Bayesian Goodness of Fit Testing with Mixtures of Triangular Distributions

Abstract

ABSTRACT. We consider the consistency of the Bayes factor in goodness of fit testing for a parametric family of densities against a nonparametric alternative. Sufficient conditions for consistency of the Bayes factor are determined and demonstrated with priors using certain mixtures of triangular densities.Running headline: Bayesian Goodness of Fit Testing

A note on Bayes factor consistency in partial linear models

“…Here, we investigate the consistency of the Bayes factor in the partial linear model, based on asymptotic results studied by McVinish et al (2009) for goodness-of-fit testing problems and the convergence rates of posterior distributions studied by Ghosal and van der Vaart (2007a) for non-i.i.d. observations.…”

“…In these regards, there have been several advances in Bayesian goodness-of-fit testing from the nonparametric Bayesian point of view. Readers are referred to Verdinelli and Wasserman (1998), Berger and Guglielmi (2001), Lenk (2003), Dass and Lee (2004), Walker et al (2004), Ghosal et al (2008), McVinish et al (2009), andTokdar et al (2010). These results on Bayes factor consistency in density estimation problems are based on verifying sufficient conditions related to posterior consistency and posterior convergence rates.…”

“…Dass andLee, 2004 andWalker et al, 2004). However, when two competing models have the Kullback-Leibler property, additional conditions are required for consistent model selection; Ghosal et al (2008) and McVinish et al (2009) investigated this issue by establishing sufficient conditions for the consistency of the Bayes factor under i.i.d. observations.…”

Sequential Monte Carlo methods for mixtures with normalized random measures with independent increments priors

“…Consistency issue are considered by Dass and Lee (2004). More recently, McVinish et al (2009) have proposed a method using mixtures of triangular distributions and considered its consistency (Fig. 2).…”

On some aspects of the asymptotic properties of Bayesian approaches in nonparametric and semiparametric models

“…The test procedures associated to the Bayes factor are thus said to be consistent if B 0/1 converges in probability to infinity under P n θ0 for all θ 0 ∈ Θ 0 and if it converges in probability to 0 under P n θ0 for all θ 0 ∈ Θ 1 . Goodness-of-fit tests have been studied in terms of their asymptotic properties among others by [Dass and Lee, 2006, R. McVinish, 2009, Rousseau, 2007, Rousseau and Choi, 2012, see also . When both hypotheses are nonparametric but one is still embedded in the other, the determination of Bayesian test procedures having good frequentist properties is quite difficult in general.…”