scholarly journals Divergent predictive states: The statistical complexity dimension of stationary, ergodic hidden Markov processes

2021 ◽  
Vol 31 (8) ◽  
pp. 083114
Author(s):  
Alexandra M. Jurgens ◽  
James P. Crutchfield
Keyword(s):  
Markov Processes ◽  
Hidden Markov ◽  
Statistical Complexity ◽  
Hidden Markov Processes

Hidden Markov Processes

2014 ◽  
Author(s):  
M. Vidyasagar
Keyword(s):  
Hidden Markov Models ◽  
Markov Processes ◽  
Viterbi Algorithm ◽  
Markov Models ◽  
Hidden Markov ◽  
Local Alignment ◽  
Biological Applications ◽  
Standard Material ◽  
Hidden Markov Processes ◽  
Genomics And Proteomics

This book explores important aspects of Markov and hidden Markov processes and the applications of these ideas to various problems in computational biology. It 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. It 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.

Generic Identication of Binary-Valued Hidden Markov Processes

2014 ◽  
Vol 5 (1) ◽  
Author(s):  
Alexander Schönhuth
Keyword(s):  
Stochastic Process ◽  
Markov Process ◽  
General Solution ◽  
Parameter Space ◽  
Markov Processes ◽  
Hidden Markov ◽  
Probability Distributions ◽  
Algebraic Statistics ◽  
Hidden Markov Processes ◽  
Lower Dimensional

The generic identication problem is to decide whether a stochastic process (Xt) is ahidden Markov process and if yes to infer its parameters for all but a subset of parametrizationsthat form a lower-dimensional subvariety in parameter space. Partial answers so far availabledepend on extra assumptions on the processes, which are usually centered around stationarity.Here we present a general solution for binary-valued hidden Markov processes. Our approach isrooted in algebraic statistics hence it is geometric in nature. We nd that the algebraic varietiesassociated with the probability distributions of binary-valued hidden Markov processes are zerosets of determinantal equations which draws a connection to well-studied objects from algebra. Asa consequence, our solution allows for algorithmic implementation based on elementary (linear)algebraic routines.

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