Inference with normal-gamma prior distributions in regression problems. (English) Zbl 1330.62128

Griffin, Jim E.; Brown, Philip J.

Bayesian Anal. 5, No. 1, 171-188 (2010).

Summary: This paper considers the effects of placing an absolutely continuous prior distribution on the regression coefficients of a linear model. We show that the posterior expectation is a matrix-shrunken version of the least squares estimate where the shrinkage matrix depends on the derivatives of the prior predictive density of the least squares estimate. The special case of the normal-gamma prior, which generalizes the Bayesian Lasso [T. Park and G. Casella, J. Am. Stat. Assoc. 103, No. 482, 681–686 (2008; Zbl 1330.62292)], is studied in depth. We discuss the prior interpretation and the posterior effects of hyperparameter choice and suggest a data-dependent default prior. Simulations and a chemometric example are used to compare the performance of the normal-gamma and the Bayesian Lasso in terms of out-of-sample predictive performance.

MSC:

62F15	Bayesian inference
62J07	Ridge regression; shrinkage estimators (Lasso)
60J22	Computational methods in Markov chains
65C60	Computational problems in statistics (MSC2010)
65C05	Monte Carlo methods

Keywords:

multiple regression; \(p>n\); normal-gamma prior; “spike-and-slab” prior; Bayesian Lasso; posterior moments; shrinkage; scale mixture of normals; Markov chain Monte Carlo

Citations:

Zbl 1330.62292