uses a Laplace approximation, the variance used is not the exact EM conditional variance. This approximation should be stated explicitly next to Proposition 3.2.
  4. [§4.4, Table 3] The HMC results under M2 appear to be dominated by a small number of divergent fits (RMSE of σ2 between 16.8 and 22.3 against a true value of 4). Reporting the number of divergent transitions or other convergence diagnostics for the HMC runs would make the comparison more interpretable and would strengthen the claim that EM-INLA is 'comparable to HMC.'
  5. [§4.3, Table 2] The QREM variance collapse under M4 is mentioned qualitatively. Please report the frequency of boundary estimates (zero variance) across replications, so the reader can assess how typical the collapse is.
  6. [Reproducibility] The code availability statement says replication code is available at GitHub but no URL is given. Please provide a link or repository identifier for journal review.
  7. [Throughout] There are minor typographical issues, e.g., 'Wepropose' in the abstract and 'understating' where 'understating uncertainty' is intended. A careful copyedit is recommended.
  8. [References] Please verify the status and URLs of the references dated 2026, since some appear to be online-first or arXiv preprints; give the most citable version.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: EM-INLA is validated against independent simulation truths and external benchmarks, and no derivation step reduces to its inputs by construction.

full rationale

The paper's central derivations are self-contained transformations of the ALD normal–exponential mixture: the E-step moments follow from the GIG kernel (Eq. 10–12), the M-step location update is an algebraic completion of the square yielding a weighted Gaussian criterion (Theorem 3.1, Eq. 13–14), and the scale and variance updates are obtained by differentiating the expected complete-data log-likelihood (Eq. 15–19). None of these steps presupposes the target estimates; the unknown scale and variance parameters are explicitly the arguments being optimized. The accuracy claim is tested against independent simulation truths and external implementations (HMC/Stan, QREM, lqmm, qrLMM), so the 'recovers the true parameters' statement is not a renamed fit. The only admitted gap—that replacing the exact M-step by an approximate INLA call means strict monotone ascent is not mathematically guaranteed—is a correctness/robustness limitation, not a circularity: the algorithm still targets a well-defined weighted Gaussian fit and the paper does not claim a monotonicity result derived from its own assumptions. No load-bearing self-citations are present, and no uniqueness theorem or ansatz is imported from the authors' prior work. The paper is accordingly a normal method-development contribution with external validation, warranting a circularity score of 0.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The model parameters σ and σ_k² are estimated by the algorithm itself; the only hand-set numbers are vague priors and an unspecified tolerance. The load-bearing axioms are standard ALD mixture results and the two domain assumptions about the accuracy of the approximate M-step and the sandwich's Gaussian/exogeneity conditions.

free parameters (4)
  • σ (ALD scale) = estimated via Eq. (17) at each EM iteration
    Scale of the working likelihood; estimated by marginal maximum likelihood; not ad hoc but affects all E-step weights.
  • σ_k² (random-effect variances) = estimated via Eq. (19)
    Between-group variances for each block; estimated from posterior moments supplied by INLA; strictly positive by construction.
  • Convergence tolerance tol = unspecified
    Stopping rule in Algorithm 1; not given a numeric value, so the number of EM iterations (and thus output) is under-specified for replication.
  • Prior variance for fixed effects = 1000
    Vague Gaussian prior on β; chosen by hand, but deliberately non-informative.
assumptions (5)
  • standard math ALD normal-exponential mixture representation (Kozumi & Kobayashi 2011)
    Eq. (6)-(8); the basis for the conditionally Gaussian latent model.
  • standard math E-step conditional moments of GIG(1/2, χ, ψ) (Jørgensen 1982)
    Eq. (10)-(12); closed forms E[v]=√(χ/ψ)+1/ψ and E[v^{-1}]=√(ψ/χ).
  • domain assumption ALD working likelihood provides valid posterior inference under misspecification (Sriram et al. 2013; Yang et al. 2016)
    Justifies using ALD posterior even when errors are non-ALD; cited in Section 2.
  • domain assumption INLA's Laplace approximations are accurate enough for the latent Gaussian M-step
    The M-step is an approximate weighted Gaussian fit; no error bound is given, and monotone EM ascent is not guaranteed.
  • domain assumption Sandwich assumptions: random effects exogenous and Gaussian, constant residual density f_i ≡ f
    Stated after Eq. (21); needed for P Var(θ)P = P and for H to use a single density estimate.

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Pith. "Pith review of Scalable Bayesian Quantile Regression via EM and INLA: The EM-INLA Algorithm." pith.science (2026). https://pith.science/paper/SZIYH5O4

@misc{pith2026260710432,
  author       = {Pith},
  title        = {Pith review of: Scalable Bayesian Quantile Regression via EM and INLA: The EM-INLA Algorithm},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SZIYH5O4}},
  note         = {Machine review of arXiv:2607.10432}
}
read the original abstract

We propose EM-INLA, a scalable algorithm for empirical-Bayes hierarchical quantile regression that combines the Expectation-Maximization (EM) algorithm with Integrated Nested Laplace Approximations (INLA). The method exploits the normal-exponential mixture representation of the Asymmetric Laplace Distribution (ALD) to reformulate each M-step as a weighted Gaussian regression handled by INLA, with the ALD scale and random-effect variances updated in closed form at each iteration. The scale and variance hyperparameters are estimated by marginal maximum likelihood via the EM algorithm, and posterior marginals for the regression and random-effect parameters are obtained from a final INLA call conditional on the converged hyperparameter estimates. The result is an algorithm that avoids all MCMC sampling while scaling to large datasets and complex hierarchical structures that are intractable for standard fully Bayesian approaches. In a simulation study across four error scenarios, EM-INLA recovers the true parameters with accuracy comparable to Hamiltonian Monte Carlo (HMC) while achieving speedups ranging from 15x to over 50x depending on the scenario. The method is applied to the 2023 Prueba Saber 11 standardized test in Colombia to model the conditional quantiles of student scores as a function of socioeconomic covariates under a hierarchical structure of students nested within schools and municipalities, with over 400,000 observations.

Figures

Figures reproduced from arXiv: 2607.10432 by the authors.

Figure 1
Figure 1. EM-INLA computation time per replication as a function of sample size 𝑛, with the random-effect dimensions (𝐽1 , 𝐽2 ) scaled proportionally. The runtime exhibits sublinear growth across all scenarios and quantile levels. T. Rodriguez & J. Cardona: Preprint submitted to Elsevier Page 15 of 27 [PITH_FULL_IMAGE:figures/full_fig_p016_1.png] view at source ↗
Figure 1
Figure 1. EM-INLA runtime versus sample size 𝑛 on log–log axes: medians, interquartile bands, and fitted slopes by scenario. 5. Application to real data: Prueba Saber 11 in Colombia 5.1. Data and model specification Standardized assessment in Colombia is carried out through the Prueba Saber 11, a national examination administered by the Colombian Institute for the Evaluation of Education (ICFES) that every student must take i… view at source ↗
Figure 2
Figure 2. Estimated coefficient paths across quantile levels (𝜏) for selected covariates. Shaded areas represent the 95% cluster bootstrap credible intervals. The remaining covariates display three distinct patterns, summarized in the clustered forest plot of [PITH_FULL_IMAGE:figures/full_fig_p019_2.png] view at source ↗
Figures from the paper (7 more)
Figure 2
Figure 2. Figure 2: Coefficient paths for selected covariates, with 95% cluster-robust sandwich bands; the dashed line marks zero. raw socioeconomic gradient. It reflects high collinearity. Because a higher-stratum household is overwhelmingly more likely to feature high socioeconomic char…
Figure 3
Figure 3. Figure 3: Forest plot of fixed-effect coefficients for selected covariates. 6. Final remarks We have proposed EM-INLA, a scalable algorithm for Bayesian hierarchical quantile regression that entirely avoids MCMC sampling. By establishing a closed-form E-step for the moments of t…
Figure 3
Figure 3. Figure 3: Estimates and 95% cluster-robust sandwich intervals for selected covariates; darker shades mark higher quantiles. T. Rodriguez & J. Cardona: Preprint submitted to Elsevier Page 24 of 29 [PITH_FULL_IMAGE:figures/full_fig_p024_3.png]
Figure 4
Figure 4. Figure 4: Estimated coefficient paths across quantile levels (𝜏) for all covariates. Shaded areas represent the 95% cluster bootstrap credible intervals. T. Rodriguez & J. Cardona: Preprint submitted to Elsevier Page 24 of 27 [PITH_FULL_IMAGE:figures/full_fig_p025_4.png]
Figure 4
Figure 4. Figure 4: Coefficient paths for all covariates, with 95% cluster-robust sandwich bands; the dashed line marks zero. T. Rodriguez & J. Cardona: Preprint submitted to Elsevier Page 26 of 29 [PITH_FULL_IMAGE:figures/full_fig_p026_4.png]
Figure 5
Figure 5. Figure 5: Forest plot of fixed-effect coefficients for all covariates. Reproducibility All computations were carried out in R 4.4.2. EM-INLA was implemented with the R-INLA package (version 24.12.11), and the fully Bayesian benchmark with rstan 2.32.7. The Stan model encodes the…
Figure 5
Figure 5. Figure 5: Estimates and 95% cluster-robust sandwich intervals for all covariates; darker shades mark higher quantiles. Reproducibility All computations were carried out in R 4.4.2. EM-INLA was implemented with the R-INLA package (version 24.12.11), and the fully Bayesian benchma…

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Pith tools

Reviewed August 2, 2026 · model on record in the stance chip above.