A Note on Probabilistic Models over Strings: The Linear Algebra Approach

Probabilistic models over strings have played a key role in developing methods that take into consideration indels as phylogenetically informative events. There is an extensive literature on using automata and transducers on phylogenies to do inference on these probabilistic models, in which an impo...

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Bibliographic Details
Published in:Bulletin of mathematical biology Vol. 75; no. 12; pp. 2529 - 2550
Main Author: Bouchard-Côté, Alexandre
Format: Journal Article
Language:English
Published: New York Springer US 01.12.2013
Springer Nature B.V
Subjects:
Algorithms
Bayes Theorem
Cell Biology
Computational Biology
Evolution, Molecular
INDEL Mutation
Life Sciences
Linear algebra
Linear Models
Mathematical and Computational Biology
Mathematical Concepts
Mathematics
Mathematics and Statistics
Models, Genetic
Models, Statistical
Original Article
Phylogeny
Alignment
Graphical models
TKF91
Automata
Phylogenetics
Probabilistic models
Factor graphs
String transducers
Indel
ISSN:0092-8240, 1522-9602, 1522-9602
Online Access:Get full text
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Summary:Probabilistic models over strings have played a key role in developing methods that take into consideration indels as phylogenetically informative events. There is an extensive literature on using automata and transducers on phylogenies to do inference on these probabilistic models, in which an important theoretical question is the complexity of computing the normalization of a class of string-valued graphical models. This question has been investigated using tools from combinatorics, dynamic programming, and graph theory, and has practical applications in Bayesian phylogenetics. In this work, we revisit this theoretical question from a different point of view, based on linear algebra. The main contribution is a set of results based on this linear algebra view that facilitate the analysis and design of inference algorithms on string-valued graphical models. As an illustration, we use this method to give a new elementary proof of a known result on the complexity of inference on the “TKF91” model, a well-known probabilistic model over strings. Compared to previous work, our proving method is easier to extend to other models, since it relies on a novel weak condition, triangular transducers, which is easy to establish in practice. The linear algebra view provides a concise way of describing transducer algorithms and their compositions, opens the possibility of transferring fast linear algebra libraries (for example, based on GPUs), as well as low rank matrix approximation methods, to string-valued inference problems.
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ISSN:0092-8240
1522-9602
1522-9602
DOI:10.1007/s11538-013-9906-6