Book contents
- Frontmatter
- Contents
- Preface
- Guide to the chapters
- Acknowledgment of support
- Part I Introduction to the four themes
- Part II Studies on the four themes
- 5 Parametric Inference
- 6 Polytope Propagation on Graphs
- 7 Parametric Sequence Alignment
- 8 Bounds for Optimal Sequence Alignment
- 9 Inference Functions
- 10 Geometry of Markov Chains
- 11 Equations Defining Hidden Markov Models
- 12 The EM Algorithm for Hidden Markov Models
- 13 Homology Mapping with Markov Random Fields
- 14 Mutagenetic Tree Models
- 15 Catalog of Small Trees
- 16 The Strand Symmetric Model
- 17 Extending Tree Models to Splits Networks
- 18 Small Trees and Generalized Neighbor-Joining
- 19 Tree Construction using Singular Value Decomposition
- 20 Applications of Interval Methods to Phylogenetics
- 21 Analysis of Point Mutations in Vertebrate Genomes
- 22 Ultra-Conserved Elements in Vertebrate and Fly Genomes
- References
- Index
20 - Applications of Interval Methods to Phylogenetics
from Part II - Studies on the four themes
Published online by Cambridge University Press: 04 August 2010
- Frontmatter
- Contents
- Preface
- Guide to the chapters
- Acknowledgment of support
- Part I Introduction to the four themes
- Part II Studies on the four themes
- 5 Parametric Inference
- 6 Polytope Propagation on Graphs
- 7 Parametric Sequence Alignment
- 8 Bounds for Optimal Sequence Alignment
- 9 Inference Functions
- 10 Geometry of Markov Chains
- 11 Equations Defining Hidden Markov Models
- 12 The EM Algorithm for Hidden Markov Models
- 13 Homology Mapping with Markov Random Fields
- 14 Mutagenetic Tree Models
- 15 Catalog of Small Trees
- 16 The Strand Symmetric Model
- 17 Extending Tree Models to Splits Networks
- 18 Small Trees and Generalized Neighbor-Joining
- 19 Tree Construction using Singular Value Decomposition
- 20 Applications of Interval Methods to Phylogenetics
- 21 Analysis of Point Mutations in Vertebrate Genomes
- 22 Ultra-Conserved Elements in Vertebrate and Fly Genomes
- References
- Index
Summary
When statistical inference is conducted in a maximum likelihood (ML) framework as discussed in Chapter 1, we are interested in the global maximum of the likelihood function over the parameter space. In practice we settle for a local optimization algorithm to numerically approximate the global solution since explicit analytical solutions for the maximum likelihood estimates (MLEs) are typically difficult to obtain. See Chapter 3 or 18 for algebraic approaches to solving such ML problems. In this chapter we will take a rigorous numerical approach to the ML problem for phylogenetic trees via interval methods. We accomplish this by first constructing an interval extension of the recursive formulation for the likelihood function of an evolutionary model on an unrooted tree. We then we use an adaptation of a widely applied global optimization algorithm using interval analysis for the phylogenetic context to rigorously enclose ML values as well as MLEs for branch lengths. The method is applied to enclose the most likely 2- and 3-taxa trees under the Jukes–Cantor model of DNA evolution. The method is general and can provide rigorous estimates when coupled with standard phylogenetic algorithms. Solutions obtained with such methods are equivalent to computer-aided proofs, unlike solutions obtained with conventional numerical methods.
Statistical inference procedures that obtain MLEs through conventional numerical methods may suffer from several major sources of errors. To fully appreciate the sources of errors we need some understanding of a number screen.
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- Algebraic Statistics for Computational Biology , pp. 359 - 374Publisher: Cambridge University PressPrint publication year: 2005
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