Convergence of latent mixing measures in finite and infinite mixture models
Convergence of latent mixing measures in finite and infinite mixture models
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DOI：
10.1214/12-aos1065
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发表时间：
2011-09
影响因子：
4.5
通讯作者：
X. Nguyen
中科院分区：
文献类型：
--
作者：
X. Nguyen
本文使用传输距离（即 Wasserstein 度量）研究有限和无限混合模型中出现的潜在混合度量的收敛行为。使用各种可辨识条件详细研究了混合测度空间上的 Wasserstein 距离与混合分布空间上的 f 散度泛函（例如 Hellinger 和 Kullback-Leibler 距离）之间的关系。离散测量的 Wasserstein 度量中的收敛意味着为测量提供支持的单个原子的收敛，从而在通常采用混合模型的聚类应用中提供簇收敛的自然解释。对于多元分布的有限混合和基于狄利克雷过程的无限混合，建立了潜在混合测度的后验分布的收敛率。
This paper studies convergence behavior of latent mixing measures that arise in finite and infinite mixture models, using transportation distances (i.e., Wasserstein metrics). The relationship between Wasserstein distances on the space of mixing measures and f-divergence functionals such as Hellinger and Kullback-Leibler distances on the space of mixture distributions is investigated in detail using various identifiability conditions. Convergence in Wasserstein metrics for discrete measures implies convergence of individual atoms that provide support for the measures, thereby providing a natural interpretation of convergence of clusters in clustering applications where mixture models are typically employed. Convergence rates of posterior distributions for latent mixing measures are established, for both finite mixtures of multivariate distributions and infinite mixtures based on the Dirichlet process.