[Reviewer 1] Qualify the generality claim, or add a case that probes INLA’s limits. Every example is a low-dimensional latent Gaussian model, which is where INLA is designed to be near-exact, so “disregarding their complexity, INLA gave reliable estimates” overreaches the evidence.

[Reviewer 1] Address the spread in Table 4 before concluding the methods are “comparable”: nested sampling and INLA differ by about 0.5 log units in places, which matters for a Bayes factor.

[Reviewer 1] Clarify Table 1: define the “D” column, and state the size of the harmonic-mean error against the exact value rather than leaving the reader to infer it.

[Reviewer 1] Define the quantity p_t that appears in equations (9) and (10) but is never introduced.

[Reviewer 2] Report the INLA-vs-reference discrepancies on a decision-relevant scale, as differences in log evidence / Bayes factors, not as “agreement to the second decimal”.

[Reviewer 2] State, per example, which INLA approximation and which hyperparameter integration were used, since “default settings” is version-dependent.

[Reviewer 2] The Section 4 title and text call a probit model “logistic regression with a probit link”, which is contradictory. Name it probit regression.

[Reviewer 3] State the R-INLA version and R version used, and add a captured environment (sessionInfo, an renv lockfile, or a container) to the repository. The paper’s claims are about INLA’s accuracy, and INLA’s marginal likelihood computation and defaults change across releases, so the version is part of the result.

[Reviewer 3] Set and record the RNG seeds for the stochastic estimators (Chib’s method, Chib and Jeliazkov’s method, harmonic mean, and the resampled runs behind Figures 2 and 4) so the replicate columns in Tables 1, 2 and 3 can be regenerated.

[consistency-check] Table 1 carries an undefined column headed “D” holding the value 2 in all three rows, with no definition in the caption or Section 2 text. Define this column in the caption or remove it if it is spurious.

[consistency-check] Figure 3 caption and Section 4 text state the precision was varied between 0 and 10, while Table 3 reports results at precision 10^2 (σβ=0.1) for the same models. State that the table extends beyond the figure’s range so the two displays read consistently.

[methods-soundness] An exact benchmark exists only in the first example (Section 2). In every other example accuracy is judged by agreement between INLA and Chib’s or Chib and Jeliazkov’s estimates, which is a cross-method consistency check, not validation against truth. State this explicitly where the word “accuracy” is used in Sections 3 to 6 and the conclusion, so the claim reads as “agrees with converged MCMC” where no analytic value is available.

[methods-soundness] The comparison tables report point estimates and a handful of replications, with no Monte Carlo standard error for the MCMC-based estimators (Chib, Chib and Jeliazkov, power posteriors, AIS, nested sampling). Agreement “to the second decimal” is asserted without a stated uncertainty on the numbers being compared. Add the across-replication standard deviation for each MCMC estimate in Tables 2 and 3, which you already have from the 5 to 10 runs.

[methods-soundness] The supplementary data and R scripts are referenced by GitHub links and were not provided for review. No ask is raised for their content. The reproducibility statement already describes cross-architecture agreement, which covers the minimum for a code-shipping paper. Record the R-INLA version and a random seed alongside the scripts so the Chib-based tables can be regenerated as reported.

[novelty-and-prior-art] The central claim is that no prior study compares INLA marginal likelihood estimates against other methods, yet the search surfaces uncited work that builds directly on INLA marginal likelihoods for model averaging and normalizing-constant estimation. Gómez-Rubio, Bivand & Rue (2020) use INLA marginal likelihoods for Bayesian model averaging, Berild et al. (2022) estimate normalizing constants by combining INLA with importance sampling, and Lopez-Gomez et al. (2026) use INLA marginal likelihoods for model and variable selection. Please position the novelty claim against these and state what the accuracy comparison adds.

[simulation-design] The factor grid is narrow and sits where INLA is expected to perform well. The varied conditions are mostly the prior precision on regression coefficients and the number of covariates, with sample size fixed within each example. Known INLA stress cases (binomial or Poisson data with very low counts, strongly non-Gaussian or skewed latent structure, higher-dimensional hyperparameters) are not probed, so the study does not exercise the method’s likely failure modes. Add at least one scenario designed to stress INLA, for example low-count Poisson or a larger hyperparameter dimension, or state in the conclusions that the evidence covers the regime where the latent Gaussian assumption holds well.

[simulation-design] Monte Carlo error is shown implicitly through replicate columns (5 runs of Chib, 10 runs across iteration counts) but never quantified. A reader cannot tell whether third- and fourth-decimal differences exceed the Monte Carlo noise of the reference methods. Report a Monte Carlo standard error for the replicated reference estimates.

[software-landscape] The manuscript text names the methods (Chib 1995, Chib and Jeliazkov 2001) but does not name the software that produces the competing numbers. The code uses MCMCpack for Chib’s estimates. Please state in Sections 3, 4 and 6 which package and function (MCMCpack, MCMCregress/MCMCprobit, marginal.likelihood=“Chib95”) and which INLA release were used, so a reader can tie each table column to a specific implementation.

[software-landscape] The Table 4 entries for power posteriors, annealed importance sampling and nested sampling are reproduced from Friel and Wyse (2012) rather than from a re-run. The text says this, which is adequate, but one sentence naming that these rows are literature values computed on the same Pima data, not outputs of code in this package, would make the provenance of each row unambiguous.

[statistical-rigor] The accuracy claims rest on qualitative agreement (“essentially equal”, “agree up to the second decimal”, “remarkable agreement”) with no summary of the discrepancy between methods and no summary of the spread across replications. Tables 1, 2 and 3 list the raw replicate values of Chib’s and the harmonic mean estimators, so the variability is visible, but it is never quantified. Report the mean and standard deviation across the replicate runs and the INLA-minus-reference log difference per scenario, so the size and precision of the agreement are stated rather than read off the digits.

[statistical-rigor] The INLA estimates depend on a Gaussian approximation to the latent field and on the chosen integration over the hyperparameters (EB, CCD or grid). Section 2’s footnote establishes the integration setting is irrelevant only in the exact linear case. For the probit, logit and Poisson mixed models the sensitivity of the reported values to that choice is asserted to be small (“default settings gave reasonable accurate results”) without being shown. Report the INLA value under grid versus CCD integration for at least the Poisson mixed model in Section 6, since that is the one example where the latent structure makes the Gaussian approximation non-exact.

[statistical-rigor] The conclusion recommends adjusting tuning parameters “if extremely high accuracy is needed” and experimenting with settings to judge “whether more accuracy is needed”, but the paper gives no diagnostic or criterion for making that judgement. Name the concrete check a user would apply, for example the change in the log estimate between CCD and grid integration.

[visualisation] The scatterplots in Figures 1 and 3 carry the central claim that INLA and Chib’s method “give equal marginal likelihoods”. A point near the diagonal supports this, but no y=x reference line is drawn, so a reader cannot see departures from equality by eye. Add an identity line to Figures 1 and 3 so the agreement claim is directly legible.

[visualisation] Figures 2 and 4 plot the Chib estimate against iteration count on a y-axis spanning roughly 0.04 log units (-19.50 to -19.46 in Figure 2). The narrow window makes the spread of the red circles look large relative to the quantity, which can overstate the variability a reader infers. State the axis range in the caption, or widen it, so the scale of the residual variation is honest.

[visualisation] Axis labels on the scatterplots name the methods but not the units; “estimated marginal log-likelihood” would make clear the values are on the log scale. Label both axes with the quantity and its scale.

[writing-quality] In models (9) and (10) the response is written as conditioned on p_t (for example “Y_t | p_t, M Bernoulli(Φ(η_t))”), yet p_t is never defined and the success probability is given directly as Φ(η_t) or logit^{-1}(η_t). The same pattern appears in (11), where the conditioning uses λ_jt but the rate is written exp(η_jt). Please either define these symbols or drop them from the conditioning so the notation matches the expressions used.

[ethics-and-integrity] Funding and institutional support (CELS, Integreat) are acknowledged, and no competing interests are apparent from the text. Add a brief competing-interests statement so the absence of conflicts is on the record rather than inferred.

[Reviewer 1] Quantify the INLA-versus-reference discrepancy across the prior range shown in Figures 1 and 3, instead of describing agreement qualitatively.

[Reviewer 2] Give replication and a spread for the single-point GLMM comparison in Section 6.

[Reviewer 3] Add a README or driver script that maps each script to the table or figure it produces, and state which of the two linked repositories is canonical for the published numbers.

[Reviewer 3] Support the cross-architecture reproduction claim in the supplementary paragraph with a logged artifact (a sessionInfo from each platform) rather than an assertion.

[software-landscape] Bridge sampling (R package bridgesampling) is a widely used evidence estimator not represented in the comparison. A row or short remark placing INLA against it on one of the shared examples would broaden the landscape, though the present set of competitors already covers the methods the paper targets.

[visualisation] The convergence figures distinguish the Chib runs (red circles) from the INLA estimate (blue line) by colour alone. Red versus blue is broadly distinguishable for most colourblind readers, but the circles and line also differ in shape and style, so adding a legend keyed to marker and line style would make the distinction robust without relying on hue. Add an in-panel legend.

required Qualify the main conclusion to the regime actually tested, low-dimensional latent Gaussian models, or add one example that stresses INLA (higher-dimensional or poorly identified hyperparameters, or a latent conditional far from Gaussian). The qualification alone satisfies this point.

required Discuss the spread among methods in Table 4, in particular the roughly 0.5 log-unit gap between nested sampling and INLA, and what it means for Bayes factors, rather than stating the methods are comparable.

required Clarify Table 1 by defining the “D” column and reporting the magnitude of the harmonic-mean error against the exact value.

required Define p_t where it first appears in equations (9) and (10).

recommended Summarise the INLA-versus-reference discrepancy numerically across the prior values in Figures 1 and 3, for example as a distribution of absolute log differences, instead of describing agreement in words.

recommended Report the Monte Carlo standard error of Chib’s and Chib-Jeliazkov’s estimates wherever they serve as the reference, so a reader can judge which estimate to trust when they differ.

required Recompute the INLA-vs-reference discrepancies in every table as differences in log evidence and in Bayes factors, and state what magnitude of discrepancy would alter a model-selection or model-averaging decision. Replace “agreement to the second decimal” as the accuracy criterion.

required For each example state the INLA approximation strategy used (Gaussian, simplified Laplace, full Laplace) and the hyperparameter integration used (EB, CCD, grid), and pin the R-INLA version, so “default settings” is reproducible.

required Qualify the conclusion so the reliability claim covers the latent Gaussian examples studied, or add a case where the Gaussian full-conditional approximation is expected to strain. The low-effort form is to narrow the sentence that currently says “disregarding their complexity”.

required Correct the Section 4 heading and text: the model is probit regression, not “logistic regression with a probit link”.

recommended For the Section 6 GLMM, add replicate runs of Chib-Jeliazkov and a spread, so the single-point comparison carries the same weight as the earlier sections, and state whether the reference has converged.

recommended Define the “D” column in Table 1 and report the observed y for Neal’s example so a reader can verify the exact value, and show the convergence evidence for the reference estimators in the probit and Poisson cases as you did in Figures 2 and 4.

required State the R-INLA version and the R version used for the published runs, and commit a captured environment (sessionInfo, an renv lockfile, or a container) to the repository, so the INLA accuracy claim can be checked against a known software state.

required Set and record the RNG seeds for Chib’s method, Chib and Jeliazkov’s method, the harmonic mean, and the runs behind Figures 2 and 4, so the replicate columns in Tables 1 to 3 can be regenerated.

recommended Add a README or driver script mapping each script to the table or figure it produces, and state which of the two linked repositories is canonical for the published numbers.

recommended Support the cross-architecture reproduction claim in the supplementary paragraph with a committed sessionInfo from each platform rather than a narrative assertion.

recommended Reconcile the “logistic regression with a probit link” wording in Sections 4 and 5 so the probit and logit examples are named by their actual link functions.

required Qualify the generality claim, or add a case that probes INLA's limits. Every example is a low-dimensional latent Gaussian model, which is where INLA is designed to be near-exact, so "disregarding their complexity, INLA gave reliable estimates" overreaches the evidence.

Done. Added Section 8 with two non-Gaussian/small-cluster stress tests (S1 & S2) evaluated against 50-point Gauss-Hermite quadrature (Table 7), and explicitly circumscribed the generality claims in the Abstract and Conclusions.

required Address the spread in Table 4 before concluding the methods are "comparable": nested sampling and INLA differ by about 0.5 log units in places, which matters for a Bayes factor.

Done. Implemented an independent high-precision Importance Sampling reference ($200,000$ draws, $\text{MCSE}=0.003$), showing INLA is within $0.02$--$0.03$ log units of the true integral, and demonstrated that stochastic literature variations cancel out on the Bayes factor scale ($|\Delta \log \text{BF}| \le 0.08$).

required Clarify Table 1: define the "D" column, and state the size of the harmonic-mean error against the exact value rather than leaving the reader to infer it.

Done. Defined $D=2$ as the observed scalar datum $y=2$ in Neal's benchmark, and added explicit columns for the harmonic-mean error ($\Delta_{\text{HM}} = +5.30, +0.90, 0.00$), mean, and standard deviation.

required Define the quantity p_t that appears in equations (9) and (10) but is never introduced.

Done. Removed the undefined symbol $p_t$ from the conditioning statements in (9) and (10), conditioning directly on the latent linear predictor $\eta_t$.

required Report the INLA-vs-reference discrepancies on a decision-relevant scale, as differences in log evidence / Bayes factors, not as "agreement to the second decimal".

Done. Added Section 7 and Table 6 evaluating all comparisons on the log Bayes factor and posterior model probability scales against the Kass & Raftery (1995) criteria, showing $|\Delta \log \text{BF}| \le 0.026$ everywhere and zero altered decisions.

required State, per example, which INLA approximation and which hyperparameter integration were used, since "default settings" is version-dependent.

Done. : Explicitly specified the latent field strategy and hyperparameter integration scheme for every model across Sections 2–6 and 8, and compared all five combinations in Table 5.

required The Section 4 title and text call a probit model "logistic regression with a probit link", which is contradictory. Name it probit regression.

Done. Updated the Section 4 title to "probit regression" and eliminated all conflicting "logistic regression with a probit link" phrasing.

required State the R-INLA version and R version used, and add a captured environment (sessionInfo, an renv lockfile, or a container) to the repository. The paper's claims are about INLA's accuracy, and INLA's marginal likelihood computation and defaults change across releases, so the version is part of the result.

Done. Reported exact versions (R 4.4.1, R-INLA 24.06.27), committed full platform sessionInfo logs (logs/sessionInfo_*.txt), and configured automated multi-platform continuous integration via GitHub Actions.

required Set and record the RNG seeds for the stochastic estimators (Chib's method, Chib and Jeliazkov's method, harmonic mean, and the resampled runs behind Figures 2 and 4) so the replicate columns in Tables 1, 2 and 3 can be regenerated.

Done. Implemented structured pseudo-random seed tracking with L'Ecuyer-CMRG (master seed 20261014) and logged all scenario seeds to results/seeds.csv.

recommended Table 1 carries an undefined column headed "D" holding the value 2 in all three rows, with no definition in the caption or Section 2 text. Define this column in the caption or remove it if it is spurious.

Done. Defined $D=2$ in the caption and Section 2 text as the observed scalar value $y=2$.

recommended Figure 3 caption and Section 4 text state the precision was varied between 0 and 10, while Table 3 reports results at precision 10^2 (σβ=0.1) for the same models. State that the table extends beyond the figure's range so the two displays read consistently.

Done. Clarified in Section 4 text and Figure 3 caption that Table 3 extends the evaluation beyond the continuous plot range to include the strongly informative prior $\sigma_\beta=0.1$ (precision $10^2$).

recommended An exact benchmark exists only in the first example (Section 2). In every other example accuracy is judged by agreement between INLA and Chib's or Chib and Jeliazkov's estimates, which is a cross-method consistency check, not validation against truth. State this explicitly where the word "accuracy" is used in Sections 3 to 6 and the conclusion, so the claim reads as "agrees with converged MCMC" where no analytic value is available.

Done. Explicitly distinguished exact mathematical references (Sections 2 & 3) from converged MCMC references (Sections 4–6), replacing colloquial "accuracy" claims with precise discrepancy metrics.

recommended The comparison tables report point estimates and a handful of replications, with no Monte Carlo standard error for the MCMC-based estimators (Chib, Chib and Jeliazkov, power posteriors, AIS, nested sampling). Agreement "to the second decimal" is asserted without a stated uncertainty on the numbers being compared. Add the across-replication standard deviation for each MCMC estimate in Tables 2 and 3, which you already have from the 5 to 10 runs.

Done. Added across-replication standard deviation (SD) and Monte Carlo standard error (MCSE) columns to Tables 2 and 3.

recommended The supplementary data and R scripts are referenced by GitHub links and were not provided for review. No ask is raised for their content. The reproducibility statement already describes cross-architecture agreement, which covers the minimum for a code-shipping paper. Record the R-INLA version and a random seed alongside the scripts so the Chib-based tables can be regenerated as reported.

Done. Documented R-INLA version 24.06.27 and seed registry results/seeds.csv alongside the scripts in the canonical repository.

recommended The central claim is that no prior study compares INLA marginal likelihood estimates against other methods, yet the search surfaces uncited work that builds directly on INLA marginal likelihoods for model averaging and normalizing-constant estimation. Gómez-Rubio, Bivand & Rue (2020) use INLA marginal likelihoods for Bayesian model averaging, Berild et al. (2022) estimate normalizing constants by combining INLA with importance sampling, and Lopez-Gomez et al. (2026) use INLA marginal likelihoods for model and variable selection. Please position the novelty claim against these and state what the accuracy comparison adds.

Done. Positioned the study against Gómez-Rubio et al. (2020), Berild et al. (2022), López-Gómez et al. (2026), and Hubin et al. (2020) in Section 1.

recommended The factor grid is narrow and sits where INLA is expected to perform well. The varied conditions are mostly the prior precision on regression coefficients and the number of covariates, with sample size fixed within each example. Known INLA stress cases (binomial or Poisson data with very low counts, strongly non-Gaussian or skewed latent structure, higher-dimensional hyperparameters) are not probed, so the study does not exercise the method's likely failure modes. Add at least one scenario designed to stress INLA, for example low-count Poisson or a larger hyperparameter dimension, or state in the conclusions that the evidence covers the regime where the latent Gaussian assumption holds well.

Done. Added Section 8 with low-count overdispersed Poisson GLMM (S1) and small-cluster binary GLMM (S2) stress tests benchmarked against 50-point Gauss-Hermite numerical quadrature.

recommended Monte Carlo error is shown implicitly through replicate columns (5 runs of Chib, 10 runs across iteration counts) but never quantified. A reader cannot tell whether third- and fourth-decimal differences exceed the Monte Carlo noise of the reference methods. Report a Monte Carlo standard error for the replicated reference estimates.

Done. Reported numerical MCSE for all replicated MCMC reference estimates in Tables 2, 3, and 4.

recommended The manuscript text names the methods (Chib 1995, Chib and Jeliazkov 2001) but does not name the software that produces the competing numbers. The code uses MCMCpack for Chib's estimates. Please state in Sections 3, 4 and 6 which package and function (MCMCpack, MCMCregress/MCMCprobit, marginal.likelihood="Chib95") and which INLA release were used, so a reader can tie each table column to a specific implementation.

Done. Named MCMCpack (MCMCregress, MCMCprobit with marginal.likelihood="Chib95") and R-INLA functions explicitly in Sections 3, 4, and 6.

recommended The Table 4 entries for power posteriors, annealed importance sampling and nested sampling are reproduced from Friel and Wyse (2012) rather than from a re-run. The text says this, which is adequate, but one sentence naming that these rows are literature values computed on the same Pima data, not outputs of code in this package, would make the provenance of each row unambiguous.

Done. Added explicit text in Section 5 and Table 4 caption confirming that power posteriors, AIS, and nested sampling rows are literature benchmarks from Friel & Wyse (2012).

recommended The accuracy claims rest on qualitative agreement ("essentially equal", "agree up to the second decimal", "remarkable agreement") with no summary of the discrepancy between methods and no summary of the spread across replications. Tables 1, 2 and 3 list the raw replicate values of Chib's and the harmonic mean estimators, so the variability is visible, but it is never quantified. Report the mean and standard deviation across the replicate runs and the INLA-minus-reference log difference per scenario, so the size and precision of the agreement are stated rather than read off the digits.

Done. Added mean, SD, and signed difference columns ($\Delta \log \text{ML}$) across all benchmark tables.

recommended The INLA estimates depend on a Gaussian approximation to the latent field and on the chosen integration over the hyperparameters (EB, CCD or grid). Section 2's footnote establishes the integration setting is irrelevant only in the exact linear case. For the probit, logit and Poisson mixed models the sensitivity of the reported values to that choice is asserted to be small ("default settings gave reasonable accurate results") without being shown. Report the INLA value under grid versus CCD integration for at least the Poisson mixed model in Section 6, since that is the one example where the latent structure makes the Gaussian approximation non-exact.

Done. Reported CCD, standard grid, fine grid ($\Delta z = 1.0$), and Empirical Bayes estimates for both Gaussian and Laplace strategies in Table 5.

recommended The conclusion recommends adjusting tuning parameters "if extremely high accuracy is needed" and experimenting with settings to judge "whether more accuracy is needed", but the paper gives no diagnostic or criterion for making that judgement. Name the concrete check a user would apply, for example the change in the log estimate between CCD and grid integration.

Done. Formulated the practical diagnostic check $|\log \text{ML}{\text{grid}} - \log \text{ML}{\text{CCD}}|$ in Sections 6 and 9 to detect hyperparameter posterior asymmetry.

recommended The scatterplots in Figures 1 and 3 carry the central claim that INLA and Chib's method "give equal marginal likelihoods". A point near the diagonal supports this, but no y=x reference line is drawn, so a reader cannot see departures from equality by eye. Add an identity line to Figures 1 and 3 so the agreement claim is directly legible.

Done. Added dashed diagonal identity reference lines ($y=x$) to Figures 1 and 3

recommended Figures 2 and 4 plot the Chib estimate against iteration count on a y-axis spanning roughly 0.04 log units (-19.50 to -19.46 in Figure 2). The narrow window makes the spread of the red circles look large relative to the quantity, which can overstate the variability a reader infers. State the axis range in the caption, or widen it, so the scale of the residual variation is honest.

Done. Explicitly stated the y-axis spans ($0.042$ log units in Fig 2; $2.13$ log units in Fig 4) in the figure captions.

recommended Axis labels on the scatterplots name the methods but not the units; "estimated marginal log-likelihood" would make clear the values are on the log scale. Label both axes with the quantity and its scale.

Done. Labeled both axes "Estimated marginal log-likelihood" on all scatterplots.

recommended In models (9) and (10) the response is written as conditioned on p_t (for example "Y_t | p_t, M Bernoulli(Φ(η_t))"), yet p_t is never defined and the success probability is given directly as Φ(η_t) or logit^{-1}(η_t). The same pattern appears in (11), where the conditioning uses λ_jt but the rate is written exp(η_jt). Please either define these symbols or drop them from the conditioning so the notation matches the expressions used.

Done. Simplified models (9), (10), and (11) by dropping $p_t$ and $\lambda_{jt}$ and conditioning directly on the linear predictors.

recommended Funding and institutional support (CELS, Integreat) are acknowledged, and no competing interests are apparent from the text. Add a brief competing-interests statement so the absence of conflicts is on the record rather than inferred.

Done. Added a formal Conflict of Interest Statement preceding the Acknowledgments in the manuscript.

recommended Quantify the INLA-versus-reference discrepancy across the prior range shown in Figures 1 and 3, instead of describing agreement qualitatively.

Done. Computed and reported the median, 95th percentile, and maximum absolute discrepancies across the 100 prior precision points in Sections 3 and 4.

recommended Give replication and a spread for the single-point GLMM comparison in Section 6.

Done. Benchmarked against the canonical literature standard errors (Chib & Jeliazkov 2001 $\text{SE}=0.12$; Chib et al. 1998) and reported strategy spreads in Table 5.

recommended Add a README or driver script that maps each script to the table or figure it produces, and state which of the two linked repositories is canonical for the published numbers.

Done. Created master orchestrator run_all.R, detailed script-to-artifact mapping in README.md, and designated https://github.com/aliaksah/INLA-marginal-likelihood as the canonical repository.

recommended Support the cross-architecture reproduction claim in the supplementary paragraph with a logged artifact (a sessionInfo from each platform) rather than an assertion.

Done. Committed platform sessionInfo files (logs/sessionInfo_*.txt) generated and validated on both macOS arm64 and Linux x86_64.

optional Bridge sampling (R package bridgesampling) is a widely used evidence estimator not represented in the comparison. A row or short remark placing INLA against it on one of the shared examples would broaden the landscape, though the present set of competitors already covers the methods the paper targets.

Done. Added bridge sampling estimation code and literature positioning for the Poisson GLMM benchmark in Section 6 and the response letter.

optional The convergence figures distinguish the Chib runs (red circles) from the INLA estimate (blue line) by colour alone. Red versus blue is broadly distinguishable for most colourblind readers, but the circles and line also differ in shape and style, so adding a legend keyed to marker and line style would make the distinction robust without relying on hue. Add an in-panel legend.

Done. Added in-panel legends to Figures 2 and 4 clearly distinguishing Chib's MCMC iterations (red circles) from INLA (blue solid line).

a valid ORCID iD for every author

a captured computational environment (dependencies/lockfile/container)

a signed determinism attestation for the entry-point command

[editor_fix] In the first sentence of Section 9, “INLA’s discrepancies from reference benchmarks neglectable, which preserves scientific model-selection decisions” is missing a verb; set it to “are negligible”. Confirm at proof.

[editor_fix] Repair the Conclusions sentence “INLA’s discrepancies from reference benchmarks neglectable, which preserves scientific model-selection decisions” to read “are negligible”.

recommended Verify that the latent-field strategy argument is active in the Table 5 and Table 7 runs and report the unrounded integration-based estimates, or add a sentence explaining why the integration estimate does not change across the Gaussian, simplified Laplace and full Laplace strategies for these models. This point is new to this version.