By L. Pachter, B. Sturmfels
The quantitative research of organic series info is predicated on equipment from data coupled with effective algorithms from laptop technological know-how. Algebra presents a framework for unifying some of the likely disparate suggestions utilized by computational biologists. This ebook deals an creation to this mathematical framework and describes instruments from computational algebra for designing new algorithms for specified, actual effects. those algorithms may be utilized to organic difficulties comparable to aligning genomes, discovering genes and developing phylogenies. the 1st a part of this ebook contains 4 chapters at the issues of statistics, Computation, Algebra and Biology, delivering quickly, self-contained introductions to the rising box of algebraic statistics and its purposes to genomics. within the moment half, the 4 issues are mixed and built to take on genuine difficulties in computational genomics. because the first booklet within the intriguing and dynamic zone, it is going to be welcomed as a textual content for self-study or for complex undergraduate and starting graduate classes.
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Additional info for Algebraic Statistics for Computational Biology
Computing the entire polytope is what we call parametric inference. This computation can be done eﬃciently in the polytope algebra which is a natural generalization of tropical arithmetic. , phylogenetic trees, with an emphasis on the neighbor-joining algorithm. 43 44 L. Pachter and B. 1 Tropical arithmetic and dynamic programming Dynamic programming was introduced by Bellman in the 1950s to solve sequential decision problems with a compositional cost structure. Dynamic programming oﬀers eﬃcient methods for progressively building a set of scores or probabilities in order to solve a problem, and many discrete algorithms for biological sequence analysis are based on the principles of dynamic programming.
61), we translate each of the probabilities above into a linear form in the unknowns pi1 i2 ···in . Namely, Prob(A = a, B = b, C = c) is replaced by a marginalization which is the sum of all pi1 i2 ···in that satisfy • for all Xα ∈ A, the Xα -coordinate of a equals iα , Statistics 35 • for all Xβ ∈ B, the Xβ -coordinate of b equals iβ , and • for all Xγ ∈ C, the Xγ -coordinate of c equals iγ . We deﬁne QA⊥⊥B | C to be the set of quadratic forms in the unknowns pi1 i2 ···in which result from this substitution.
46) indicates that the set of optimal solutions to the maximum likelihood problem is the disjoint union of three “surfaces of explanations”. 46) is actually true? Does running the EM algorithm 100, 000 times without converging to a parameter vector whose likelihood is larger constitute a mathematical proof? Can it be turned into a mathematical proof? 3. For a numerical approach see Chapter 20. 4 Markov models We now introduce Markov chains, hidden Markov models and Markov models on trees, using the algebraic notation of the previous sections.
Algebraic Statistics for Computational Biology by L. Pachter, B. Sturmfels