Permutree Process

Masahiro Nakano, Daiki Chijiwa, Ryohei Shibue, Yasuhiro Fujiwara, Ryo Nishikimi, Tomoharu Iwata, Akisato Kimura, Takeshi Yamada, Naonori Ueda

Keywords: Bayesian nonparametrics, Permutree, Marked point process

TL;DR: Stochastic process on permutrees, which can represent permutations, trees, partitions, and binary sequences in a unified manner

Abstract:

The permutree is an innovative mathematical concept recently introduced in the field of combinatorics. It encompasses permutations, trees, partitions, and binary sequences as its special cases, while also allowing for interpolations between them. In this paper, we present the permutree notion within the context of Bayesian machine learning. We exploit the fact that permutrees have a one-to-one correspondence with special permutations to propose a stochastic process on permutrees, and further propose two data modeling strategies analogous to the stick-breaking process and Chinese restaurant process that are frequently used in Bayesian nonparametrics.
Permutations, trees, partitions, and binary sequences frequently appear as building blocks in Bayesian nonparametric models, and these models have been studied and developed independently. However, in practical situations, there are many complicated problems that require master craftsmanship to combine these individual models into a single giant model. Our models provide a framework for modeling such complicated tasks in a unified manner. As a significant application, we demonstrate the potential of our models for phylogenetic analysis of lineages, which involve coalescence, recombination, multiple ancestors, and mutation.

Primary Area: Probabilistic methods (for example: variational inference, Gaussian processes)

Submission Number: 17176