On the use of Non-Local Prior Densities in Bayesian Hypothesis Tests

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.

Moment Priors for Bayesian Model Choice with Applications to Directed Acyclic Graphs*

“…It is clear from Dawid's proof that, by forcing the prior density under M 1 to vanish on Θ 0 , one can speed up the decrease of BF 10 (y (n) ) when M 0 holds. This is indeed the approach taken by Johnson and Rossell (2010) when defining non-local priors. We focus here on a specific family of non-local priors.…”

“…Essentially, the asymptotic learning rate is exponential when the larger model holds, while it behaves only as a power of the sample size when the smaller model is assumed to be true. To countervail this phenomenon, Johnson and Rossell (2010) recently suggested that priors for nested model comparison should be non-local (thus vanishing on the null) and showed that such priors can be effectively constructed (in particular as moment priors). The main advantages of non-local priors can be summarized under two headings: from a descriptive viewpoint, they embody a notion of separation between the larger and the smaller model; from an inferential perspective, they produce an accelerated learning behaviour when the smaller model holds.…”

Bayesian Evaluation of Informative Hypotheses

“…This is a common property of the Bayes factor caused by the fact that it is easier to find support against H 0 instead of finding support for H 0 because H 0 is a precise hypothesis while H 1 is a composite hypothesis. Interested readers are referred to Johnson and Rossell (2010) who proposed a method where the evidence for the true hypothesis accumulates with the same rate under H 0 and H 1 . Table 4.4: Choices of b, g, r and error probabilities under two distributions of standardized effect θ and a sample size of n = 100.…”

Reply to Gelman, Gaudart, Pericchi: More reasons to revise standards for statistical evidence

“…Because evidence against false null hypotheses can be accumulated exponentially fast (8)(9)(10), it is likely that any decision-theoretic analyses of optimal evidence thresholds would lead to thresholds substantially greater than 5. Thus, even if it were possible to routinely conduct the analyses suggested in ref.…”