非参数隐马尔可夫模型与独立同分布模型中的聚类风险
Clustering risk in Non-parametric Hidden Markov and I.I.D. Models
AI总结：
本文深入分析隐马尔可夫与独立同分布模型中的聚类贝叶斯风险，揭示其与分类风险的关系，证明贝叶斯分类器在聚类中近似最优，并利用HMM可辨识性建立非参数设定下聚类超额风险的界限。
摘要：
我们对隐马尔可夫模型和独立同分布模型背景下的聚类贝叶斯风险进行了深入分析。在这两种设定中，我们识别了聚类风险与分类贝叶斯风险相当的情形，以及其最小化器（即贝叶斯聚类器）可由贝叶斯分类器推导出的情形。虽然我们证明了基于贝叶斯分类器的聚类并不总是与最优贝叶斯聚类器一致，但我们也表明这种差异主要是理论上的，贝叶斯分类器在聚类中仍接近最优。一个关键量应运而生，它捕捉了分类和聚类任务的基本难度。此外，通过利用隐马尔可夫模型的可辨识性，我们在一般非参数设定下建立了插件贝叶斯分类器聚类超额风险的界限，为其在实践中的广泛使用提供了理论依据。模拟进一步说明了我们的发现。
英文摘要：
We conduct an in-depth analysis of the Bayes risk of clustering in the context of Hidden Markov and i.i.d. models. In both settings, we identify the situations where this risk is comparable to the Bayes risk of classification and those where its minimizer, the Bayes clusterer, can be derived from the Bayes classifier. While we demonstrate that clustering based on the Bayes classifier does not always match the optimal Bayes clusterer, we show that this difference is primarily theoretical and that the Bayes classifier remains nearly optimal for clustering. A key quantity emerges, capturing the fundamental difficulty of both classification and clustering tasks. Furthermore, by leveraging the identifiability of HMMs, we establish bounds on the clustering excess risk of a plug-in Bayes classifier in the general nonparametric setting, offering theoretical justification for its widespread use in practice. Simulations further illustrate our findings.
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