On Probability Distributions for Trees: Representations, Inference and Learning

TL;DR

This paper explores probability distributions over tree structures using algebraic representations, enabling new learning algorithms and extensions to unranked trees relevant for modern web and document applications.

Contribution

It introduces an algebraic framework for modeling and learning probability distributions over trees, including unranked trees, with applications to automata-based inference.

Findings

Algebraic representation facilitates learning algorithms for rational tree series.

Extension to unranked trees applicable to XML and web data.

Supports nondeterministic automata for inference tasks.

Abstract

We study probability distributions over free algebras of trees. Probability distributions can be seen as particular (formal power) tree series [Berstel et al 82, Esik et al 03], i.e. mappings from trees to a semiring K . A widely studied class of tree series is the class of rational (or recognizable) tree series which can be defined either in an algebraic way or by means of multiplicity tree automata. We argue that the algebraic representation is very convenient to model probability distributions over a free algebra of trees. First, as in the string case, the algebraic representation allows to design learning algorithms for the whole class of probability distributions defined by rational tree series. Note that learning algorithms for rational tree series correspond to learning algorithms for weighted tree automata where both the structure and the weights are learned. Second, the algebraic representation can be easily extended to deal with unranked trees (like XML trees where a symbol may have an unbounded number of children). Both properties are particularly relevant for applications: nondeterministic automata are required for the inference problem to be relevant (recall that Hidden Markov Models are equivalent to nondeterministic string automata); nowadays applications for Web Information Extraction, Web Services and document processing consider unranked trees.