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Bayesian hypothesis testing: A reference approach

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Summary For any probability model M={p(x|θ, ω), θeΘ, ωeΩ} assumed to describe the probabilistic behaviour of data xeX, it is argued that testing whether or not the available data are compatible with the hypothesis H0={θ=θ0} is best considered as a formal decision problem on whether to use (a0), or not to use (a0), the simpler probability model (or null model) M0={p(x|θ0, ω), ωeΩ}, where the loss difference L(a0, θ, ω) –L(a0, θ, ω) is proportional to the amount of information δ(θ0, ω), which would be lost if the simplified model M0 were used as a proxy for the assumed model M. For any prior distribution π(θ, ω), the appropriate normative solution is obtained by rejecting the null model M0 whenever the corresponding posterior expectation ∫∫δ(θ0, θ, ω)π(θ, ω|x)dθdω is sufficiently large. Specification of a subjective prior is always difficult, and often polemical, in scientific communication. Information theory may be used to specify a prior, the reference prior, which only depends on the assumed model M, and mathematically describes a situation where no prior information is available about the quantity of interest. The reference posterior expectation, d(θ0, x) =∫δπ(δ|x)dδ, of the amount of information δ(θ0, θ, ω) which could be lost if the null model were used, provides an attractive nonnegative test function, the intrinsic statistic, which is invariant under reparametrization. The intrinsic statistic d(θ0, x) is measured in units of information, and it is easily calibrated (for any sample size and any dimensionality) in terms of some average log-likelihood ratios. The corresponding Bayes decision rule, the Bayesian reference criterion (BRC), indicates that the null model M0 should only be rejected if the posterior expected loss of information from using the simplified model M0 is too large or, equivalently, if the associated expected average log-likelihood ratio is large enough. The BRC criterion provides a general reference Bayesian solution to hypothesis testing which does not assume a probability mass concentrated on M0 and, hence, it is immune to Lindley's paradox. The theory is illustrated within the context of multivariate normal data, where it is shown to avoid Rao's paradox on the inconsistency between univariate and multivariate frequentist hypothesis testing. Resume Pour un modele probabiliste M={p(x|θ, ω) θeΘ, ωeΩ} cense decrire le comportement probabiliste de donnees xeX, nous soutenons que tester si les donnees sont compatibles avec une hypothese H0={θ=θ0 doit etre considere comme un probleme decisionnel concernant l'usage du modele M0={p(x|θ0, ω) ωeΩ}, avec une fonction de cout qui mesure la quantite d'information qui peut etre perdue si le modele simplifieM0 est utilise comme approximation du veritable modele M. Le cout moyen, calcule par rapport a une loi a priori de reference idoine fournit une statistique de test pertinente, la statistique intrinseque d(θ0, x), invariante par reparametrisation. La statistique intrinseque d(θ0, x) est mesuree en unites d'information, et sa calibrage, qui est independante de la taille de lechantillon et de la dimension du parametre, ne depend pas de sa distribution a l'echantillonage. La regle de Bayes correspondante, le critere de Bayes de reference (BRC), indique que H0 doit seulement eetre rejete si le cout a posteriori moyen de la perte d'information a utiliser le modele simplifieM0 est trop grande. Le critere BRC fournit une solution bayesienne generale et objective pour les tests d'hypotheses precises qui ne reclame pas une masse de Dirac concentree sur M0. Par consequent, elle echappe au paradoxe de Lindley. Cette theorie est illustree dans le contexte de variables normales multivariees, et on montre qu'elle evite le paradoxe de Rao sur l'inconsistence existant entre tests univaries et multivaries.