Bayesian models for missing and misclassified variables using integrated nested Laplace approximations

Emma Skarstein; H{\aa}vard Rue; Leonardo Soares Bastos; Stefanie Muff

REVIEW 2 major objections 4 minor 41 references

Bayesian models for missing and misclassified variables using integrated nested Laplace approximations

T0 review · 2 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read Bayesian models with misclassified categorical covariates can be fit with INLA by importance-sampling the latent true covariate; a misclassified binary response is handled with a shifted, scaled logit link.

desk verdict Useful, code-bearing adaptation of INLA+importance sampling to misclassification, but the reported credible intervals are built on a quantile-of-mixture shortcut that needs fixing. read the letter →

arxiv 2411.16311 v1 pith:FMQ6NDKI submitted 2024-11-25 stat.ME

Emma Skarstein , Leonardo Soares Bastos , H{aa}vard Rue , Stefanie Muff This is my paper · ORCID

classification stat.ME MSC 62F1562J12

keywords misclassificationintegratednestedLaplaceapproximationimportancesamplingcategoricalcovariatebinaryresponsemissingdataBayesianinferencemeasurementerror

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

The paper claims that Bayesian regression with a misclassified binary covariate, previously awkward for INLA because the true covariate is a latent discrete variable, can be handled by sampling the latent covariate from its posterior given the observed misclassified version, fitting the regression model conditional on each sample with INLA, and averaging the resulting conditional posteriors with importance weights proportional to the conditional marginal likelihood. It also claims that a misclassified binary response can be handled directly in INLA by replacing the logit link with a shifted, scaled logit that encodes sensitivity and specificity. If these claims hold, users of the fast INLA machinery can correct the attenuation and bias caused by misclassification without abandoning the framework.

What carries the argument

The misclassification matrix M, the exposure model logit(p_x) = alpha0 + Z_alpha, and the derived Bernoulli posterior for the latent covariate, whose success probability p_{x|w,Z} is a simple Bayes update of the exposure probability using the misclassification matrix. The marginalization identity pi(theta_y | y,w,Z) = integral pi(theta_y | x,y,Z) pi(x | y,w,Z) dx is approximated by importance sampling, with weights proportional to the conditional marginal likelihood pi(y | x^(j), Z) computed by INLA. For the response case, the central object is the shifted and scaled logit link logit((p_s - (1 - pi00))/(pi11 - (1 - pi00))) = beta0 + Z_i^T beta_z, which converts a misclassified binary response into an ordinary Bernoulli model with a corrected link.

What would settle it

Simulate datasets with a known true binary covariate and misclassification matrix, then run the proposed importance-sampling INLA procedure with a deliberately wrong but plausible misclassification matrix and examine coverage of the posterior intervals for the covariate coefficient across many replications; if coverage drops well below the nominal level, the known-M assumption is demonstrably load-bearing.

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Extended reading notes

Core claim

The central claim is that the marginal posterior of the regression coefficients in a model with a misclassified categorical covariate can be approximated as a weighted average of INLA conditional posteriors, with weights given by the normalized conditional marginal likelihoods. The true covariate is treated as a latent Bernoulli variable, sampled from the posterior that combines the exposure model and the misclassification matrix; the conditional model given each sampled covariate is then fitted in INLA. For a misclassified binary response, the paper shows that the observed outcome still follows a Bernoulli model, but with a shifted and scaled logit link that explicitly contains the sensitivity and specificity, so the correction can be implemented without a latent discrete variable, directly in INLA. In the special case where the misclassified variable can be interpreted as a discretization of a latent Gaussian variable with measurement error, the paper shows that the whole model becomes a probit measurement-error model and can be fitted using only INLA.

Load-bearing premise

The load-bearing premise is that the exposure model coefficients and the misclassification probabilities are known in advance; if they are misspecified, the importance-weighted posterior will be biased, and the method offers no internal mechanism to detect or correct that misspecification.

Editorial extensions

If this is right

A naive regression using the misclassified covariate attenuates the covariate effect, while the proposed importance-sampling correction recovers the true coefficient when the exposure model and misclassification probabilities are correctly specified.
Missing categorical covariate values can be imputed within the same importance-sampling framework by treating missingness as 50% misclassification, so the exposure model alone determines the sampled value.
A misclassified binary response can be corrected in INLA with a fixed or uncertain sensitivity and specificity, and the latter can be propagated by averaging over a grid of plausible values.
When the misclassified binary variable arises from dichotomizing a mismeasured continuous variable, the entire model can be fitted in INLA alone, avoiding the slower importance-sampling loop.
The choice between differential and nondifferential misclassification can substantially change the estimated coefficients, as the cervical-cancer and herpes example demonstrates.

Reading between the lines

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

Because the method treats the exposure model coefficients and the misclassification matrix as known inputs, real applications will likely need sensitivity analyses over these inputs; the paper's grid-weighting idea for response misclassification suggests a natural template for such analyses.
The block proposal of the entire latent covariate vector is flagged as inefficient; a component-wise or sequential proposal could both speed convergence and open the door to estimating the exposure and misclassification parameters outside INLA.
The numerical instability of the shifted logit link at low sensitivity and specificity is especially consequential because that is exactly the regime where correcting for misclassification matters most; a more stable reparameterization of the link would expand the method's practical reach.
The same imputation logic could be extended to models with multiple categorical covariates that are each partially missing or misclassified, since each latent covariate can be sampled and weighted in the same conditional-marginal-likelihood scheme.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

You should know two things upfront. First, this is the first paper I've seen that actually makes misclassified categorical covariates and responses work inside INLA, with a GitHub package and full reproduction code. Second, the credible intervals in the key figures are built on a quantile approximation that is not valid, and that needs to be fixed before the central inference claims can be taken at face value.

What's new: the importance-sampling-with-INLA machinery is from Berild et al. (2022), and the missing-data analogue is Skarstein et al. (2023), but this paper adapts it to misclassification, derives the conditional Bernoulli sampling probability (Eq. 5) cleanly, and implements the response-link reparameterization (Eq. 7) as a custom 'sslogit' link in R-INLA. The differential-misclassification case study (cervical cancer/HSV-2) is a good real-data illustration, and the code and package make the methods immediately usable.

The main soft spot is not the known-exposure-model assumption, which the authors admit, but something they don't flag. In Section 3.2.1 they state that posterior α-quantiles are approximated by the weighted average of the component posterior quantiles. That is not correct. The posterior is a mixture of the conditional posteriors; the α-quantile of a mixture is the inverse of the mixture CDF at α, not the mixture of the component quantiles. The figures (1,2,6,7) report 95% credible intervals, and if those were computed with that formula, coverage is likely off, potentially by a lot when the imputed covariate shifts βx. Posterior means are fine — expectation is linear in the mixture — but the interval logic is not. This is fixable: compute the mixture CDF and invert it (or mix the INLA marginal densities directly) rather than averaging quantiles.

The other limitations are acknowledged in the text: exposure-model and misclassification probabilities are assumed known, and the response-link method goes numerically unstable for low sensitivity/specificity. Those are real but scoped. The running time is heavy (7 hours for 200k iterations), though the discussion is honest about that.

Bottom line: this is a genuinely useful methods paper for the INLA/epidemiology crowd, with reproducible code and a clear derivation. The quantile issue is a load-bearing flaw in the presentation of intervals, but it is fixable and doesn't undermine the point estimates. I'd send it to a serious referee; a competent referee will catch the quantile problem and the authors can address it in revision.

Referee Report

2 major / 4 minor

Summary. The paper proposes two ways to fit regression models with misclassified categorical variables using the integrated nested Laplace approximation (INLA). For covariate misclassification, the true binary covariate x is treated as latent and sampled from its conditional distribution given the observed misclassified version w and an assumed exposure model; each sampled x is used to fit the model in INLA, and the conditional posterior distributions of the regression coefficients are combined by importance sampling weights proportional to the conditional marginal likelihoods. For response misclassification, the paper derives a modified logit link, called sslogit, that expresses the success probability of the observed response in terms of the true response probability and known sensitivity and specificity, and implements it directly in INLA. The paper also covers missing categorical covariates, a special case where misclassification arises from dichotomizing a mismeasured continuous variable, and illustrates all methods with simulations and two applied examples using the accompanying R package inlamisclass.

Significance. If the proposed methods are correct, they meaningfully extend the class of Bayesian models that can be fit efficiently with INLA to settings with discrete latent variables induced by misclassification and missingness. The paper is accompanied by reproducible code and an R package, which is a practical strength. The response-misclassification link is a clean reparameterization that is straightforward to implement. However, the central covariate-misclassification algorithm has a load-bearing technical flaw in its treatment of posterior quantiles: it averages conditional quantiles rather than computing quantiles of the weighted mixture, which can produce mis-calibrated credible intervals. The method also requires the exposure model and misclassification probabilities to be known, a limitation that is acknowledged but not addressed with sensitivity analyses.

major comments (2)

[Section 3.2.1, unnumbered display for q_alpha(beta_x | y, w, Z)] The approximation q_alpha(beta_x | y,w,Z) approx sum_j omega_j q_alpha(beta_x | x^(j), y, Z) is not a valid way to compute a quantile of a mixture distribution. If F_j(t) are the conditional posterior CDFs, the mixture CDF is F(t) = sum_j omega_j F_j(t), and the alpha-quantile is F^{-1}(alpha), not sum_j omega_j F_j^{-1}(alpha). Quantiles are nonlinear functionals, so the displayed formula is generally wrong. For example, for two equally weighted conditional posteriors N(-1,1) and N(1,1), the 0.025 quantile of the mixture lies below -1.96, whereas the weighted average of the two 0.025 quantiles equals -1.96, producing an interval that is too narrow. Since Figures 1, 2, 6, and 7 present 95% credible intervals and the manuscript does not state any alternative computation, the reported interval coverage is not guaranteed. The authors should replace this step with numerical inversion of the weighted mixture CDF (or sampling from the mixture) and update all simulation and application results accordingly.
[Section 3.2.1 and Algorithm 1] The covariate-misclassification method assumes that the exposure-model coefficients alpha and the misclassification matrix M are known; this is stated in the text but the implications are not explored. In the simulations and applications, alpha and M are either known by design or chosen hypothetically (as in the birth weight example), and no sensitivity analysis is reported for plausible misspecification of these inputs. Because in many real applications these quantities are not known precisely, the paper would be substantially strengthened by a systematic study of how mis-specified alpha or M affect the resulting posterior distributions and credible intervals, or by a demonstration of how these parameters can be estimated within the importance-sampling framework.

minor comments (4)

[Section 5.1] The statement 'With 200,000 iterations, each model took us around seven hours to run' should clarify whether this is per simulated dataset or for the entire set of ten datasets, and whether all models were run sequentially or in parallel.
[Section 3.2.2] The notation in this section introduces x_c and x_d in addition to the earlier binary x, which can be confusing; a brief sentence clarifying that x_c is the latent continuous covariate and x_d is its discretized version would help the reader.
[Section 4.1] The grid approximation for p(py | s) uses weights omega(pi00(k), pi11(k)) derived from a joint distribution for sensitivity and specificity, but the manuscript does not specify how this joint distribution is chosen or whether a product of independent Beta distributions is intended; the supporting information should be referenced within the main text.
[Abstract and Section 7] The paper repeatedly refers to 'detailed code in the supporting information' and a GitHub repository; for archival purposes, the code should be deposited in a permanent repository with a versioned DOI, since GitHub links may not be stable.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the covariate-misclassification posterior is derived from a standard Bayes/marginal-likelihood identity, and the response-misclassification link is an algebraic reparameterization; self-citations are implementation groundwork, not load-bearing.

full rationale

The central derivation in Section 3.2.1 starts from the exact identity pi(theta_y | y,w,Z) = integral pi(theta_y | x,y,Z) pi(x | y,w,Z) dx and approximates it by importance sampling with weights proportional to the conditional marginal likelihoods pi(y | x(j),Z). This is a standard Monte Carlo identity, not a fitted quantity renamed as a prediction. The exposure-model coefficients alpha and misclassification matrix M are stated assumptions ('we assume the parameters of the exposure model and the misclassification probabilities to be known'), and the paper explicitly notes that estimating them would require an MCMC extension. The response-misclassification result in Section 4 follows by algebra: p_si = pi11 p_yi + (1-pi00)(1-p_yi) is inverted to define the shifted, scaled logit link in Eq. (7); this is a reparameterization of the same Bernoulli model, presented as an implementation device for INLA rather than as an empirical prediction. The paper relies on earlier work by the same group (Berild et al. 2022 for INLA-within-importance-sampling; Muff et al. 2015 for measurement-error INLA; Skarstein et al. 2023 for missing-data extensions), but these are external, published methodological building blocks and are not used as a uniqueness theorem or to forbid alternatives. All simulation studies compare against known true parameter values, providing external checks. The only notable technical concern in the manuscript, approximating posterior quantiles of the mixture by weighted averages of conditional quantiles, is a mathematical correctness issue (quantiles are not linear functionals of mixture components), not a circularity: the claimed approximation does not equal its input by construction, and the paper's core weighting identity does not depend on it. For these reasons the circularity score is low.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central method rests on known exposure models and misclassification probabilities, which are external inputs rather than estimated quantities. No new physical entities are introduced.

free parameters (3)

Misclassification probabilities (sensitivity, specificity or matrix entries) = Birth weight example: pi00=0.95, pi11=0.80; HSV example: differential matrices bM1 and bM0
These are assumed known inputs to the model, chosen by the authors based on substantive assumptions or estimated from a validation subset. The proposed importance sampling method does not estimate them.
Exposure model coefficients alpha = Birth weight case 2: alpha0=-0.4, alpha_z=0.02; simulations: alpha=(-0.5, 0.25)
The model for the latent covariate x given other covariates is assumed known; the paper states this assumption in Section 3.2.1.
Measurement error standard deviation sigma_u in the discretized Gaussian model = Fixed to 1 in the simulation of Section 5.2
In the latent Gaussian approach, sigma_u must be specified; the paper fixes it to 1, and the result depends on this choice.

assumptions (5)

domain assumption The exposure model for the latent covariate is known and correctly specified (Eq. 3)
Section 3.2.1 assumes the coefficients alpha are known; if misspecified, the posterior correction is biased.
domain assumption Covariate misclassification is nondifferential given the true covariate, i.e., the observed misclassified variable is conditionally independent of the response given the true covariate (Section 2, paragraph on nondifferential misclassification)
This gives the factorization used in Eq. (4). The HSV application later relaxes this with differential matrices.
domain assumption INLA provides accurate approximations to conditional posterior marginals and marginal likelihoods
The weighting uses INLA's approximate marginal likelihood for each imputed covariate vector.
domain assumption The importance sampling proposal is adequate with a finite number of samples
The paper uses 100k-200k draws but does not report effective sample size or Monte Carlo error.
domain assumption Missing categorical covariate values are missing completely at random (Section 5.3)
The simulation generates MCAR missingness, and the method's validity under other missingness mechanisms is not demonstrated.

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Cite this review

Pith. "Pith review of Bayesian models for missing and misclassified variables using integrated nested Laplace approximations." pith.science (2026). https://pith.science/paper/FMQ6NDKI

@misc{pith2026241116311,
  author       = {Pith},
  title        = {Pith review of: Bayesian models for missing and misclassified variables using integrated nested Laplace approximations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FMQ6NDKI}},
  note         = {Machine review of arXiv:2411.16311}
}

read the original abstract

Misclassified variables used in regression models, either as a covariate or as the response, may lead to biased estimators and incorrect inference. Even though Bayesian models to adjust for misclassification error exist, it has not been shown how these models can be implemented using integrated nested Laplace approximation (INLA), a popular framework for fitting Bayesian models due to its computational efficiency. Since INLA requires the latent field to be Gaussian, and the Bayesian models adjusting for covariate misclassification error necessarily introduce a latent categorical variable, it is not obvious how to fit these models in INLA. Here, we show how INLA can be combined with importance sampling to overcome this limitation. We also discuss how to account for a misclassified response variable using INLA directly without any additional sampling procedure. The proposed methods are illustrated through a number of simulations and applications to real-world data, and all examples are presented with detailed code in the supporting information.

Figures

Figures reproduced from arXiv: 2411.16311 by the authors.

**Figure 2.** Box plots of the posterior means for the estimated coefficients from ten different [PITH_FULL_IMAGE:figures/full_fig_p017_2.png] view at source ↗

**Figure 3.** Posterior means and 95% credible intervals for the example where the misclas [PITH_FULL_IMAGE:figures/full_fig_p019_3.png] view at source ↗

Figures from the paper (4 more)

**Figure 4.** Figure 4: Posterior means and 95% credible intervals for the estimated coefficients from [PITH_FULL_IMAGE:figures/full_fig_p020_4.png]

**Figure 5.** Figure 5: Posterior marginal distributions of the response success probability for models [PITH_FULL_IMAGE:figures/full_fig_p021_5.png]

**Figure 6.** Figure 6: Posterior means and 95% credible intervals for the naive and error-corrected [PITH_FULL_IMAGE:figures/full_fig_p023_6.png]

**Figure 7.** Figure 7: Posterior means and 95% credible intervals for the naive and error-corrected [PITH_FULL_IMAGE:figures/full_fig_p025_7.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

O., Martino, S., Gómez-Rubio, V., and Rue, H

Berild, M. O., Martino, S., Gómez-Rubio, V., and Rue, H. (2022). Importance sampling with the integrated nested laplace approximation. Journal of Computational and Graphical Statistics , 31(4):1225--1237

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Buonaccorsi, J. P., Laake, P., and Veierød, M. B. (2005). On the effect of misclassification on bias of perfectly measured covariates in regression. Biometrics , 61(3):831--836

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Carroll, R., Gail, M. H., and Lubin, J. H. (1993). Case-control studies with errors in covariates. Journal of the American Statistical Association , 88(421):185--199

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Felsenstein, J. (2005). Using the quantitative genetic threshold model for inferences between and within species. Philosophical Transactions of the Royal Society B: Biological Sciences , 360(1459):1427--1434

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