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We study probability distributions over free algebras of trees. Probabilitydistributions can be seen as particular (formal power) tree series [Berstel etal 82, Esik et al 03], i.e. mappings from trees to a semiring K . A widelystudied class of tree series is the class of rational (or recognizable) treeseries which can be defined either in an algebraic way or by means ofmultiplicity tree automata. We argue that the algebraic representation is veryconvenient to model probability distributions over a free algebra of trees.First, as in the string case, the algebraic representation allows to designlearning algorithms for the whole class of probability distributions defined byrational tree series. Note that learning algorithms for rational tree seriescorrespond to learning algorithms for weighted tree automata where both thestructure and the weights are learned. Second, the algebraic representation canbe easily extended to deal with unranked trees (like XML trees where a symbolmay have an unbounded number of children). Both properties are particularlyrelevant for applications: nondeterministic automata are required for theinference problem to be relevant (recall that Hidden Markov Models areequivalent to nondeterministic string automata); nowadays applications for WebInformation Extraction, Web Services and document processing consider unrankedtrees.

On Probability Distributions for Trees: Representations, Inference and Learning