On variance estimation for Bayesian variable selection
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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 argue that the use conjugate continuous shrinkage priors for Bayesian variable selection can have detrimental consequences for such error variance estimation. Instead, we recommend the use of priors which treat the regression coefficients and error variance as independent a priori. We revisit the canonical reference for invariant priors, Jeffreys (1961), and highlight a caveat with their use that Jeffreys himself noted. For the case study of Bayesian ridge regression, we demonstrate that these scale-invariant priors severely underestimate the variance. More generally, we discuss how these priors also interfere with the mechanics of the Bayesian global-local shrinkage framework. With these insights, we extend the Spike-and-Slab Lasso of Rockova and George (2016) to the unknown variance case, using an independent prior for the error variance. Our procedure outperforms both alternative penalized likelihood methods and the fixed variance case on simulated data.
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