• Bernoulli
  • Volume 22, Number 4 (2016), 2080-2100.

Consistency of Bayes factor for nonnested model selection when the model dimension grows

Min Wang and Yuzo Maruyama

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


Zellner’s $g$-prior is a popular prior choice for the model selection problems in the context of normal regression models. Wang and Sun [J. Statist. Plann. Inference 147 (2014) 95–105] recently adopt this prior and put a special hyper-prior for $g$, which results in a closed-form expression of Bayes factor for nested linear model comparisons. They have shown that under very general conditions, the Bayes factor is consistent when two competing models are of order $O(n^{\tau})$ for $\tau <1$ and for $\tau=1$ is almost consistent except a small inconsistency region around the null hypothesis. In this paper, we study Bayes factor consistency for nonnested linear models with a growing number of parameters. Some of the proposed results generalize the ones of the Bayes factor for the case of nested linear models. Specifically, we compare the asymptotic behaviors between the proposed Bayes factor and the intrinsic Bayes factor in the literature.

Article information

Bernoulli, Volume 22, Number 4 (2016), 2080-2100.

Received: December 2014
First available in Project Euclid: 3 May 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Bayes factor growing number of parameters model selection consistency nonnested linear models Zellner’s $g$-prior


Wang, Min; Maruyama, Yuzo. Consistency of Bayes factor for nonnested model selection when the model dimension grows. Bernoulli 22 (2016), no. 4, 2080--2100. doi:10.3150/15-BEJ720.

Export citation


  • [1] Bayarri, M.J., Berger, J.O., Forte, A. and García-Donato, G. (2012). Criteria for Bayesian model choice with application to variable selection. Ann. Statist. 40 1550–1577.
  • [2] Berger, J.O., Ghosh, J.K. and Mukhopadhyay, N. (2003). Approximations and consistency of Bayes factors as model dimension grows. J. Statist. Plann. Inference 112 241–258.
  • [3] Casella, G., Girón, F.J., Martínez, M.L. and Moreno, E. (2009). Consistency of Bayesian procedures for variable selection. Ann. Statist. 37 1207–1228.
  • [4] Cox, D.R. (1962). Further results on tests of separate families of hypotheses. J. Roy. Statist. Soc. Ser. B 24 406–424.
  • [5] Fernández, C., Ley, E. and Steel, M.F.J. (2001). Benchmark priors for Bayesian model averaging. J. Econometrics 100 381–427.
  • [6] Fujikoshi, Y. (1993). Two-way ANOVA models with unbalanced data. Discrete Math. 116 315–334.
  • [7] Girón, F.J., Martínez, M.L., Moreno, E. and Torres, F. (2006). Objective testing procedures in linear models: Calibration of the $p$-values. Scand. J. Stat. 33 765–784.
  • [8] Girón, F.J., Moreno, E., Casella, G. and Martínez, M.L. (2010). Consistency of objective Bayes factors for nonnested linear models and increasing model dimension. Rev. R. Acad. Cienc. Exactas FíS. Nat. Ser. A Math. RACSAM 104 57–67.
  • [9] Gustafson, P., Hossain, S. and MacNab, Y.C. (2006). Conservative prior distributions for variance parameters in hierarchical models. Canad. J. Statist. 34 377–390.
  • [10] Hoel, P.G. (1947). On the choice of forecasting formulas. J. Amer. Statist. Assoc. 42 605–611.
  • [11] Kass, R.E. and Raftery, A.E. (1995). Bayes factors. J. Amer. Statist. Assoc. 90 773–795.
  • [12] Kass, R.E. and Vaidyanathan, S.K. (1992). Approximate Bayes factors and orthogonal parameters, with application to testing equality of two binomial proportions. J. Roy. Statist. Soc. Ser. B 54 129–144.
  • [13] Ley, E. and Steel, M.F.J. (2012). Mixtures of $g$-priors for Bayesian model averaging with economic applications. J. Econometrics 171 251–266.
  • [14] Liang, F., Paulo, R., Molina, G., Clyde, M.A. and Berger, J.O. (2008). Mixtures of $g$ priors for Bayesian variable selection. J. Amer. Statist. Assoc. 103 410–423.
  • [15] Maruyama, Y. (2013). A Bayes factor with reasonable model selection consistency for ANOVA model. Available at arXiv:0906.4329v2 [stat.ME].
  • [16] Maruyama, Y. and George, E.I. (2011). Fully Bayes factors with a generalized $g$-prior. Ann. Statist. 39 2740–2765.
  • [17] Moreno, E. and Girón, F.J. (2008). Comparison of Bayesian objective procedures for variable selection in linear regression. TEST 17 472–490.
  • [18] Moreno, E., Girón, F.J. and Casella, G. (2010). Consistency of objective Bayes factors as the model dimension grows. Ann. Statist. 38 1937–1952.
  • [19] Moreno, E., Girón, F.J. and Casella, G. (2014). Posterior model consistency in variable selection as the model dimension grows. Preprint.
  • [20] Pesaran, M.H. and Weeks, M. (1999). Non-nested hypothesis testing: An overview. Cambridge Working Papers in Economics 9918, Faculty of Economics, Univ. of Cambridge.
  • [21] Wang, M. and Sun, X. (2013). Bayes factor consistency for unbalanced ANOVA models. Statistics 47 1104–1115.
  • [22] Wang, M. and Sun, X. (2014). Bayes factor consistency for nested linear models with a growing number of parameters. J. Statist. Plann. Inference 147 95–105.
  • [23] Wang, M., Sun, X. and Lu, T. (2015). Bayesian structured variable selection in linear regression models. Comput. Statist. 30 205–229.
  • [24] Watnik, M., Johnson, W. and Bedrick, E.J. (2001). Nonnested linear model selection revisited. Comm. Statist. Theory Methods 30 1–20.
  • [25] Watnik, M.R. and Johnson, W.O. (2002). The behaviour of linear model selection tests under globally non-nested hypotheses. Sankhyā Ser. A 64 109–138.
  • [26] Wetzels, R., Grasman, R.P.P.P. and Wagenmakers, E.-J. (2012). A default Bayesian hypothesis test for ANOVA designs. Amer. Statist. 66 104–111.
  • [27] Zellner, A. (1986). On assessing prior distributions and Bayesian regression analysis with $g$-prior distributions. In Bayesian Inference and Decision Techniques. Stud. Bayesian Econometrics Statist. 6 233–243. Amsterdam: North-Holland.