Variance prior forms for high-dimensional Bayesian variable selection

TL;DR

This paper investigates the impact of conjugate priors on Bayesian high-dimensional variable selection and proposes an independent prior approach for better variance estimation, improving predictive accuracy.

Contribution

It highlights the drawbacks of conjugate priors for variance estimation and extends the Spike-and-Slab Lasso to incorporate unknown variance with improved performance.

Findings

Conjugate priors can underestimate variance in high-dimensional models.

Independent priors for coefficients and variance improve estimation accuracy.

Extended Spike-and-Slab Lasso outperforms fixed variance and other methods on real data.

Abstract

Consider the problem of high dimensional variable selection for the Gaussian linear model when the unknown error variance is also of interest. In this paper, we show that the use of conjugate shrinkage priors for Bayesian variable selection can have detrimental consequences for such variance estimation. Such priors are often motivated by the invariance argument of Jeffreys (1961). Revisiting this work, however, we highlight a caveat that Jeffreys himself noticed; namely that biased estimators can result from inducing dependence between parameters a priori. In a similar way, we show that conjugate priors for linear regression, which induce prior dependence, can lead to such underestimation in the Bayesian high-dimensional regression setting. Following Jeffreys, we recommend as a remedy to treat regression coefficients and the error variance as independent a priori. Using such an independence prior framework, we extend the Spike-and-Slab Lasso of Rockova and George (2018) to the unknown variance case. This extended procedure outperforms both the fixed variance approach and alternative penalized likelihood methods on simulated data. On the protein activity dataset of Clyde and Parmigiani (1998), the Spike-and-Slab Lasso with unknown variance achieves lower cross-validation error than alternative penalized likelihood methods, demonstrating the gains in predictive accuracy afforded by simultaneous error variance estimation.