Infinite mixtures of trees

Finite mixtures of tree-structured distributions have been shown to be efficient and effective in modeling multivariate distributions. Using Dirichlet processes, we extend this approach to allow countably many tree-structured mixture components. The resulting Bayesian framework allows us to deal with the problem of selecting the number of mixture components by computing the posterior distribution over the number of components and integrating out the components by Bayesian model averaging. We apply the proposed framework to identify the number and the properties of predominant precipitation patterns in historical archives of climate data.

[1]  Gregory F. Cooper,et al.  A Bayesian Method for the Induction of Probabilistic Networks from Data , 1992 .

[2]  D. Rubin,et al.  Maximum likelihood from incomplete data via the EM - algorithm plus discussions on the paper , 1977 .

[3]  C. N. Liu,et al.  Approximating discrete probability distributions with dependence trees , 1968, IEEE Trans. Inf. Theory.

[4]  David Maxwell Chickering,et al.  Learning Bayesian Networks is NP-Complete , 2016, AISTATS.

[5]  R. K. Shyamasundar,et al.  Introduction to algorithms , 1996 .

[6]  Tommi S. Jaakkola,et al.  Tractable Bayesian learning of tree belief networks , 2000, Stat. Comput..

[7]  Andrei Z. Broder,et al.  Generating random spanning trees , 1989, 30th Annual Symposium on Foundations of Computer Science.

[8]  T. Ferguson.  A Bayesian Analysis of Some Nonparametric Problems , 1973 .

[9]  Jirí Grim.  On structural approximating multivariate discrete probability distributions , 1984, Kybernetika.

[10]  M. West,et al.  Hyperparameter estimation in Dirichlet process mixture models , 1992 .

[11]  Michael I. Jordan,et al.  Learning with Mixtures of Trees , 2001, J. Mach. Learn. Res..

[12]  D. Blackwell,et al.  Ferguson Distributions Via Polya Urn Schemes , 1973 .

[13]  Charles J. Colbourn,et al.  Two Algorithms for Unranking Arborescences , 1996, J. Algorithms.

[14]  Nathan Srebro,et al.  Maximum likelihood bounded tree-width Markov networks , 2001, Artif. Intell..

[15]  C. Antoniak.  Mixtures of Dirichlet Processes with Applications to Bayesian Nonparametric Problems , 1974 .

[16]  Michael I. Jordan,et al.  Hierarchical Dirichlet Processes , 2006 .

[17]  Wai Lam,et al.  LEARNING BAYESIAN BELIEF NETWORKS: AN APPROACH BASED ON THE MDL PRINCIPLE , 1994, Comput. Intell..

[18]  M. Escobar,et al.  Markov Chain Sampling Methods for Dirichlet Process Mixture Models , 2000 .