# High-dimensional Bayesian Tobit regression for censored response with Horseshoe prior

arXiv: https://arxiv.org/abs/2505.08288 · HTML: https://mlsanity.com/paper/2505.08288/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 61 references that are in the map.
- First posted on arXiv 13 May 2025.

## Builds on (most cited first, 15 of 15)

- [A General Framework for Updating Belief Distributions](https://mlsanity.com/paper/1306.6430/) (2013)
- [Concentration of tempered posteriors and of their variational approximations](https://mlsanity.com/paper/1706.09293/) (2017)
- [Sparsity information and regularization in the horseshoe and other shrinkage priors](https://mlsanity.com/paper/1707.01694/) (2017)
- [Bayesian linear regression with sparse priors](https://mlsanity.com/paper/1403.0735/) (2014)
- [abess: A Fast Best Subset Selection Library in Python and R](https://mlsanity.com/paper/2110.09697/) (2021)
- [Empirical Bayes posterior concentration in sparse high-dimensional linear models](https://mlsanity.com/paper/1406.7718/) (2014)
- [High-dimensional sparse classification using exponential weighting with empirical hinge loss](https://mlsanity.com/paper/2312.12952/) (2023)
- [Bayesian conjugacy in probit, tobit, multinomial probit and extensions: A review and new results](https://mlsanity.com/paper/2206.08118/) (2022)
- [Concentration of a sparse Bayesian model with Horseshoe prior in estimating high-dimensional precision matrix](https://mlsanity.com/paper/2406.14269/) (2024)
- [Model selection and minimax estimation in generalized linear models](https://mlsanity.com/paper/1409.8491/) (2014)
- [Concentration properties of fractional posterior in 1-bit matrix completion](https://mlsanity.com/paper/2404.08969/) (2024)
- [Empirical priors for prediction in sparse high-dimensional linear regression](https://mlsanity.com/paper/1903.00961/) (2019)
- [Bayesian Endogenous Tobit Quantile Regression](https://mlsanity.com/paper/1505.07541/) (2015)
- [High-dimensional Censored Regression via the Penalized Tobit Likelihood](https://mlsanity.com/paper/2203.02601/) (2022)
- [Adaptive posterior concentration rates for sparse high-dimensional linear regression with random design and unknown error variance](https://mlsanity.com/paper/2405.19016/) (2024)

## Built on by (most cited first, 4 of 4)

- [Handling bounded response in high dimensions: a Horseshoe prior Bayesian Beta regression approach](https://mlsanity.com/paper/2505.22211/) (2025)
- [Heavy Lasso: sparse penalized regression under heavy-tailed noise via data-augmented soft-thresholding](https://mlsanity.com/paper/2506.07790/) (2025)
- [Bayesian Pliable Lasso with Horseshoe Prior for Interaction Effects in GLMs with Missing Responses](https://mlsanity.com/paper/2509.07501/) (2025)
- [Mixture-of-experts Wishart model for covariance matrices with an application to Cancer drug screening](https://mlsanity.com/paper/2602.13888/) (2026)

## Abstract (arXiv)

Censored response variables--where outcomes are only partially observed due to known bounds--arise in numerous scientific domains and present serious challenges for regression analysis. The Tobit model, a classical solution for handling left-censoring, has been widely used in economics and beyond. However, with the increasing prevalence of high-dimensional data, where the number of covariates exceeds the sample size, traditional Tobit methods become inadequate. While frequentist approaches for high-dimensional Tobit regression have recently been developed, notably through Lasso-based estimators, the Bayesian literature remains sparse and lacks theoretical guarantees. In this work, we propose a novel Bayesian framework for high-dimensional Tobit regression that addresses both censoring and sparsity. Our method leverages the Horseshoe prior to induce shrinkage and employs a data augmentation strategy to facilitate efficient posterior computation via Gibbs sampling. We establish posterior consistency and derive concentration rates under sparsity, providing the first theoretical results for Bayesian Tobit models in high dimensions. Numerical experiments show that our approach outperforms favorably with the recent Lasso-Tobit method. Our method is implemented in the R package tobitbayes, which can be found on Github.