  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.
