Default Bayes Factors for Nonnested Hypothesis Testing

Equality and inequality constrained multivariate linear models: Objective model selection using constrained posterior priors

“…For a thorough discussion of the Bayes factor, see, for instance, Kass and Raftery (1995). The (in)equality constrained models discussed in the previous section are all nested in the unconstrained model M 0 , which does not assume any constraints on the model parameters H. For this reason, we can specify a so-called encompassing prior p 0 for H under the unconstrained model M 0 (Berger and Mortera, 1999;Klugkist and Hoijtink, 2007).…”

“…In Fig. 1 (dashed line), the geometric IBF based on these so-called constrained posterior priors, which are inspired by the symmetrical intrinsic priors in Berger and Mortera (1999), are displayed for the Example 2 discussed above. The figure shows that the least complex model M 1 is preferred over the more complex model M 0 with a factor two when the fit of M 1 and M 0 is approximately equal.…”

“…In another example by Berger and Mortera (1999) (their Example 1), the following models were under evaluation default Bayes factors and was symmetrical around m ¼ 0. Furthermore, the outcomes of these Bayes factors were satisfactory in the sense that the Bayes factor went to infinity in favor of the correct model as n-1.…”

Alternative Bayes factors: Sample size determination and discriminatory power assessment

“…This is considered a strong justification for the use of these alternative Bayes factors. For this topic, see, for instance, Berger and Pericchi (1996), Dmochowski (1996), De Santis and Spezzaferri (1997), Moreno et al (1998), Berger and Mortera (1999). In the following, we will conventionally refer to alternative Bayes factors defined with either proper or improper priors as robust Bayes factors and default Bayes factors, respectively.…”

Bayesian model selection of informative hypotheses for repeated measurements

“…In order to use the Bayes factor as selection criterion between a set of equality and inequality constrained models, a prior distribution must be specified for the parameters under each model of interest. When the constrained models under evaluation are nested in a larger encompassing model, Berger and Mortera (1999) specified a prior under this encompassing model from which truncated distributions were constructed under the constrained models. Independently, the idea of an encompassing prior was successfully applied by Klugkist, Laudy, and Hoijtink (2005b) in the context of analysis of (co)variance.…”