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arxiv: 2606.24357 · v1 · pith:DX5I6BW6new · submitted 2026-06-23 · 📊 stat.ME · math.ST· stat.CO· stat.TH

Bayesian Variable Selection in Generalized Linear Models

Lucia Filippozzi , I\~nigo Urteaga , Claudio Agostinelli This is my paper

Pith reviewed 2026-06-25 22:38 UTC · model grok-4.3

classification 📊 stat.ME math.STstat.COstat.TH
keywords Bayesian variable selectionGeneralized linear modelsPosterior consistencyConjugate priorsGibbs samplingExponential familyCovariate selectionHierarchical model
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The pith

A fully conjugate Bayesian framework selects covariates in generalized linear models for any exponential family distribution while proving posterior consistency for both inclusions and active coefficients.

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

The paper develops a Bayesian method for choosing which covariates to include in generalized linear models that works for any distribution in the exponential family. It uses a hierarchical model with binary indicators for each covariate's inclusion directly in the linear predictor. This setup allows full conjugacy so that variable selection and coefficient estimation occur together within one posterior. The authors prove that the posterior concentrates on the correct inclusions and coefficients as data grows. A reader would care because earlier Bayesian approaches were either non-conjugate, limited to linear regression, or lacked consistency guarantees, limiting reliable inference that accounts for selection uncertainty.

Core claim

We propose a fully Bayesian hierarchical and conjugate framework for covariate selection in GLMs, applicable to any distribution in the exponential family, based on modeling a binary inclusion indicator that directly encodes covariate inclusion in the linear predictor. In our approach, variable selection and parameter estimation are performed simultaneously, incorporating both sources of uncertainty in posterior inference. Consequently, our methodology provides a valid post-model Bayesian selection procedure. We present theoretical guarantees of the proposed fully conjugate Bayesian variable selection for GLMs, establishing posterior consistency of both the inclusion indicators and the activ

What carries the argument

binary inclusion indicator that directly encodes covariate inclusion in the linear predictor to achieve a fully conjugate hierarchical formulation applicable to any exponential family distribution

If this is right

  • Variable selection and parameter estimation performed simultaneously while incorporating both sources of uncertainty.
  • Provides a valid post-model Bayesian selection procedure.
  • Efficient Gibbs Sampling algorithm with corresponding R package implementation.
  • Competitive predictive and inferential performance on synthetic and real-world datasets.
  • Theoretical guarantees establishing posterior consistency of the inclusion indicators and the active regression coefficients.

Where Pith is reading between the lines

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

  • The framework could be applied to high-dimensional data where many covariates are present but only a few are relevant.
  • It might allow direct uncertainty quantification in predictions that accounts for which variables were selected.
  • Extensions to non-exponential-family responses would require new conjugacy constructions.
  • The Gibbs sampler could serve as a baseline for comparing mixing speed against other MCMC methods for GLMs.

Load-bearing premise

Modeling a binary inclusion indicator directly in the linear predictor achieves a fully conjugate hierarchical formulation for any exponential family distribution.

What would settle it

A simulation in which the posterior probability of the true inclusion vector fails to approach one as the sample size tends to infinity for some member of the exponential family.

Figures

Figures reproduced from arXiv: 2606.24357 by Claudio Agostinelli, I\~nigo Urteaga, Lucia Filippozzi.

Figure 1
Figure 1. Figure 1: Poisson Model. Component-wise accuracy of the inclusion variable [PITH_FULL_IMAGE:figures/full_fig_p014_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Poisson Model. With p = 20 total covariates, d = 10 noisy and 2c = 4 correlated variables. The accuracy of the inclusion variable z increases for larger values of the n/p ratio. Shifting our focus to the recovery of the regression coefficients β in Setting C, we are particularly interested in two aspects: (i) whether the model correctly estimates the coefficients associated with the relevant covariates, an… view at source ↗
Figure 3
Figure 3. Figure 3: Poisson Model. Posterior distribution of [PITH_FULL_IMAGE:figures/full_fig_p016_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Poisson Model. Error metrics of the active coefficients [PITH_FULL_IMAGE:figures/full_fig_p017_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Poisson Model. Posterior distribution of some non-active coefficients ( [PITH_FULL_IMAGE:figures/full_fig_p017_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Crabs dataset. random samples added as noise (as noted in the dataset documentation), are discarded through the variable selector z. This indicates that, as we expected, the model effectively identifies and excludes pure noise covariates. For evaluating prediction accuracy, we calculated the Mean Absolute Error (MAE) and Root Mean Squared Error (RootMSE) on test set outcomes of interest. The distribution o… view at source ↗
Figure 7
Figure 7. Figure 7: Crabs dataset. Prediction error metrics across folds. Predictive accuracy for our [PITH_FULL_IMAGE:figures/full_fig_p019_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Heart Disease dataset. estimated three models under identical conditions: BayesVS-GLM, our proposed Bayesian GLM with variable selection; BayesGLM, a Bayesian GLM with all variables included (no sparsity); and its frequentist counterpart MLE. The prior hyperparameters for BayesVS-GLM model were fixed to a0 = 0.01, ξ0 i.i.d. ∼ Bern(0.5), α = 1. To illustrate the performance of BayesVS-GLM in variable select… view at source ↗
Figure 9
Figure 9. Figure 9: Heart Disease dataset. Prediction error metrics across folds: Balanced accuracy [PITH_FULL_IMAGE:figures/full_fig_p021_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Pollution dataset. Across the model space of dimension 2p −1 = 32767, the Gibbs sampler explored 3025 distinct models during the MCMC run. After discarding those visited only 1 or 2 times, 954 models remain. The most frequent models consistently include nonw, while there’s a disagreement regarding sox and educ, and even more variabilty for prec and jant. Although some of these variables overlap with those… view at source ↗
Figure 11
Figure 11. Figure 11: Pollution dataset. Pairwise relative frequencies of inclusion. [PITH_FULL_IMAGE:figures/full_fig_p023_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Pollution dataset. Prediction error metrics across folds: Adjusted R2 (left), [PITH_FULL_IMAGE:figures/full_fig_p023_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Poisson Model. Component-wise accuracy of the inclusion variable [PITH_FULL_IMAGE:figures/full_fig_p061_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Poisson Model. Accuracy of the inclusion variable [PITH_FULL_IMAGE:figures/full_fig_p062_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Poisson Model. Posterior distribution of [PITH_FULL_IMAGE:figures/full_fig_p063_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Poisson Model. RelMSE of the of the active coefficients ( [PITH_FULL_IMAGE:figures/full_fig_p064_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Poisson Model. Posterior distribution of some non-active coefficients ( [PITH_FULL_IMAGE:figures/full_fig_p064_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: Poisson Model. Component-wise accuracy of the inclusion variable [PITH_FULL_IMAGE:figures/full_fig_p065_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: Poisson Model. Accuracy of the inclusion variable [PITH_FULL_IMAGE:figures/full_fig_p066_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: Poisson Model. Posterior distribution of [PITH_FULL_IMAGE:figures/full_fig_p066_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: Poisson Model. Posterior distribution of some non-active coefficients ( [PITH_FULL_IMAGE:figures/full_fig_p067_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: Poisson Model. RelMSE of the of the active coefficients ( [PITH_FULL_IMAGE:figures/full_fig_p067_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: Linear Model. Component-wise accuracy of the inclusion variable [PITH_FULL_IMAGE:figures/full_fig_p068_23.png] view at source ↗
Figure 24
Figure 24. Figure 24: Linear Model. Accuracy of the inclusion variable [PITH_FULL_IMAGE:figures/full_fig_p069_24.png] view at source ↗
Figure 25
Figure 25. Figure 25: Linear Model. Posterior distribution of β ◦ z, for n = 100, p = 10 and d noisy variables. Our posterior mean is close to the MLE. This confirms that our method is able to accurately estimate the coefficients associated with truly relevant covariates [PITH_FULL_IMAGE:figures/full_fig_p069_25.png] view at source ↗
Figure 26
Figure 26. Figure 26: Linear Model. RelMSE of the of the active coefficients ( [PITH_FULL_IMAGE:figures/full_fig_p070_26.png] view at source ↗
Figure 27
Figure 27. Figure 27: Linear Model. Posterior distribution of some non-active coefficients ( [PITH_FULL_IMAGE:figures/full_fig_p070_27.png] view at source ↗
Figure 28
Figure 28. Figure 28: Linear Model. Component-wise accuracy of the inclusion variable [PITH_FULL_IMAGE:figures/full_fig_p071_28.png] view at source ↗
Figure 29
Figure 29. Figure 29: Linear Model. Accuracy of the inclusion variable [PITH_FULL_IMAGE:figures/full_fig_p071_29.png] view at source ↗
Figure 30
Figure 30. Figure 30: Linear Model. Posterior distribution of β ◦ z, for n = 100, p = 10 and 2c correlated redundant variables [PITH_FULL_IMAGE:figures/full_fig_p072_30.png] view at source ↗
Figure 31
Figure 31. Figure 31: Linear Model. Posterior distribution of some non-active coefficients ( [PITH_FULL_IMAGE:figures/full_fig_p072_31.png] view at source ↗
Figure 32
Figure 32. Figure 32: Linear Model. RelMSE of the of the active coefficients ( [PITH_FULL_IMAGE:figures/full_fig_p073_32.png] view at source ↗
Figure 33
Figure 33. Figure 33: Linear Model. Component-wise accuracy of the inclusion variable [PITH_FULL_IMAGE:figures/full_fig_p073_33.png] view at source ↗
Figure 34
Figure 34. Figure 34: Linear Model. With p = 20 total covariates, d = 10 noisy and 2c = 4 correlated variables. Accuracy of the inclusion variable z, and increasing ratio n/p. Shifting our focus to the recovery of the regression coefficients β in Setting C [PITH_FULL_IMAGE:figures/full_fig_p074_34.png] view at source ↗
Figure 35
Figure 35. Figure 35: Linear Model. Posterior distribution of β ◦ z, for p = 20, d = 10 and 2c = 4. We then summarize estimation performance through a quantitative error metric. Fig￾ures 36a and 36b show the Relative Mean Squared Error (and standard deviation) and the Relative Max Squared Error, of the estimated active coefficient with respect to the true known coefficients β ∗(1), as n increases for fixed values of p, c, and … view at source ↗
Figure 36
Figure 36. Figure 36: Linear Model. Error metrics of the active coefficients [PITH_FULL_IMAGE:figures/full_fig_p076_36.png] view at source ↗
Figure 37
Figure 37. Figure 37: Logistic Model. Component-wise accuracy of the inclusion variable [PITH_FULL_IMAGE:figures/full_fig_p077_37.png] view at source ↗
Figure 38
Figure 38. Figure 38: Logistic Model. Accuracy of the inclusion variable [PITH_FULL_IMAGE:figures/full_fig_p077_38.png] view at source ↗
Figure 39
Figure 39. Figure 39: Logistic Model. Posterior distribution of [PITH_FULL_IMAGE:figures/full_fig_p078_39.png] view at source ↗
Figure 40
Figure 40. Figure 40: Logistic Model. RelMSE of the of the active coefficients ( [PITH_FULL_IMAGE:figures/full_fig_p078_40.png] view at source ↗
Figure 41
Figure 41. Figure 41: Logistic Model. Posterior distribution of some non-active coefficients ( [PITH_FULL_IMAGE:figures/full_fig_p079_41.png] view at source ↗
Figure 42
Figure 42. Figure 42: Logistic Model. Component-wise accuracy of the inclusion variable [PITH_FULL_IMAGE:figures/full_fig_p079_42.png] view at source ↗
Figure 43
Figure 43. Figure 43: Logistic Model. Accuracy of the inclusion variable [PITH_FULL_IMAGE:figures/full_fig_p080_43.png] view at source ↗
Figure 44
Figure 44. Figure 44: Logistic Model. Posterior distribution of [PITH_FULL_IMAGE:figures/full_fig_p080_44.png] view at source ↗
Figure 45
Figure 45. Figure 45: Logistic Model. Posterior distribution of some non-active coefficients ( [PITH_FULL_IMAGE:figures/full_fig_p081_45.png] view at source ↗
Figure 46
Figure 46. Figure 46: Logistic Model. RelMSE of the of the active coefficients ( [PITH_FULL_IMAGE:figures/full_fig_p081_46.png] view at source ↗
Figure 47
Figure 47. Figure 47: Logistic Model. Component-wise accuracy of the inclusion variable [PITH_FULL_IMAGE:figures/full_fig_p082_47.png] view at source ↗
Figure 48
Figure 48. Figure 48: Logistic Model. With p = 10 total covariates, d = 5 noisy and 2c = 2 correlated variables. Accuracy of the inclusion variable z, and increasing ratio n/p [PITH_FULL_IMAGE:figures/full_fig_p082_48.png] view at source ↗
Figure 49
Figure 49. Figure 49: Logistic Model. Posterior distribution of [PITH_FULL_IMAGE:figures/full_fig_p083_49.png] view at source ↗
Figure 50
Figure 50. Figure 50: Logistic Model. Error metrics of the active coefficients [PITH_FULL_IMAGE:figures/full_fig_p084_50.png] view at source ↗
Figure 51
Figure 51. Figure 51: Heart Disease dataset. Pairwise relative frequencies of inclusion. [PITH_FULL_IMAGE:figures/full_fig_p085_51.png] view at source ↗
Figure 52
Figure 52. Figure 52: Linear Model. Posterior inclusion probabilities Pr( [PITH_FULL_IMAGE:figures/full_fig_p086_52.png] view at source ↗
Figure 53
Figure 53. Figure 53: Linear Model. Proportion of correctly recovered entries of [PITH_FULL_IMAGE:figures/full_fig_p087_53.png] view at source ↗
Figure 54
Figure 54. Figure 54: Linear Model. Overall selection accuracy (proportion of correctly identified [PITH_FULL_IMAGE:figures/full_fig_p087_54.png] view at source ↗
Figure 55
Figure 55. Figure 55: Poisson Model. Posterior inclusion probabilities Pr( [PITH_FULL_IMAGE:figures/full_fig_p088_55.png] view at source ↗
Figure 56
Figure 56. Figure 56: Poisson Model. Proportion of correctly recovered entries of [PITH_FULL_IMAGE:figures/full_fig_p089_56.png] view at source ↗
Figure 57
Figure 57. Figure 57: Poisson Model. Overall selection accuracy (proportion of correctly identified [PITH_FULL_IMAGE:figures/full_fig_p089_57.png] view at source ↗
Figure 58
Figure 58. Figure 58: Logistic Model. Posterior inclusion probabilities Pr( [PITH_FULL_IMAGE:figures/full_fig_p090_58.png] view at source ↗
Figure 59
Figure 59. Figure 59: Logistic Model. Proportion of correctly recovered entries of [PITH_FULL_IMAGE:figures/full_fig_p091_59.png] view at source ↗
Figure 60
Figure 60. Figure 60: Logistic Model. Overall selection accuracy (proportion of correctly identified [PITH_FULL_IMAGE:figures/full_fig_p091_60.png] view at source ↗
read the original abstract

Covariate selection in Generalized Linear Models (GLMs) is a fundamental problem in statistics, as including irrelevant predictors might lead to overfitting and poor interpretability, while omitting relevant ones might result in biased estimates. Most Bayesian approaches to variable selection -- including spike-and-slab priors and continuous shrinkage priors -- have key limitations, e.g., (i) are based on non fully conjugate formulations, (ii) are restricted to a linear model, or (iii) lack posterior consistency guarantees for the variable selection procedure and model parameters. In this work, we propose a fully Bayesian hierarchical and conjugate framework for covariate selection in GLMs, applicable to any distribution in the exponential family, based on modeling a binary inclusion indicator that directly encodes covariate inclusion in the linear predictor. In our approach, variable selection and parameter estimation are performed simultaneously, incorporating both sources of uncertainty in posterior inference. Consequently, our methodology provides a valid post-model Bayesian selection procedure. We present theoretical guarantees of the proposed fully conjugate Bayesian variable selection for GLMs, establishing posterior consistency of both the inclusion indicators and the active regression coefficients. We derive an efficient Gibbs Sampling algorithm with a corresponding R package implementation. We validate the proposed method on synthetic and real-world datasets, demonstrating competitive predictive and inferential performance.

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 proposes a fully Bayesian hierarchical and conjugate framework for covariate selection in generalized linear models (GLMs) applicable to any exponential-family distribution. It models binary inclusion indicators that directly encode covariate inclusion in the linear predictor, enabling simultaneous variable selection and parameter estimation via a Gibbs sampler (with R package). The central claims are posterior consistency of the inclusion indicators and active regression coefficients, along with competitive performance on synthetic and real data.

Significance. If the conjugacy construction and consistency results hold, the work would address documented limitations of existing spike-and-slab and shrinkage approaches (non-conjugacy, restriction to linear models, lack of selection consistency) by supplying a broadly applicable conjugate formulation with explicit posterior-consistency guarantees. The open R implementation would further support reproducibility.

major comments (2)
  1. [§3] §3 (theoretical results): the abstract asserts posterior consistency for both inclusion indicators and active coefficients, yet the provided text contains no proof sketch, no statement of the required assumptions on the design matrix or link function, and no reference to the specific theorem establishing the result. Without these details the load-bearing claim cannot be verified.
  2. [§4] §4 (Gibbs sampler): the claim of a 'fully conjugate' hierarchical model for arbitrary exponential-family GLMs is central, but the text supplies neither the explicit full-conditional derivations nor the form of the prior on the inclusion indicators that would be needed to confirm conjugacy holds beyond the Gaussian case.
minor comments (2)
  1. The abstract states that the method 'provides a valid post-model Bayesian selection procedure,' but the manuscript does not clarify how the joint posterior is used for selection versus the usual marginal inclusion probabilities.
  2. Table and figure captions should explicitly state the GLM family, link function, and sample size used in each experiment so that the synthetic-data results can be reproduced from the description alone.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive comments on our manuscript. We address each of the major comments below, indicating the revisions we will make to improve clarity and completeness.

read point-by-point responses

  1. Referee: [§3] §3 (theoretical results): the abstract asserts posterior consistency for both inclusion indicators and active coefficients, yet the provided text contains no proof sketch, no statement of the required assumptions on the design matrix or link function, and no reference to the specific theorem establishing the result. Without these details the load-bearing claim cannot be verified.

    Authors: The referee is correct that the current manuscript version does not include a proof sketch or explicit assumptions in the main text. We will revise the manuscript to include a dedicated subsection in §3 with a proof sketch, a clear statement of the assumptions (including conditions on the design matrix such as bounded eigenvalues and on the link function for the exponential family), and a reference to the theorem number. This will allow verification of the posterior consistency claims for the inclusion indicators and active coefficients. revision: yes

  2. Referee: [§4] §4 (Gibbs sampler): the claim of a 'fully conjugate' hierarchical model for arbitrary exponential-family GLMs is central, but the text supplies neither the explicit full-conditional derivations nor the form of the prior on the inclusion indicators that would be needed to confirm conjugacy holds beyond the Gaussian case.

    Authors: We agree that explicit derivations are necessary to substantiate the conjugacy claim for general GLMs. In the revision, we will add an appendix containing the full-conditional distributions for all parameters, including the derivation steps, and specify the prior on the inclusion indicators (a product of independent Bernoulli distributions with success probability depending on a hyperparameter). This will demonstrate how conjugacy is achieved for arbitrary exponential-family distributions via the hierarchical structure. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The abstract and provided material present a new hierarchical model using binary inclusion indicators to achieve conjugacy for any exponential-family GLM, with separate posterior consistency results for inclusions and coefficients. No equations or steps are shown that reduce by construction to fitted inputs, self-definitions, or load-bearing self-citations; the consistency claims are stated as independent theoretical guarantees rather than re-expressions of the model inputs. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review yields no explicit free parameters, invented entities, or detailed axioms beyond the domain assumption of exponential family distributions; full text would be required to audit any additional modeling choices.

axioms (1)
  • domain assumption The response distribution belongs to the exponential family.
    The method is stated to apply to any distribution in the exponential family.

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