Posterior contractions rates (PCRs) strengthen the notion of Bayesian consistency, quantifying the speed at which the posterior distribution concentrates on arbitrarily small neighborhoods of the true model, with probability tending to 1 or almost surely, as the sample size goes to infinity. Under the Bayesian nonparametric framework, a common assumption in the study of PCRs is that the model is dominated for the observations; that is, it is assumed that the posterior can be written through the Bayes formula. In this paper, we consider the problem of establishing PCRs in Bayesian nonparametric models where the posterior distribution is not available through the Bayes formula, and hence models that are non-dominated for the observations. By means of the Wasserstein distance and a suitable sieve construction, our main result establishes PCRs in Bayesian nonparametric models where the posterior is available through a more general disintegration than the Bayes formula. To the best of our knowledge, this is the first general approach to provide PCRs in non-dominated Bayesian nonparametric models, and it relies on minimal modeling assumptions and on a suitable continuity assumption for the posterior distribution. Some refinements of our result are presented under additional assumptions on the prior distribution, and applications are given with respect to the Dirichlet process prior and the normalized extended Gamma process prior.
翻译:在巴伊西亚非参数缩缩率(PCRs)研究中,一个共同的假设是,该模型在观察中占主导地位;也就是说,假设后表可以通过贝耶斯公式书写;在本文中,我们考虑了在巴伊西亚非参数模型中建立后表分配速度的问题,因为后表分配无法通过巴伊斯公式获得,因此,随着抽样规模的无限性,可能倾向于1个或几乎肯定的模型。在巴伊西亚非参数缩缩缩缩缩率框架下,在巴伊西亚的非参数模型中,后表分配速度集中到任意的小社区,随着抽样规模的无限化而变化。在巴伊斯的非参数中,后表分布速度可能很快,因此,随着瓦塞斯坦距离和合适的筛选结构的构建,我们的主要结果就是,在巴伊亚非参数的模型中,后表可以通过比巴伊斯公式更为普遍的解体模式书写成。 据我们所知,这是第一个在巴伊西亚非参数分配后非参数模型中提供PCRR(PRCRs)的非参数模型,因此,因此,在观察中并不以主控模式为基础进行观察模型的模型的模型的模型的模型前分配,在先前的模型的扩大后,而采用后,根据以前的模型的模型的假设,其前推推后推推推后推后推后推后推后推后推后推后推后推后推后推后推后推后推后推后推后推后推后推后推后推后推后推。