Pith. sign in

REVIEW 5 major objections 7 minor 29 references

Bayesian Pliable Lasso with Horseshoe Prior for Interaction Effects in GLMs with Missing Responses

T0 review · 5 major / 7 minor · reviewed 2026-08-04 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read A Bayesian pliable lasso sharing one horseshoe local scale between each main effect and its interaction vector enforces heredity and beats the frequentist original in estimation, prediction, and selection.

desk verdict A useful Bayesian pliable lasso with a joint horseshoe prior and a working Gibbs sampler, but the paper's claim that the prior enforces strong heredity is not accurate; the shared scale gives soft coupling only. read the letter →

arxiv 2509.07501 v1 pith:4OQPWCSQ submitted 2025-09-09 stat.ME stat.APstat.COstat.ML

The Tien Mai This is my paper · ORCID
classification stat.MEstat.APstat.COstat.ML MSC 62F1562J0762J12
keywords pliablelassoBayesianinferencehorseshoepriorinteractioneffectssparsitymissingdataGibbssamplergeneralizedlinearmodels
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 sets out to give the pliable lasso — a model where each predictor's effect is a main effect plus an adjustment that varies with modifying covariates — a fully Bayesian treatment. Its device is a horseshoe prior in which each main effect β_j and its modifier-interaction vector θ_j draw on the same local shrinkage scale, so the two shrink or survive together; the author argues this automatically enforces the pliable lasso's heredity rule (an interaction enters only with its main effect), while returning sparse, interpretable fits with a full posterior. Why it would matter: the frequentist pliable lasso returns only point estimates and has no built-in route for partially missing outcomes, whereas this version supplies credible intervals for main and interaction effects, extends to binary and count responses through a generalized linear model formulation, and imputes missing responses inside the sampler. Simulations across six data configurations plus two high-dimensional settings are used to claim the Bayesian version beats the pliable lasso, plain lasso, and plain horseshoe in estimation error, prediction error, and selection accuracy, with the largest gains when the modifying variables are binary. An application to dementia and cognitive-decline data rounds out the case, with the pliable models ahead of the non-interaction baselines on test-set prediction.

What carries the argument

The load-bearing object is the shared local shrinkage parameter λ_j in the horseshoe prior assigned to the block (β_j, θ_j): β_j ~ N(0, λ²_j τ²) with θ_j ~ N(0, λ²_j τ² I_q), and half-Cauchy priors on every λ_j and on the global scale τ. A horseshoe prior is the continuous shrinkage prior that squeezes small coefficients toward zero through the global scale while its heavy-tailed local scales let strong signals escape; sharing λ_j inside each predictor block is what couples main effects to interactions, giving the Bayesian analogue of the pliable lasso's heredity constraint. Carrying the inference is an all-conjugate Gibbs sampler obtained by writing each half-Cauchy as an inverse-gamma mixt

What would settle it

Simulate a truth that violates heredity (a nonzero interaction attached to a zero main effect) alongside a pure-main-effect truth with no interactions, and check whether the shared-scale prior recovers each generating structure; if the orphan interaction is systematically shrunk away, the coupling is imposing structure the data do not support. A second check: run many replications and measure the empirical coverage of the reported 95% credible intervals — coverage far from 95% would directly falsify the uncertainty-quantification claim.

Watch

Extended reading notes

Core claim

The central claim is that a joint horseshoe prior — one global scale τ plus one per-predictor local scale λ_j shared by β_j and the whole vector θ_j — turns the pliable lasso into a Bayesian procedure without sacrificing its structured view of interactions. Because the same λ_j multiplies both the main effect and its modifier adjustments, a predictor that is noise shrinks to near zero at both levels at once, while a genuinely active predictor keeps its main effect together with its interactions; the paper presents this coupling as an automatic enforcement of the strong heredity constraint, in contrast to priors placed separately on β and θ that would let interactions float free of their main

Load-bearing premise

The construction stands or falls on the premise that forcing a main effect and its interactions to share one shrinkage-strength parameter is an adequate stand-in for the rule 'no interaction without its main effect' — yet continuous shrinkage never produces exact zeros, so that rule is only approximated, and if the approximation is not faithful the heredity guarantee does not hold.

Editorial extensions

If this is right

  • Practitioners get a pliable-lasso-style interaction model with credible intervals, so a reported interaction comes with a quantified uncertainty statement instead of a bare point estimate.
  • One conjugate Gibbs cycle covers Gaussian, binary, and count responses plus partially missing outcomes — missing values are imputed from their Gaussian full conditional, under a missing-at-random assumption that the paper itself flags — so no new sampling machinery is needed for these complications.
  • If the simulation comparisons hold, the method's edge over the frequentist pliable lasso is largest when the modifying variables are binary and when p exceeds n, exactly the regimes where the optimization-based original is least stable.
  • When the truth contains no interactions, the pliable horseshoe stays competitive in estimation and prediction while retaining top selection accuracy, so the added interaction flexibility does not appear to cost much when it is unused.
  • On the dementia data, both pliable methods beat the plain lasso and plain horseshoe in test-set prediction (with the frequentist pliable lasso best on this particular dataset), suggesting the interaction structure itself, rather than the estimation scheme, drives the gain.

Reading between the lines

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

  • A consequence the paper leaves implicit: the heredity guarantee it advertises is soft, not literal — a continuous shrinkage prior assigns zero posterior probability to exactly zero coefficients, so a user who needs the strict rule 'no interaction without its main effect' would have to threshold the posterior draws or adopt a constrained spike-and-slab prior.
  • Because the missing-response imputation assumes ignorable missingness, a natural extension the paper does not build is an explicit missingness model for cases where nonresponse depends on the outcome itself; the sampler's Gaussian step would then be replaced by a joint model of outcome and missingness.
  • The posterior draws of θ_j make possible a decision-theoretic use the paper does not develop — reporting posterior probabilities that each interaction is active, or a credible set over interacting predictors — which would convert the selection-accuracy comparisons into a calibrated risk statement.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

5 major / 7 minor

Summary. The paper proposes a Bayesian version of the pliable lasso, replacing the frequentist penalty with a horseshoe prior on main effects β_j and interaction effects θ_j. The prior in Eq. (5) shares a local scale λ_j between β_j and θ_j, and the paper claims that this hierarchy "enforces the heredity constraint" of Eq. (1). The authors develop a Gibbs sampler for the Gaussian linear model, extend it to missing responses via data augmentation, and provide a Polya-Gamma based sampler for logistic regression in an appendix. Simulation studies across six settings (continuous/binary X and Z, correlated/independent predictors, n=200-1000, and two high-dimensional configurations) compare the method against the pliable lasso, lasso, and plain horseshoe, reporting estimation error, prediction error, and variable-selection metrics. A real-data application to the OASIS brain data is included.

Significance. If the central claims are correct, the paper delivers a useful practical tool: a fully Bayesian treatment of the pliable lasso with uncertainty quantification and missing-response support, alongside a ready-to-use R package (hspliable). The Gibbs conditionals in Section 2.3 are algebraically transparent and appear correct for the Gaussian case, which is a concrete strength; the logistic appendix is also coherent. The simulation results are systematic and consistently favorable to the proposed method. However, the paper's key advertised property—that the prior enforces strong heredity—is not actually achieved by a continuous horseshoe prior. This does not necessarily destroy the proposed method (soft coupling can still be useful), but it changes what the paper delivers relative to its abstract and Discussion, and it weakens the interpretability guarantee that motivates the paper. The significance is therefore conditional: the method may be a useful shrinkage-based alternative to the pliable lasso, but it is not a Bayesian analogue that imposes the heredity structure of Eq. (1).

major comments (5)
  1. [Section 2.2, Eq. (5)] The paper repeatedly states that the shared λ_j prior "enforces the heredity constraint" (Eq. (1)) and that the structure "naturally imposes" it (Abstract, Section 2.2, Section 6). This is incorrect for a continuous prior. Conditional on λ_j and τ, β_j and θ_j are independent zero-mean normals; the shared scale does not impose a one-way constraint. Since the horseshoe is continuous, P(β_j=0|y)=P(θ_j=0|y)=0, so the exact logical statement "θ_j nonzero only if β_j nonzero" is satisfied by no posterior draw. What is achieved is soft, symmetric shrinkage coupling. This is a load-bearing mismatch because the paper's motivation and interpretability claims rely on strong heredity. Please rephrase the claims as soft shrinkage coupling, or modify the prior to include a spike/slab or hard constraint (e.g., setting θ_j=0 when β_j is effectively zero) if strong heredity is truly intended.
  2. [Section 4.2, Tables 2-5] The simulation protocol compares the proposed pHS against pLasso, Lasso, and HS. For pHS, the estimate used for β and θ is not fully specified: the text says "where β̂ and θ̂ are the estimate from the considered methods" but does not state whether these are posterior means, medians, or some thresholded quantities. This matters because the horseshoe posterior mean is not sparse (all coefficients are nonzero with probability one), so the reported variable-selection metrics (accuracy, FDR, FPR) must be based on some thresholding or credible-interval rule. The paper does not define that rule. Without this, the FDR/FPR values near 0 and accuracy near 1 in the tables cannot be reproduced or interpreted. Please specify the exact post-processing rule for selection and for point estimates.
  3. [Section 4.2.1, Table 5] In the high-dimensional simulations (n=100, p=120 and n=200, p=250), pHS reports FDR = 0.01 (0.07) and accuracy = 1.00 (0.00) with Est.(β)=0.25 (0.98). The standard deviation of Est.(β) is almost four times the mean, indicating that some simulations produce very large errors, yet the selection metrics remain perfect (accuracy=1.00, FDR≈0). This is suspicious: if a method has perfect variable-selection performance in every replication, its estimation error should be small unless the magnitudes are poorly estimated. A possible explanation is that the selection threshold used is extremely conservative (e.g., only large coefficients count as positives), while the posterior mean still includes small nonzero values for all coefficients. Please clarify the relationship between the point estimate and the selection rule, and report the distribution of Est.(β) (e.g., median, quantiles) to ensure t
  4. [Section 5, Table 8] The real-data section (Section 5) claims "none of these interactions is deemed significant, as the corresponding 90% credible intervals include zero." However, the table only reports credible intervals for the selected covariates (nWBV and ASF for pHS). The table does not provide credible intervals for the interaction effects θ_nWBV and θ_ASF, nor does it report the actual 90% credible intervals for those effects. If the interactions are not significant, this weakens the claim that the method recovers meaningful interaction structures. Please include the credible intervals for all reported interaction effects, or state explicitly if they were omitted for space.
  5. [Section 3 (missing data)] The missing-data extension assumes the missingness mechanism is ignorable (MAR conditional on covariates). The paper states this explicitly in the last paragraph, which is good. However, the simulation study in Section 4.2.2 removes responses completely at random, which is a special case of MCAR. The method's performance under this ideal condition does not validate the method under MAR (e.g., missingness depending on observed covariates but not the response) or MNAR mechanisms, which are more common in practice. The authors should either add simulations with a covariate-dependent missingness mechanism, or temper the claim that the method "effectively handles missing responses" to be specifically about ignorable mechanisms.
minor comments (7)
  1. [Abstract] The phrase "enforces heredity constraints" is repeated from the abstract. As discussed in the major comments, this should be softened to "softly enforces" or "encourages" to avoid overclaiming.
  2. [Section 2.2] Typo: "shringkage" should be "shrinkage" and "hierarchial" should be "hierarchical" in the paragraph describing the horseshoe prior.
  3. [Section 2.3, Step 1] In the definition of the partial residual r_{(-j)} for β_j, the term Z_j θ_j is subtracted, but Z_j is defined as diag(x_j) Z. In the notation, x_j is a column vector; it is clearer to write Z_j = x_j * Z (elementwise product) to avoid confusion with matrix multiplication. Also, in Step 2, r_{(-j)} is redefined omitting x_j β_j; the notation could be clarified by using a different symbol (e.g., r_{(-j),β}) for the two partial residuals.
  4. [Section 4.1] The simulation setup states "We set the number of predictors to p=10 and the number of modifying covariates to q=4." However, for Setting V and VI, the paper does not explicitly state the distribution of X_i when it is binary (only "Ber(0.5)" in Table 1). Please clarify in the text.
  5. [Table 6] The formatting of Table 6 appears corrupted: the columns for Est.(β) and Est.(θ) show values like "0.03 0.02 0.11 0.04" and "1.16 0.22" that are not aligned with the numbers that follow. This likely stems from a LaTeX table misalignment. Please check and reformat.
  6. [Appendix B] In the logistic regression appendix, the notation d is used for the dimension of γ_j (which is 1+q), but earlier in the paper q is used. Please be consistent (e.g., let d = q+1) to avoid confusion.
  7. [References] Reference [4] (Bhadra et al., 2019) is cited as "Lasso meets horseshoe: A survey" but the full title is "Lasso Meets Horseshoe: A Survey". Also, the R package 'svreg' is cited via the GitHub link in the text but not included in the references; if the package has a formal citation, please add it.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the prior, sampler, and evaluations are self-contained; self-citations are not load-bearing.

full rationale

The paper's derivation chain is self-contained. The joint horseshoe prior in Eq. (5) is stated directly as a modeling choice, and the Gibbs conditionals in Section 2.3 are derived from that prior and the Gaussian/GLM likelihood using the standard Makalic--Schmidt auxiliary-variable reparameterization, which is an external, well-known result. Missing-response handling in Section 3 is standard data augmentation: missing values are drawn from their Gaussian full conditional given the current parameters, and no predictive claim is fitted from the missing-data results. The simulations use independently generated training and test sets, and the reported prediction/estimation errors are out-of-sample evaluations of a fixed prior model, not quantities obtained by fitting the model to those targets. The paper cites several works by the same author (refs. [17], [18], [19]) but only as examples of horseshoe prior usage in the literature; none of these citations is used to justify the central pliable-heredity claim or the sampler. The only notable issue is wording: the abstract, Section 2.2, and Section 6 say the hierarchical structure 'enforces' or 'naturally imposes' the strong heredity constraint of Eq. (1), whereas a continuous horseshoe prior assigns positive prior mass to configurations with beta_j = 0 and theta_j nonzero, so the exact logical constraint has zero posterior probability. This is an overstatement of model behavior and a correctness/interpretation concern, not a circularity: the claimed constraint is not an input that is later disguised as an output. No load-bearing step reduces to its own inputs, and no prediction is forced by construction.

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

The central claim rests on the pliable structure, the heredity target, ignorability, and the standard scale-mixture representation. No new physical or probabilistic entities are introduced. The fixed hyperparameters sigma0_sq, a0, b0 and an unspecified ridge term are hand-set inputs.

free parameters (3)
  • sigma0_sq = 1
    Prior variance for the intercept and modifier intercept, fixed by hand as a standard weakly informative choice.
  • a0, b0 = 1e-2
    Shape and rate of the inverse-gamma prior on noise variance, fixed by hand as a vague prior.
  • ridge stabilization term in logistic sampler = not specified
    The appendix states matrix inversions are stabilized with a small ridge term but gives no value, an implementation choice that affects reproducibility.
assumptions (5)
  • domain assumption The linear predictor is exactly eta_i = beta_0 + Z_i^T theta_0 + sum_j x_ij (beta_j + Z_i^T theta_j), so all interactions are of the form X_j times Z_k and no other interactions exist.
    Stated in Section 2.1, Eq. (3). The method inherits the pliable lasso's structural restriction.
  • domain assumption Strong heredity (theta_j nonzero only if beta_j nonzero) is a valid constraint and is the target of the prior.
    Invoked in Eq. (1) and Section 2.2; the paper assumes this is the right regularization, though it is not exactly enforced.
  • domain assumption Missingness is ignorable (missing at random conditional on observed covariates and model parameters).
    Stated in Section 3 remark; needed for the imputation step to target the correct posterior.
  • standard math A half-Cauchy distribution can be represented as a scale mixture of inverse-gammas with auxiliary variables.
    Used in Section 2.3, following Makalic and Schmidt (2016).
  • domain assumption The response distribution belongs to the exponential family with canonical link as in Eq. (2).
    Section 2.1, Eq. (2); necessary for the GLM framing, though detailed sampling is only given for Gaussian and logistic cases.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Bayesian Pliable Lasso with Horseshoe Prior for Interaction Effects in GLMs with Missing Responses." pith.science (2026). https://pith.science/paper/4OQPWCSQ

@misc{pith2026250907501,
  author       = {Pith},
  title        = {Pith review of: Bayesian Pliable Lasso with Horseshoe Prior for Interaction Effects in GLMs with Missing Responses},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4OQPWCSQ}},
  note         = {Machine review of arXiv:2509.07501}
}
read the original abstract

Sparse regression problems, where the goal is to identify a small set of relevant predictors, often require modeling not only main effects but also meaningful interactions through other variables. While the pliable lasso has emerged as a powerful frequentist tool for modeling such interactions under strong heredity constraints, it lacks a natural framework for uncertainty quantification and incorporation of prior knowledge. In this paper, we propose a Bayesian pliable lasso that extends this approach by placing sparsity-inducing priors, such as the horseshoe, on both main and interaction effects. The hierarchical prior structure enforces heredity constraints while adaptively shrinking irrelevant coefficients and allowing important effects to persist. We extend this framework to Generalized Linear Models (GLMs) and develop a tailored approach to handle missing responses. To facilitate posterior inference, we develop an efficient Gibbs sampling algorithm based on a reparameterization of the horseshoe prior. Our Bayesian framework yields sparse, interpretable interaction structures, and principled measures of uncertainty. Through simulations and real-data studies, we demonstrate its advantages over existing methods in recovering complex interaction patterns under both complete and incomplete data. Our method is implemented in the package \texttt{hspliable} available on Github.

Figures

Figures reproduced from arXiv: 2509.07501 by the authors.

Figure 1
Figure 1. Histogram plots for comparison of posterior distributions by parameter for Hoseshoe and pliable Horseshoe methods in Setting I with n = 200, p = 10, q = 4. The true value for β1 is 2, for β2 is −2, and for β5, β6 are 0. where ybi = βb0 + Z ⊤ test,iθb0 + Pp j=1 Xtest,i  βbj + Z ⊤ test,iθbj  is the prediction on testing data for pliable lasso and pliable horseshoe. For lasso and horseshoe method the prediction is wi… view at source ↗
Figure 2
Figure 2. Trace plots and ACF plots for β1, β2, β7, β8 (from left to right) in Setting I with n = 200, p = 10, q = 4. The true values are β1 = 2, β2 = −2, β7 = β8 = 0 [PITH_FULL_IMAGE:figures/full_fig_p019_2.png] view at source ↗
Figure 3
Figure 3. Trace plots and ACF plots for θ1,2; θ2,2; θ3,2; θ4,2 (from left to right) in Setting I with n = 200, p = 10, q = 4. The true values are θ1,2 = 1, θ2,2 = −2, θ3,2 = 2, θ4,2 = 0. B Simulations with binary response We developed a Gibbs sampling algorithm for Bayesian inference in the logistic pliable lasso model with a hierarchical Horseshoe prior. Let yi ∈ {0, 1} denote the binary response for observation i, X ∈ R n×p… view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

29 extracted references · 29 canonical work pages

  1. [1]

    and Rizopoulos, D

    Andrinopoulou, E.-R. and Rizopoulos, D. (2016). Bayesian shrinkage approach for a joint model of longitudinal and survival outcomes assuming different association structures.Statistics in medicine, 35(26):4813–4823. 16

    work page 2016

  2. [2]

    Q., Wang, P., and Zhang, H

    Asenso, T. Q., Wang, P., and Zhang, H. (2024). Pliable lasso for the support vector machine. Communications in Statistics-Simulation and Computation, 53(2):786–798

    work page 2024

  3. [3]

    Q., Zhang, H., and Liang, Y

    Asenso, T. Q., Zhang, H., and Liang, Y. (2022). Pliable lasso for the multinomial logistic regression.Communications in Statistics-Theory and Methods, 51(11):3596–3611

    work page 2022

  4. [4]

    G., and Willard, B

    Bhadra, A., Datta, J., Polson, N. G., and Willard, B. T. (2019). Lasso meets horseshoe: A survey.Statistical Science, 34(3):405–427

    work page 2019

  5. [5]

    (2025).sail: Sparse Additive Interaction Learning

    Bhatnagar, S., Yang, Y., and Greenwood, C. (2025).sail: Sparse Additive Interaction Learning. R package version 0.1.0, commit f5c4b09a396c667aedc4e167b4fca5e65f5c5f51

    work page 2025

  6. [6]

    Bien, J., Taylor, J., and Tibshirani, R. (2013). A lasso for hierarchical interactions.Annals of statistics, 41(3):1111

    work page 2013

  7. [7]

    and Van De Geer, S

    B¨ uhlmann, P. and Van De Geer, S. (2011).Statistics for high-dimensional data: methods, theory and applications. Springer Science & Business Media

    work page 2011

  8. [8]

    M., Polson, N

    Carvalho, C. M., Polson, N. G., and Scott, J. G. (2009). Handling sparsity via the horseshoe. Journal of Machine Learning Research W&CP, 5:73–80

    work page 2009

Show all 29 references
  1. [9]

    Q., and Zucknick, M

    D’Alessandro, M., Asenso, T. Q., and Zucknick, M. (2025). Integrating multiple data sources with interactions in multi-omics using cooperative learning.Statistics in Medicine, 44(13- 14):e70148

    2025

  2. [10]

    and Tibshirani, R

    Du, W. and Tibshirani, R. (2018). A pliable lasso for the cox model.arXiv preprint arXiv:1807.06770

    2018 arXiv

  3. [11]

    Friedman, J., Hastie, T., and Tibshirani, R. (2010). Regularization paths for generalized linear models via coordinate descent.Journal of Statistical Software, 33(1):1–22

    2010

  4. [12]

    (2021).Introduction to high-dimensional statistics

    Giraud, C. (2021).Introduction to high-dimensional statistics. Chapman and Hall/CRC

    2021

  5. [13]

    H., and Friedman, J

    Hastie, T., Tibshirani, R., Friedman, J. H., and Friedman, J. H. (2009).The elements of statistical learning: data mining, inference, and prediction, volume 2. Springer

    2009

  6. [14]

    M., Riley, R

    Hoogland, J., IntHout, J., Belias, M., Rovers, M. M., Riley, R. D., E. Harrell Jr, F., Moons, K. G., Debray, T. P., and Reitsma, J. B. (2021). A tutorial on individualized treatment effect prediction from randomized trials with a binary endpoint.Statistics in medicine, 40(26):...

    2021

  7. [15]

    Kim, R., M¨ uller, S., and Garcia, T. P. (2021). svreg: Structural varying-coefficient regression to differentiate how regional brain atrophy affects motor impairment for huntington disease severity groups.Biometrical Journal, 63(6):1254–1271

    2021

  8. [16]

    and Hastie, T

    Lim, M. and Hastie, T. (2015). Learning interactions via hierarchical group-lasso regularization. Journal of Computational and Graphical Statistics, 24(3):627–654

    2015

  9. [17]

    Mai, T. T. (2024). Concentration of a Sparse Bayesian Model With Horseshoe Prior in Esti- mating High-Dimensional Precision Matrix.Stat, 13(4):e70008. 17

    2024

  10. [18]

    Mai, T. T. (2025a). Handling bounded response in high dimensions: a Horseshoe prior Bayesian Beta regression approach.arXiv preprint arXiv:2505.22211

    arXiv

  11. [19]

    Mai, T. T. (2025b). High-dimensional Bayesian Tobit regression for censored response with Horseshoe prior.arXiv preprint arXiv:2505.08288

    arXiv

  12. [20]

    and Schmidt, D

    Makalic, E. and Schmidt, D. F. (2016). A simple sampler for the horseshoe estimator.IEEE Signal Processing Letters, 23(1):179–182

    2016

  13. [21]

    S., Wang, T

    Marcus, D. S., Wang, T. H., Parker, J., Csernansky, J. G., Morris, J. C., and Buckner, R. L. (2007). Open access series of imaging studies (oasis): cross-sectional mri data in young, middle aged, nondemented, and demented older adults.Journal of cognitive neuroscience, 19(9):1...

    2007

  14. [22]

    and Nelder, J

    McCullagh, P. and Nelder, J. A. (1989).Generalized linear models. Monographs on Statistics and Applied Probability. Chapman & Hall, London, 2nd edition

    1989

  15. [23]

    G., Scott, J

    Polson, N. G., Scott, J. G., and Windle, J. (2013). Bayesian inference for logistic models using p´ olya–gamma latent variables.Journal of the American statistical Association, 108(504):1339– 1349

    2013

  16. [24]

    Sun, N., Chu, J., He, Q., Wang, Y., Han, Q., Yi, N., Zhang, R., and Shen, Y. (2024). Bhaft: Bayesian heredity-constrained accelerated failure time models for detecting gene-environment interactions in survival analysis.Statistics in Medicine, 43(21):4013–4026

    2024

  17. [25]

    and Friedman, J

    Tibshirani, R. and Friedman, J. (2020). A pliable lasso.Journal of Computational and Graph- ical Statistics, 29(1):215–225

    2020

  18. [26]

    (2019).horseshoe: Imple- mentation of the Horseshoe Prior

    van der Pas, S., Scott, J., Chakraborty, A., and Bhattacharya, A. (2019).horseshoe: Imple- mentation of the Horseshoe Prior. R package version 0.2.0

    2019

  19. [27]

    van der Pas, S., Szab´ o, B., and van der Vaart, A. (2017). Adaptive posterior contraction rates for the horseshoe.Electronic Journal of Statistics, 11:3196–3225

    2017

  20. [28]

    Wu, C., Cui, Y., and Ma, S. (2014). Integrative analysis of gene–environment interactions under a multi-response partially linear varying coefficient model.Statistics in medicine, 33(28):4988– 4998

    2014

  21. [29]

    Wu, M., Huang, J., and Ma, S. (2018). Identifying gene-gene interactions using penalized tensor regression.Statistics in medicine, 37(4):598–610. 18 A Additional plots for the simulation Figure 2: Trace plots and ACF plots forβ 1, β2, β7, β8 (from left to right) in Setting I w...

    2018

Pith tools

Reviewed August 4, 2026 · model on record in the stance chip above.