, Volume 17, Issue 3, pp 585–605 | Cite as

Compatible priors for Bayesian model comparison with an application to the Hardy–Weinberg equilibrium model

  • Guido Consonni
  • Eduardo Gutiérrez-PeñaEmail author
  • Piero Veronese
Original Paper


Suppose we entertain Bayesian inference under a collection of models. This requires assigning a corresponding collection of prior distributions, one for each model’s parameter space. In this paper we address the issue of relating priors across models, and provide both a conceptual and a pragmatic justification for this task. Specifically, we consider the notion of “compatible” priors across models, and discuss and compare several strategies to construct such distributions. To explicate the issues involved, we refer to a specific problem, namely, testing the Hardy–Weinberg Equilibrium model, for which we provide a detailed analysis using Bayes factors.


Bayes factor Conjugate prior Dirichlet distribution Jeffreys conditioning Kullback–Leibler projection Nested model Predictive distribution Trinomial distribution 

Mathematics Subject Classification (2000)

62F15 62P10 92D25 


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  1. Bernardo JM, Rueda R (2002) Bayesian hypothesis testing: a reference approach. Int Stat Rev 70:351–372 zbMATHCrossRefGoogle Scholar
  2. Bernardo JM, Smith AFM (1994) Bayesian theory. Wiley, Chichester zbMATHCrossRefGoogle Scholar
  3. Casella G, Moreno E (2006) Objective Bayesian variable selection. J Am Stat Assoc 101:157–167 zbMATHCrossRefMathSciNetGoogle Scholar
  4. Consonni G, Gutiérrez-Peña E, Veronese P (2005) Compatible priors for Bayesian model comparison with an application to the Hardy–Weinberg equilibrium model. Technical Report, Dipartimento di Economia Politica e Metodi Quantitativi, University of Pavia.
  5. Dawid AP, Lauritzen SL (2001) Compatible prior distributions. In: George E (ed) Bayesian methods with applications to science, policy and official statistics. Monographs of official statistics Office for official publications of the European Communities, Luxembourg, pp 109–118.
  6. Dupuis JA, Robert CP (2003) Variable selection in qualitative models via an entropic explanatory power. J Stat Plan Inference 111:77–94 zbMATHCrossRefMathSciNetGoogle Scholar
  7. Emigh TH (1980) A comparison of tests for Hardy–Weinberg equilibrium. Biometrics 36:627–642 zbMATHCrossRefMathSciNetGoogle Scholar
  8. Goutis C, Robert CP (1998) Model choice in generalised linear models: a Bayesian approach via Kullback–Leibler projections. Biometrika 85:29–37 zbMATHCrossRefMathSciNetGoogle Scholar
  9. Gutiérrez-Peña E, Smith AFM (1995) Conjugate parameterizations for natural exponential families. J Am Stat Assoc 90:1347–1356 zbMATHCrossRefGoogle Scholar
  10. Haldane JBS (1954) An exact test for randomness of mating. J Genet 52:631–635 Google Scholar
  11. Hernández JL, Weir BS (1989) A disequilibrium coefficient approach to Hardy–Weinberg testing. Biometrics 45:53–70 zbMATHCrossRefMathSciNetGoogle Scholar
  12. Ibrahim JG (1997) On properties of predictive priors in linear models. Am Stat 51:333–337 CrossRefMathSciNetGoogle Scholar
  13. Kotz S, Balakrishnan N, Johnson NL (2000) Continuous multivariate distributions. Models and Applications, vol 1, 2nd edn. Wiley, New York zbMATHGoogle Scholar
  14. Kass RE, Raftery AE (1995) Bayes factors. J Am Stat Assoc 90:773–795 zbMATHCrossRefGoogle Scholar
  15. Lindley DV (1988) Statistical inference concerning Hardy–Weinberg equilibrium. In: Bernardo JM, DeGroot MH, Lindley DV, Smith. AFM (eds) Bayesian statistics 3. University Press, Oxford, pp 307–326 Google Scholar
  16. McCulloch RE, Rossi PE (1992) Bayes factor for nonlinear hypotheses and likelihood distributions. Biometrika 79:663–676 zbMATHCrossRefMathSciNetGoogle Scholar
  17. Morris CN (1982) Natural exponential families with quadratic variance functions. Ann Stat 10:65–80 zbMATHCrossRefGoogle Scholar
  18. Roverato A, Consonni G (2004) Compatible prior distributions for DAG models. J Roy Stat Soc B 66:47–61 zbMATHCrossRefMathSciNetGoogle Scholar
  19. Shoemaker J, Painter I, Weir SB (1998) A Bayesian characterization of Hardy–Weinberg disequilibrium. Genetics 149:2079–2088 Google Scholar
  20. Spiegelhalter DJ, Smith AFM (1980) Bayes factor and choice criteria for linear models. J Roy Stat Soc B 42:215–220 MathSciNetGoogle Scholar
  21. Wald A (1949) Note on the consistency of the maximum likelihood estimate. Ann Math Stat 20:595–601 zbMATHCrossRefMathSciNetGoogle Scholar
  22. Weir BS (1996) Genetic data analysis. Sinuer, Sunderland Google Scholar

Copyright information

© Sociedad de Estadística e Investigación Operativa 2007

Authors and Affiliations

  • Guido Consonni
    • 1
  • Eduardo Gutiérrez-Peña
    • 2
    Email author
  • Piero Veronese
    • 3
  1. 1.Università di PaviaPaviaItaly
  2. 2.IIMAS, Universidad Nacional Autónoma de MéxicoMexico D.F.Mexico
  3. 3.Università L. BocconiMilanoItaly

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