# Mean-field Variational Bayes for Sparse Probit Regression

arXiv: https://arxiv.org/abs/2601.21765 · HTML: https://mlsanity.com/paper/2601.21765/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 8 of its 20 references that are in the map.
- First posted on arXiv 29 Jan 2026.

## Builds on (most cited first, 8 of 8)

- [Variational Inference: A Review for Statisticians](https://mlsanity.com/paper/1601.00670/) (2016)
- [Conjugate Bayes for probit regression via unified skew-normal distributions](https://mlsanity.com/paper/1802.09565/) (2018)
- [Scalable and Accurate Variational Bayes for High-Dimensional Binary Regression Models](https://mlsanity.com/paper/1911.06743/) (2019)
- [Leave Pima Indians alone: binary regression as a benchmark for Bayesian computation](https://mlsanity.com/paper/1506.08640/) (2015)
- [Bayesian conjugacy in probit, tobit, multinomial probit and extensions: A review and new results](https://mlsanity.com/paper/2206.08118/) (2022)
- [Ultimate P'olya Gamma Samplers -- Efficient MCMC for possibly imbalanced binary and categorical data](https://mlsanity.com/paper/2011.06898/) (2020)
- [Scalable Expectation Propagation for Mixed-Effects Regression](https://mlsanity.com/paper/2409.14646/) (2024)
- [Scalable Variable Selection and Model Averaging for Latent Regression Models Using Approximate Variational Bayes](https://mlsanity.com/paper/2509.11751/) (2025)

## Abstract (arXiv)

We consider Bayesian variable selection for binary outcomes under a probit link with a spike-and-slab prior on the regression coefficients. Motivated by the computational challenges encountered by Markov chain Monte Carlo (MCMC) samplers in high-dimensional regimes, we develop a mean-field variational Bayes approximation in which all variational factors admit closed-form updates, and the evidence lower bound is available in closed form. This, in turn, allows the development of an efficient coordinate ascent variational inference algorithm to find the optimal values of the variational parameters. The approach produces posterior inclusion probabilities and parameter estimates, enabling interpretable selection and prediction within a single framework. As shown in both simulated and real data applications, the proposed method successfully identifies the important variables and is orders of magnitude faster than MCMC, while maintaining comparable accuracy.