Lectures on Probability Theory and Statistics

Ecole d’Eté de Probabilités de Saint-Flour XXXI - 2001

verfasst von: Simon Tavaré, Ofer Zeitouni

herausgegeben von: Jean Picard

Verlag: Springer Berlin Heidelberg

Buchreihe : Lecture Notes in Mathematics

Enthalten in: Professional Book Archive

Einloggen, um Zugang zu erhalten

Über dieses Buch

This volume contains lectures given at the 31st Probability Summer School in Saint-Flour (July 8-25, 2001). Simon Tavaré’s lectures serve as an introduction to the coalescent, and to inference for ancestral processes in population genetics. The stochastic computation methods described include rejection methods, importance sampling, Markov chain Monte Carlo, and approximate Bayesian methods. Ofer Zeitouni’s course on "Random Walks in Random Environment" presents systematically the tools that have been introduced to study the model. A fairly complete description of available results in dimension 1 is given. For higher dimension, the basic techniques and a discussion of some of the available results are provided. The contribution also includes an updated annotated bibliography and suggestions for further reading. Olivier Catoni's course appears separately.

Inhaltsverzeichnis

Frontmatter

Part I: Ancestral Inference in Population Genetics

Contents.

1 Introduction
- 1.1 Genealogical processes
- 1.2 Organization of the notes
- 1.3 Acknowledgements
2 The Wright-Fisher model
- 2.1 Random drift
- 2.2 The genealogy of the Wright-Fisher model
- 2.3 Properties of the ancestral process
- 2.4 Variable population size
3 The Ewens Sampling Formula
- 3.1 The effects of mutation
- 3.2 Estimating the mutation rate
- 3.3 Allozyme frequency data
- 3.4 Simulating an infinitely-many alleles sample
- 3.5 A recursion for the ESF
- 3.6 The number of alleles in a sample
- 3.7 Estimating \(\theta \)
- 3.8 Testing for selective neutrality
4 The Coalescent
- 4.1 Who is related to whom?
- 4.2 Genealogical trees
- 4.3 Robustness in the coalescent
- 4.4 Generalizations
- 4.5 Coalescent reviews
5 The Infinitely-many-sites Model
- 5.1 Measures of diversity in a sample
- 5.2 Pairwise difference curves
- 5.3 The number of segregating sites
- 5.4 The infinitely-many-sites model and the coalescent
- 5.5 The tree structure of the infinitely-many-sites model
- 5.6 Rooted genealogical trees
- 5.7 Rooted genealogical tree probabilities
- 5.8 Unrooted genealogical trees
- 5.9 Unrooted genealogical tree probabilities
- 5.10 A numerical example
- 5.11 Maximum likelihood estimation
6 Estimation in the Infinitely-many-sites Model
- 6.1 Computing likelihoods
- 6.2 Simulating likelihood surfaces
- 6.3 Combining likelihoods
- 6.4 Unrooted tree probabilities
- 6.5 Methods for variable population size models
- 6.6 More on simulating mutation models
- 6.7 Importance sampling
- 6.8 Choosing the weights
7 Ancestral Inference in the Infinitely-many-sites Model
- 7.1 Samples of size two
- 7.2 No variability observed in the sample
- 7.3 The rejection method
- 7.4 Conditioning on the number of segregating sites
- 7.5 An importance sampling method
- 7.6 Modeling uncertainty in N and \(\mu \)
- 7.7 Varying mutation rates
- 7.8 The time to the MRCA of a population given data from a sample
- 7.9 Using the full data
8 The Age of a Unique Event Polymorphism
- 8.1 UEP trees
- 8.2 The distribution of \(T_\Delta \)
- 8.3 The case \(\mu = 0\)
- 8.4 Simulating the age of an allele
- 8.5 Using intra-allelic variability
9 Markov Chain Monte Carlo Methods
- 9.1 K-Allele models
- 9.2 A biomolecular sequence model
- 9.3 A recursion for sampling probabilities
- 9.4 Computing probabilities on trees
- 9.5 The MCMC approach
- 9.6 Some alternative updating methods
- 9.7 Variable population size
- 9.8 A Nuu Chah Nulth data set
- 9.9 The age of a UEP
- 9.10 A Yakima data set
10 Recombination
- 10.1 The two locus model
- 10.2 The correlation between tree lengths
- 10.3 The continuous recombination model
- 10.4 Mutation in the ARG
- 10.5 Simulating samples
- 10.6 Linkage disequilibrium and haplotype sharing
11 ABC: Approximate Bayesian Computation
- 11.1 Rejection methods
- 11.2 Inference in the fossil record
- 11.3 Using summary statistics
- 11.4 MCMC methods
- 11.5 The genealogy of a branching process
12 Afterwords
- 12.1 The effects of selection
- 12.2 The combinatorics connection
- 12.3 Bugs and features
References

Simon Tavaré

Part II: Random Walks in Random Environment

Contents.

1 Introduction
- 1.1 Model
- 1.2 Examples
2 RWRE - d = 1
- 2.1 Ergodic theorems
- 2.2 CLT for ergodic environments
- 2.3 Large deviations
- 2.4 The subexponential regime
- 2.5 Sinai’s model: non standard limit laws and aging properties
3 RWRE - d > 1
- 3.1 Ergodic Theorems
- 3.2 A Law of Large Numbers in \(\mathbb{Z}^d\)
- 3.3 CLT for walks in balanced environments
- 3.4 Large deviations for nestling walks
- 3.5 Kalikow’s condition
References

Ofer Zeitouni

Backmatter

Titel: Lectures on Probability Theory and Statistics
verfasst von: Simon Tavaré
Ofer Zeitouni
herausgegeben von: Jean Picard
Verlag: Springer Berlin Heidelberg
Electronic ISBN: 978-3-540-39874-5
Print ISBN: 978-3-540-20832-7
DOI: https://doi.org/10.1007/b95197