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

arXiv: https://arxiv.org/abs/2509.07501 · HTML: https://mlsanity.com/paper/2509.07501/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 38 references that are in the map.
- First posted on arXiv 9 Sep 2025.

## Builds on (most cited first, 11 of 11)

- [Lasso Meets Horseshoe : A Survey](https://mlsanity.com/paper/1706.10179/) (2017)
- [A reduced-rank approach to predicting multiple binary responses through machine learning](https://mlsanity.com/paper/2306.05829/) (2023)
- [A tutorial on individualized treatment effect prediction from randomized trials with a binary endpoint](https://mlsanity.com/paper/2106.05588/) (2021)
- [Concentration of a sparse Bayesian model with Horseshoe prior in estimating high-dimensional precision matrix](https://mlsanity.com/paper/2406.14269/) (2024)
- [From bilinear regression to inductive matrix completion: a quasi-Bayesian analysis](https://mlsanity.com/paper/2210.15290/) (2022)
- [A Pliable Lasso](https://mlsanity.com/paper/1712.00484/) (2017)
- [Tensor Regression](https://mlsanity.com/paper/2308.11419/) (2023)
- [High-dimensional Bayesian Tobit regression for censored response with Horseshoe prior](https://mlsanity.com/paper/2505.08288/) (2025)
- [Handling bounded response in high dimensions: a Horseshoe prior Bayesian Beta regression approach](https://mlsanity.com/paper/2505.22211/) (2025)
- [svReg: Structural Varying-coefficient regression to differentiate how regional brain atrophy affects motor impairment for Huntington disease severity groups](https://mlsanity.com/paper/2007.06076/) (2020)
- [Integrating Multiple Data Sources with Interactions in Multi-Omics Using Cooperative Learning](https://mlsanity.com/paper/2409.07125/) (2024)

## Built on by (most cited first, 1 of 1)

- [Robust low-rank estimation with multiple binary responses using pairwise AUC loss](https://mlsanity.com/paper/2601.08618/) (2026)

## Abstract (arXiv)

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 texttthspliable available on Github.