🤖 AI Summary
This paper investigates the fundamental relationship between clustering risk and classification risk under nonparametric hidden Markov models (HMMs) and i.i.d. settings. Using Bayesian decision theory and nonparametric statistical analysis, we establish—for the first time in a nonparametric HMM framework—an upper bound on the clustering excess risk of a plug-in Bayesian classifier. We further introduce a unified information-theoretic quantity that characterizes the intrinsic difficulty of both tasks. Theoretically, we prove that, under broad regularity conditions, the Bayesian classifier is nearly optimal for clustering: its clustering performance loss is both controlled and universally bounded. Extensive simulations confirm the tightness of the derived bounds and the practical efficacy of the approach. Our core contribution lies in revealing an intrinsic equivalence between classification and clustering risks, thereby providing a theoretically grounded, supervised-style solution to unsupervised clustering.
📝 Abstract
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.
