To improve the predictability of complex computational models in the experimentally-unknown domains, we propose a Bayesian statistical machine learning framework utilizing the Dirichlet distribution that combines results of several imperfect models. This framework can be viewed as an extension of Bayesian stacking. To illustrate the method, we study the ability of Bayesian model averaging and mixing techniques to mine nuclear masses. We show that the global and local mixtures of models reach excellent performance on both prediction accuracy and uncertainty quantification and are preferable to classical Bayesian model averaging. Additionally, our statistical analysis indicates that improving model predictions through mixing rather than mixing of corrected models leads to more robust extrapolations.
We propose an exact slice sampler for Hierarchical Dirichlet process (HDP) and its associated mixture models (Teh et al., 2006). Although there are existing MCMC algorithms for sampling from the HDP, a slice sampler has been missing from the literature. Slice sampling is well-known for its desirable properties including its fast mixing and its natural potential for parallelization. On the other hand, the hierarchical nature of HDPs poses challenges to adopting a full-fledged slice sampler that automatically truncates all the infinite measures involved without ad-hoc modifications. In this work…
Copula-based methods provide a flexible approach to build missing data imputation models of multivariate data of mixed types. However, the choice of copula function is an open question. We consider a Bayesian nonparametric approach by using an infinite mixture of elliptical copulas induced by a Dirichlet process mixture to build a flexible copula function. A slice sampling algorithm is used to sample from the infinite dimensional parameter space. We extend the work on prior parallel tempering used in finite mixture models to the Dirichlet process mixture model to overcome the mixing issue in m…
In this work, we develop a novel Bayesian estimation method for the Dirichlet process (DP) mixture of the inverted Dirichlet distributions, which has been shown to be very flexible for modeling vectors with positive elements. The recently proposed extended variational inference (EVI) framework is adopted to derive an analytically tractable solution. The convergency of the proposed algorithm is theoretically guaranteed by introducing single lower bound approximation to the original objective function in the VI framework. In principle, the proposed model can be viewed as an infinite inverted Dir…
When working with multimodal Bayesian posterior distributions, Markov chain Monte Carlo (MCMC) algorithms have difficulty moving between modes, and default variational or mode-based approximate inferences will understate posterior uncertainty. And, even if the most important modes can be found, it is difficult to evaluate their relative weights in the posterior. Here we propose an approach using parallel runs of MCMC, variational, or mode-based inference to hit as many modes or separated regions as possible and then combine these using Bayesian stacking, a scalable method for constructing a we…
Posterior concentration rates for empirical Bayes procedures, with applications to Dirichlet Process mixtures
在本文中，我们提供了一般的条件来检查模型和先验数据依赖先验（或经验贝叶斯方法）的后验集中率。我们的目标是提供与Ghosal和van der Vaart（2007 a）的开创性论文中提供的条件接近的条件。然后，我们将一般定理应用到两个不同的设置：使用Dirichlet过程混合高斯随机变量的密度估计与基础措施取决于一些经验量和估计的强度下的Aalen模型的计数过程。非齐次泊松过程的模拟研究也说明了我们的结果。在前一种情况下，我们也得到一些结果的估计的混合密度和反卷积问题。在后者中，我们提供了一个一般定理后的浓度率计数过程与Aalen乘法强度与先验不依赖于数据。
We develop a flexible Erlang mixture model for survival analysis. The model for the survival density is built from a structured mixture of Erlang densities, mixing on the integer shape parameter with a common scale parameter. The mixture weights are constructed through increments of a distribution function on the positive real line, which is assigned a Dirichlet process prior. The model has a relatively simple structure, balancing flexibility with efficient posterior computation. Moreover, it implies a mixture representation for the hazard function that involves time-dependent mixture weights,…
Bayesian hierarchical models are used to share information between related samples and obtain more accurate estimates of sample-level parameters, common structure, and variation between samples. When the parameter of interest is the distribution or density of a continuous variable, a hierarchical model for continuous distributions is required. A number of such models have been described in the literature using extensions of the Dirichlet process and related processes, typically as a distribution on the parameters of a mixing kernel. We propose a new hierarchical model based on the Pólya tree, …
Mixtures of product distributions are a powerful device for learning about heterogeneity within data populations. In this class of latent structure models, de Finetti's mixing measure plays the central role for describing the uncertainty about the latent parameters representing heterogeneity. In this paper posterior contraction theorems for de Finetti's mixing measure arising from finite mixtures of product distributions will be established, under the setting the number of exchangeable sequences of observed variables increases while sequence length(s) may be either fixed or varied. The role of…
Bayesian Mixture Models for Heterogeneous Extremes
Unité Biostatistique et Processus Spatiaux, INRAE, France（生物统计与空间过程单元，法国国家农业食品与环境研究院）；School of Mathematics, University of Edinburgh, UK（爱丁堡大学数学学院）；Department of Mathematics, University of Aveiro, Portugal（阿威罗大学数学系）；Department of Environmental Sciences, Informatics and Statistics, Ca’Foscari University of Venice, Italy（威尼斯卡福斯卡里大学环境科学、信息学与统计学系）
We study the reknown deconvolution problem of recovering a distribution function from independent replicates (signal) additively contaminated with random errors (noise), whose distribution is known. We investigate whether a Bayesian nonparametric approach for modelling the latent distribution of the signal can yield inferences with asymptotic frequentist validity under the $L^1$-Wasserstein metric. When the error density is ordinary smooth, we develop two inversion inequalities relating either the $L^1$ or the $L^1$-Wasserstein distance between two mixture densities (of the observations) to th…
We study Bayesian estimation of finite mixture models in a general setup where the number of components is unknown and allowed to grow with the sample size. An assumption on growing number of components is a natural one as the degree of heterogeneity present in the sample can grow and new components can arise as sample size increases, allowing full flexibility in modeling the complexity of data. This however will lead to a high-dimensional model which poses great challenges for estimation. We novelly employ the idea of a sample size dependent prior in a Bayesian model and establish a number of…
