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This book explores important aspects of Markov and hidden Markov processes and the applications of these ideas to various problems in computational biology. The book starts from first principles, so that no previous knowledge of probability is necessary. However, the work is rigorous and mathematical, making it useful to engineers and mathematicians, even those not interested in biological applications. A range of exercises is provided, including drills to familiarize the reader with concepts and more advanced problems that require deep thinking about the theory. Biological applications are taken from post-genomic biology, especially genomics and proteomics. The topics examined include standard material such as the Perron-Frobenius theorem, transient and recurrent states, hitting probabilities and hitting times, maximum likelihood estimation, the Viterbi algorithm, and the Baum-Welch algorithm. The book contains discussions of extremely useful topics not usually seen at the basic level, such as ergodicity of Markov processes, Markov Chain Monte Carlo (MCMC), information theory, and large deviation theory for both i.i.d and Markov processes. The book also presents state-of-the-art realization theory for hidden Markov models. Among biological applications, it offers an in-depth look at the BLAST (Basic Local Alignment Search Technique) algorithm, including a comprehensive explanation of the underlying theory. Other applications such as profile hidden Markov models are also explored.

Computational biology. --- Markov processes. --- BLAST theory. --- BaumЗelch algorithm. --- Bayes' rule. --- Cramr's theorem. --- GENSCAN algorithm. --- GLIMMER algorithm. --- Hankel matrix. --- Hankel rank condition. --- Hoeffding's inequality. --- KullbackЌeibler divergence. --- Markov chain. --- Markov process. --- Markov property. --- Monte Carlo simulation. --- PerronІrobenius theorem. --- Probability theory. --- Sanov's theorem. --- Viterbi algorithm. --- alignment. --- alpha-mixing process. --- amino acids. --- canonical form. --- complete realization problem. --- computational biology. --- concave function. --- conditional entropy. --- convex function. --- entropy function. --- entropy. --- ergodicity. --- expected value. --- finite alphabet. --- gene-finding problem. --- genomics. --- hidden Markov model. --- hidden Markov processes. --- hitting probability. --- information theory. --- irreducible matrices. --- large deviation property. --- large deviation theory. --- likelihood estimation. --- likelihood. --- lower semi-continuous function. --- lower semi-continuous relaxation. --- maximal segmental score. --- maximum likelihood estimate. --- mean hitting time. --- moment generating function. --- nonnegative matrices. --- nucleotide. --- optimal gapped alignment. --- periodic irreducible matrices. --- post-genomic biology. --- primitive matrices. --- probability distribution. --- probability. --- protein classification. --- proteomics. --- quasi-realization. --- random variable. --- rate function. --- recurrent state. --- relative entropy rate. --- relative entropy. --- sequence alignment. --- sequence. --- state transition matrix. --- stationary distribution. --- total variation distance. --- transient state. --- ultra-mixing process.