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On the use of non‐local prior densities in Bayesian hypothesis tests

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  • Valen E. Johnson
  • David Rossell
Abstract
Summary. We examine philosophical problems and sampling deficiencies that are associated with current Bayesian hypothesis testing methodology, paying particular attention to objective Bayes methodology. Because the prior densities that are used to define alternative hypotheses in many Bayesian tests assign non‐negligible probability to regions of the parameter space that are consistent with null hypotheses, resulting tests provide exponential accumulation of evidence in favour of true alternative hypotheses, but only sublinear accumulation of evidence in favour of true null hypotheses. Thus, it is often impossible for such tests to provide strong evidence in favour of a true null hypothesis, even when moderately large sample sizes have been obtained. We review asymptotic convergence rates of Bayes factors in testing precise null hypotheses and propose two new classes of prior densities that ameliorate the imbalance in convergence rates that is inherited by most Bayesian tests. Using members of these classes, we obtain analytic expressions for Bayes factors in linear models and derive approximations to Bayes factors in large sample settings.

Suggested Citation

  • Valen E. Johnson & David Rossell, 2010. "On the use of non‐local prior densities in Bayesian hypothesis tests," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 72(2), pages 143-170, March.
    • Handle: RePEc:bla:jorssb:v:72:y:2010:i:2:p:143-170
    DOI: 10.1111/j.1467-9868.2009.00730.x
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    References listed on IDEAS

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    1. Elías Moreno, 2005. "Objective Bayesian methods for one-sided testing," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 14(1), pages 181-198, June.
    2. Casella, George & Moreno, Elias, 2006. "Objective Bayesian Variable Selection," Journal of the American Statistical Association, American Statistical Association, vol. 101, pages 157-167, March.
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    4. Juan Cano & Mathieu Kessler & Elías Moreno, 2004. "On intrinsic priors for nonnested models," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 13(2), pages 445-463, December.
    5. Liang, Feng & Paulo, Rui & Molina, German & Clyde, Merlise A. & Berger, Jim O., 2008. "Mixtures of g Priors for Bayesian Variable Selection," Journal of the American Statistical Association, American Statistical Association, vol. 103, pages 410-423, March.
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