This text is based on a set of not es produced for courses given for gradu­ ate students in mathematics, computer science and biochemistry during the academic year 1998-1999 at the University of Turku in Turku and at the Royal Institute of Technology (KTH) in Stockholm. The course in Turku was organized by Professor Mats Gyllenberg's groupl and was also included 2 within the postgraduate program ComBi , a Graduate School in Compu­ tational Biology, Bioinformatics, and Biometry, directed by Professor Esko Ukkonen at the University of Helsinki. The purpose of the courses was to give a thorough and systematic intro duc ti on to probabilistic modelling in bioinformatics for advanced undergraduate and graduate students who had a fairly limited background in prob ability theory, but were otherwise well trained in mathematics and were already familiar with at least some of the techniques of algorithmic sequence analysis. Portions of the material have also been lectured at shorter graduate courses and seminars both in Finland and in Sweden. The initial set of notes circulated also for a time outside those two countries via the World Wide Web. The intermediate course in probability theory and techniques of discrete mathematics held by the author at the University College of Södertörn (Hud­ dinge, Sweden) during the academic year 1997-1998 has also influenced the presentation. The opportunity to give this course is hereby gratefully ac­ knowledged.

1 Prerequisites in probability calculus.- 2 Information and the Kullback Distance.- 3 Probabilistic Models and Learning.- 4 EM Algorithm.- 5 Alignment and Scoring.- 6 Mixture Models and Profiles.- 7 Markov Chains.- 8 Learning of Markov Chains.- 9 Markovian Models for DNA sequences.- 10 Hidden Markov Models an Overview.- 11 HMM for DNA Sequences.- 12 Left to Right HMM for Sequences.- 13 Derin's Algorithm.- 14 Forward-Backward Algorithm.- 15 Baum-Welch Learning Algorithm.- 16 Limit Points of Baum-Welch.- 17 Asymptotics of Learning.- 18 Full Probabilistic HMM.