Skip to main content
SpringerLink
Log in

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

  • Original Paper
  • Published:
TEST Aims and scope Submit manuscript

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  • Bernardo JM, Rueda R (2002) Bayesian hypothesis testing: a reference approach. Int Stat Rev 70:351–372

    Article  MATH  Google Scholar 

  • Bernardo JM, Smith AFM (1994) Bayesian theory. Wiley, Chichester

    Book  MATH  Google Scholar 

  • Casella G, Moreno E (2006) Objective Bayesian variable selection. J Am Stat Assoc 101:157–167

    Article  MATH  MathSciNet  Google Scholar 

  • 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. http://economia.unipv.it/~gconsonni/www/papers/CGV_TEST-TechRep.pdf

  • 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. http://www.stat.cmu.edu/ISBA/index.html

  • Dupuis JA, Robert CP (2003) Variable selection in qualitative models via an entropic explanatory power. J Stat Plan Inference 111:77–94

    Article  MATH  MathSciNet  Google Scholar 

  • Emigh TH (1980) A comparison of tests for Hardy–Weinberg equilibrium. Biometrics 36:627–642

    Article  MATH  MathSciNet  Google Scholar 

  • Goutis C, Robert CP (1998) Model choice in generalised linear models: a Bayesian approach via Kullback–Leibler projections. Biometrika 85:29–37

    Article  MATH  MathSciNet  Google Scholar 

  • Gutiérrez-Peña E, Smith AFM (1995) Conjugate parameterizations for natural exponential families. J Am Stat Assoc 90:1347–1356

    Article  MATH  Google Scholar 

  • Haldane JBS (1954) An exact test for randomness of mating. J Genet 52:631–635

    Google Scholar 

  • Hernández JL, Weir BS (1989) A disequilibrium coefficient approach to Hardy–Weinberg testing. Biometrics 45:53–70

    Article  MATH  MathSciNet  Google Scholar 

  • Ibrahim JG (1997) On properties of predictive priors in linear models. Am Stat 51:333–337

    Article  MathSciNet  Google Scholar 

  • Kotz S, Balakrishnan N, Johnson NL (2000) Continuous multivariate distributions. Models and Applications, vol 1, 2nd edn. Wiley, New York

    MATH  Google Scholar 

  • Kass RE, Raftery AE (1995) Bayes factors. J Am Stat Assoc 90:773–795

    Article  MATH  Google Scholar 

  • 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 

  • McCulloch RE, Rossi PE (1992) Bayes factor for nonlinear hypotheses and likelihood distributions. Biometrika 79:663–676

    Article  MATH  MathSciNet  Google Scholar 

  • Morris CN (1982) Natural exponential families with quadratic variance functions. Ann Stat 10:65–80

    Article  MATH  Google Scholar 

  • Roverato A, Consonni G (2004) Compatible prior distributions for DAG models. J Roy Stat Soc B 66:47–61

    Article  MATH  MathSciNet  Google Scholar 

  • Shoemaker J, Painter I, Weir SB (1998) A Bayesian characterization of Hardy–Weinberg disequilibrium. Genetics 149:2079–2088

    Google Scholar 

  • Spiegelhalter DJ, Smith AFM (1980) Bayes factor and choice criteria for linear models. J Roy Stat Soc B 42:215–220

    MathSciNet  Google Scholar 

  • Wald A (1949) Note on the consistency of the maximum likelihood estimate. Ann Math Stat 20:595–601

    Article  MATH  MathSciNet  Google Scholar 

  • Weir BS (1996) Genetic data analysis. Sinuer, Sunderland

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Università di Pavia, Pavia, Italy

    Guido Consonni

  2. IIMAS, Universidad Nacional Autónoma de México, Apartado Postal 20-726, 01000, Mexico D.F., Mexico

    Eduardo Gutiérrez-Peña

  3. Università L. Bocconi, Milano, Italy

    Piero Veronese

Authors
  1. Guido Consonni
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Eduardo Gutiérrez-Peña
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Piero Veronese
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Eduardo Gutiérrez-Peña.

Additional information

G.C.’s research was partially supported by MIUR, Rome, (PRIN 2003138887) and the University of Pavia. P.V.’s research was partially supported by MIUR, Rome, (PRIN 2003138887) and by L. Bocconi University. The research of E.G. was partially supported by CONACyT (Grant 32256-E) and SNI, Mexico. The authors also benefited from partial support through Project No. 4 of the Bilateral Executive Programme for Technological and Scientific Co-operation Between Italy and Mexico 2003–2005.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Consonni, G., Gutiérrez-Peña, E. & Veronese, P. Compatible priors for Bayesian model comparison with an application to the Hardy–Weinberg equilibrium model. TEST 17, 585–605 (2008). https://doi.org/10.1007/s11749-007-0057-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11749-007-0057-7

Keywords

Mathematics Subject Classification (2000)

Search

Navigation