scholarly journals Shannon Entropy Rate of Hidden Markov Processes

2021 ◽  
Vol 183 (2) ◽  
Author(s):  
Alexandra M. Jurgens ◽  
James P. Crutchfield
Keyword(s):  
Markov Processes ◽  
Shannon Entropy ◽  
Statistical Models ◽  
Hidden Markov ◽  
Entropy Rate ◽  
Minimal Set ◽  
Fundamental Physics ◽  
Hidden Markov Processes ◽  
Finite State ◽  
Finite Expression

AbstractHidden Markov chains are widely applied statistical models of stochastic processes, from fundamental physics and chemistry to finance, health, and artificial intelligence. The hidden Markov processes they generate are notoriously complicated, however, even if the chain is finite state: no finite expression for their Shannon entropy rate exists, as the set of their predictive features is generically infinite. As such, to date one cannot make general statements about how random they are nor how structured. Here, we address the first part of this challenge by showing how to efficiently and accurately calculate their entropy rates. We also show how this method gives the minimal set of infinite predictive features. A sequel addresses the challenge’s second part on structure.

scholarly journals L-cumulants, L-cumulant embeddings and algebraic statis- tics

2012 ◽  
Vol 3 (1) ◽  
Author(s):  
Piotr Zwiernik
Keyword(s):  
Markov Processes ◽  
Statistical Models ◽  
Markov Models ◽  
Hidden Markov ◽  
Random Variables ◽  
Algebraic Varieties ◽  
Main Motivation ◽  
Hidden Markov Processes ◽  
Probabilistic Setting ◽  
Tree Models

Focusing on the discrete probabilistic setting we generalize the combinatorial denitionof cumulants to L-cumulants. This generalization keeps all the desired properties of the classicalcumulants like semi-invariance and vanishing for independent blocks of random variables. Theseproperties make L-cumulants useful for the algebraic analysis of statistical models. We illustratethis for general Markov models and hidden Markov processes in the case when the hidden processis binary. The main motivation of this work is to understand cumulant-like coordinates in algebraicstatistics and to give a more insightful explanation why tree cumulants give such an elegantdescription of binary hidden tree models. Moreover, we argue that L-cumulants can be used in theanalysis of certain classical algebraic varieties.

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.

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