Bayesian inference with dependent normalized completely random measures
Bayesian inference with dependent normalized completely random measures
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具有相关归一化完全随机测量的贝叶斯推理
DOI：
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发表时间：
2014
期刊：
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通讯作者：
Igor Prunster
中科院分区：
文献类型：
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作者：
Antonio Lijoi;Bernardo Nipoti;Igor Prunster
相关先验过程的提出和研究一直是近年来贝叶斯非参数文献的主要研究热点。本文引入了一类灵活的非参数相关先验，研究了它们的性质，并推导了一种适合的抽样方案，使它们能够具体实现。所提出的类是通过规范化依赖的完全随机测度获得的，其中依赖是通过完全随机测度基础上的泊松随机测度的适当构造而产生的。我们首先为整个依赖的完全随机度量类提供一般的分布结果，然后我们将它们专一化为两个特定的先验，它们代表了具体实现的自然候选，因为它们的分析可追踪性：二元Dirichlet和归一化$\sigma$稳定过程。我们的分析结果，特别是部分可交换的分割概率函数，也构成了确定用于绘制后向推理的马尔可夫链蒙特卡罗算法的基础，该算法在单变量情况下可简化为著名的Blackwell—MacQueen P\ {o}lya urn格式。该算法可用于密度估计和分析数据的聚类结构，并通过一个真实的双样本数据集示例进行了说明。
The proposal and study of dependent prior processes has been a major research focus in the recent Bayesian nonparametric literature. In this paper, we introduce a flexible class of dependent nonparametric priors, investigate their properties and derive a suitable sampling scheme which allows their concrete implementation. The proposed class is obtained by normalizing dependent completely random measures, where the dependence arises by virtue of a suitable construction of the Poisson random measures underlying the completely random measures. We first provide general distributional results for the whole class of dependent completely random measures and then we specialize them to two specific priors, which represent the natural candidates for concrete implementation due to their analytic tractability: the bivariate Dirichlet and normalized $\sigma$-stable processes. Our analytical results, and in particular the partially exchangeable partition probability function, form also the basis for the determination of a Markov Chain Monte Carlo algorithm for drawing posterior inferences, which reduces to the well-known Blackwell--MacQueen P\'{o}lya urn scheme in the univariate case. Such an algorithm can be used for density estimation and for analyzing the clustering structure of the data and is illustrated through a real two-sample dataset example.