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

Abstract

Standard-Nutzungsbedingungen:Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden.Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen.Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. We propose a new method for the objective comparison of two nested models based on non-local priors. More specifically, starting with a default prior under each of the two models, we construct a moment prior under the larger model, and then use the fractional Bayes factor for a comparison. Non-local priors have been recently introduced to obtain a better separation between nested models, thus accelerating the learning behaviour, relative to currently used local priors, when the smaller model holds. Although the argument showing the superior performance of non-local priors is asymptotic, the improvement they produce is already apparent for small to moderate samples sizes, which makes them a useful and practical tool. As a by-product, it turns out that routinely used objective methods, such as ordinary fractional Bayes factors, are alarmingly slow in learning that the smaller model holds. On the downside, when the larger model holds, non-local priors exhibit a weaker discriminatory power against sampling distributions close to the smaller model. However, this drawback becomes rapidly negligible as the sample size grows, because the learning rate of the Bayes factor under the larger model is exponentially fast, whether one uses local or non-local priors. We apply our methodology to directed acyclic graph models having a Gaussian distribution. Because of the recursive nature of the joint density, and the assumption of global parameter independence embodied in our prior, calculations need only be performed for individual vertices admitting a distinct parent structure under the two graphs; additionally we obtain closed-form expressions as in the ordinary conjugate case. We provide illustrations of our method for a simple three-variable case, as well as for a more elaborate seven-variable situation. Although we concentrate on pairwise comparisons of nested models, our procedure can be implemented to carry-out a search over the space of all models. Terms of use: Documents inJune 2010.

Toward a Multisubject Analysis of Neural Connectivity

“…The function m ( G ) in the multiple graphical model prior (Eqn. 4) provides an adjustment for the fact that the size of the space 𝒢 grows super-exponentially with the number P of vertices (Consonni and La Rocca, 2010). In this paper we follow Chen and Chen (2008); Scott and Berger (2010); Foygel and Drton (2013) and control multiplicity using the binomial correction $$mfalse(Gfalse)=\prod _{i=1}^P{true(\begin{array}{c}centerP\\ center|G_i|\end{array}true)}^{-1}false[false|G_{i}false|\le d_{\text{max}}false].$$ Here d max is a fixed upper bound on the in-degree of vertices in G that encodes prior knowledge on the support of the graphical models (e.g.…”

“…Secondly, conventional model selection criteria for graphical models are often biased towards selecting more complex models (i.e. more edges), since there are typically very many models in which the data-generating model is nested; these models are also able to fit the data well (albeit with some coefficients close or equal to zero; Consonni and La Rocca, 2010). Consequently many more data are required to exclude more complex alternatives.…”

Nonlocal Priors for High-Dimensional Estimation

“…Rossell et al (2013) found analogous τ for probit regression, and also considered learning τ either via a hyper-prior or minimizing posterior predictive loss (Gelfand and Ghosh, 1998). Consonni and La Rocca (2010) devised objective Bayes strategies. Yet another possibility is to match the unit information prior e.g.…”

Bayesian Model Selection in High-Dimensional Settings

“…The parameter r is called the order of the density. Consonni and La Rocca (2010) proposed a similar class of prior densities for application to graphical models, though in their proposal the densities corresponding to Equation (2) are not proper.…”