The Annals of Statistics

Consistency of objective Bayes factors as the model dimension grows

Elías Moreno, F. Javier Girón, and George Casella

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In the class of normal regression models with a finite number of regressors, and for a wide class of prior distributions, a Bayesian model selection procedure based on the Bayes factor is consistent [Casella and Moreno J. Amer. Statist. Assoc. 104 (2009) 1261–1271]. However, in models where the number of parameters increases as the sample size increases, properties of the Bayes factor are not totally understood. Here we study consistency of the Bayes factors for nested normal linear models when the number of regressors increases with the sample size. We pay attention to two successful tools for model selection [Schwarz Ann. Statist. 6 (1978) 461–464] approximation to the Bayes factor, and the Bayes factor for intrinsic priors [Berger and Pericchi J. Amer. Statist. Assoc. 91 (1996) 109–122, Moreno, Bertolino and Racugno J. Amer. Statist. Assoc. 93 (1998) 1451–1460].

We find that the the Schwarz approximation and the Bayes factor for intrinsic priors are consistent when the rate of growth of the dimension of the bigger model is O(nb) for b < 1. When b = 1 the Schwarz approximation is always inconsistent under the alternative while the Bayes factor for intrinsic priors is consistent except for a small set of alternative models which is characterized.

Article information

Ann. Statist., Volume 38, Number 4 (2010), 1937-1952.

First available in Project Euclid: 11 July 2010

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62F05: Asymptotic properties of tests
Secondary: 62J15: Paired and multiple comparisons

Bayes factors BIC intrinsic priors linear models multiplicity of parameters rate of growth


Moreno, Elías; Girón, F. Javier; Casella, George. Consistency of objective Bayes factors as the model dimension grows. Ann. Statist. 38 (2010), no. 4, 1937--1952. doi:10.1214/09-AOS754.

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