Comparing an HMM and an SCFG

1 Baskin Center for Computer S ien e and Engineering, University of California, Santa Cruz, CA 95064, U.S.A. E-mail: {jagota,rlyngsoe} se.u s .edu 2 BRICS⋆ ⋆ ⋆, Department of Computer S ien e, University of Aarhus, Ny Munkegade, DK-8000 Århus C, Denmark. E-mail: storm bri s.dk. Abstra t Sto hasti models are ommonly used in bioinformati s, e.g. hidden Markov models for modeling sequen e families or sto hasti ontext-free grammars for modeling RNA se ondary stru ture formation. Comparing data is a ommon task in bioinformati s, and it is thus natural to onsider how to ompare sto hasti models. In this paper we present the rst study of the problem of omparing a hidden Markov model and a sto hasti ontext-free grammar. We des ribe how to ompute their o-emission or ollision probability, i.e. the probability that they independently generate the same sequen e. We also onsider the related problem of nding a run through a hidden Markov model and derivation in a grammar that generate the same sequen e and have maximal joint probability by a generalization of the CYK algorithm for parsing a sequen e by a sto hasti ontext-free grammar. We illustrate the methods by an experiment on RNA se ondary stru tures. 1 Introdu tion The basi hain-like stru ture of the key biomole ules, DNA, RNA, and proteins, allows an abstra t view of these as strings, or sequen es, over nite alphabets, obviously of nite length. Furthermore, these sequen es are not ompletely random, but exhibit various kinds of stru tures in di erent ontexts. E.g. a family of homologous proteins is likely to have similar amino a id residues in equivalent positions; an RNA sequen e will have pairs of omplementary subsequen es to form base pairing heli es. Hen e, it is natural to onsider applying models from formal language theory to model di erent lasses of biologi al sequen es. Though not ompletely random, biologi al sequen es an still possess inherent sto hasti traits, e.g. due to mutations in a family of homologous sequen es or a la k of knowledge (and omputing power) to orre tly model all aspe ts of RNA se ondary stru ture formation. Thus, it is often better to use sto hasti models giving a probability distribution over all sequen es, where a high