# On the Beta Prime Prior for Scale Parameters in High-Dimensional Bayesian Regression Models

arXiv: https://arxiv.org/abs/1807.06539 · HTML: https://mlsanity.com/paper/1807.06539/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 17 Jul 2018.

## Built on by (most cited first, 2 of 2)

- [A spectral mixture representation of isotropic kernels with application to random Fourier features](https://mlsanity.com/paper/2411.02770/) (2024)
- [GRASP: Grouped Regression with Adaptive Shrinkage Priors](https://mlsanity.com/paper/2506.18092/) (2025)

## Abstract (arXiv)

We study high-dimensional Bayesian linear regression with a general beta prime distribution for the scale parameter. Under the assumption of sparsity, we show that appropriate selection of the hyperparameters in the beta prime prior leads to the (near) minimax posterior contraction rate when p gg n. For finite samples, we propose a data-adaptive method for estimating the hyperparameters based on marginal maximum likelihood (MML). This enables our prior to adapt to both sparse and dense settings, and under our proposed empirical Bayes procedure, the MML estimates are never at risk of collapsing to zero. We derive efficient Monte Carlo EM and variational EM algorithms for implementing our model, which are available in the R package NormalBetaPrime. Simulations and analysis of a gene expression data set illustrate our model's self-adaptivity to varying levels of sparsity and signal strengths.