Consistency of Bayesian linear model selection with a growing number of parameters

Zuofeng Shang, Murray K. Clayton

Research output: Contribution to journalArticlepeer-review

19 Scopus citations


Linear models with a growing number of parameters have been widely used in modern statistics. One important problem about this kind of model is the variable selection issue. Bayesian approaches, which provide a stochastic search of informative variables, have gained popularity. In this paper, we will study the asymptotic properties related to Bayesian model selection when the model dimension p is growing with the sample size n. We consider p≤n and provide sufficient conditions under which: (1) with large probability, the posterior probability of the true model (from which samples are drawn) uniformly dominates the posterior probability of any incorrect models; and (2) the posterior probability of the true model converges to one in probability. Both (1) and (2) guarantee that the true model will be selected under a Bayesian framework. We also demonstrate several situations when (1) holds but (2) fails, which illustrates the difference between these two properties. Finally, we generalize our results to include g-priors, and provide simulation examples to illustrate the main results.

Original languageEnglish (US)
Pages (from-to)3463-3474
Number of pages12
JournalJournal of Statistical Planning and Inference
Issue number11
StatePublished - Nov 2011
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Statistics and Probability
  • Statistics, Probability and Uncertainty
  • Applied Mathematics


  • Bayesian model selection
  • Consistency of Bayes factor
  • Consistency of posterior odds ratio
  • G-priors
  • Gibbs sampling
  • Growing number of parameters
  • Posterior model consistency


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