Improving the Convergence Properties of the Data Augmentation Algorithm with an Application to Bayesian Mixture Modeling
Improving the Convergence Properties of the Data Augmentation Algorithm with an Application to Bayesian Mixture Modeling
复制标题
应用贝叶斯混合模型提高数据增强算法的收敛性
DOI：
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发表时间：
2009
期刊：
影响因子：
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通讯作者：
C. Robert
中科院分区：
文献类型：
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作者：
J. Hobert;Vivekananda Roy;C. Robert
驱动数据增强 (DA) 和三明治算法的可逆马尔可夫链定义了自伴算子，其谱编码了算法的收敛特性。当目标分布具有不可数的支持时（实践中几乎总是如此），通常很难掌握这些谱。我们证明，如果增广空间是有限的，那么（在正则条件下）DA 和三明治链定义的算子是紧凑的，并且谱是 $[0,1)$ 的有限子集。此外，我们证明了夹心算子的谱支配了DA算子的谱，因为前者的有序元素都小于或等于后者的相应元素。作为一个具体的例子，我们研究了一种广泛使用的 DA 算法，用于探索与贝叶斯混合模型相关的后验密度 [J.罗伊.国家主义者。苏克。序列。 B 56（1994）363--375]。特别是，我们将这种混合 DA 算法与 Fr\"{u}hwirth-Schnatter [J. Amer. Statist. Assoc. 96 (2001) 194--209] 提出的基于随机标签切换的替代算法进行了比较。
The reversible Markov chains that drive the data augmentation (DA) and sandwich algorithms define self-adjoint operators whose spectra encode the convergence properties of the algorithms. When the target distribution has uncountable support, as is nearly always the case in practice, it is generally quite difficult to get a handle on these spectra. We show that, if the augmentation space is finite, then (under regularity conditions) the operators defined by the DA and sandwich chains are compact, and the spectra are finite subsets of $[0,1)$. Moreover, we prove that the spectrum of the sandwich operator dominates the spectrum of the DA operator in the sense that the ordered elements of the former are all less than or equal to the corresponding elements of the latter. As a concrete example, we study a widely used DA algorithm for the exploration of posterior densities associated with Bayesian mixture models [J. Roy. Statist. Soc. Ser. B 56 (1994) 363--375]. In particular, we compare this mixture DA algorithm with an alternative algorithm proposed by Fr\"{u}hwirth-Schnatter [J. Amer. Statist. Assoc. 96 (2001) 194--209] that is based on random label switching.