%%  Eric Anderson's Dissertation Preliminary Pages


\Title{Monte Carlo Methods for Inference in \\ Population Genetic
Models}

\Author{Eric C. Anderson}

\Program{Interdisciplinary Program in Quantitative Ecology and Resource
Management}

\Year{2001}


\titlepage



\Chair{Elizabeth A. Thompson}{Professor}{Department of Statistics}

\Signature{Elizabeth A. Thompson}
\Signature{Joe Felsenstein}
%\Signature{Richard Fenske}
%\Signature{Matthew Stephens}
\Signature{Robin S. Waples}


\signaturepage




\doctoralquoteslip



\abstract{
This dissertation describes novel applications of Monte Carlo and
Markov chain Monte Carlo (MCMC) techniques to statistical inference in
problems from the field of conservation genetics.  The inference problems
are motivated by issues arising in the conservation
and management of trout and salmon. The first half of the
thesis deals with estimating effective population size and related
quantities from temporally spaced samples of genetic data.   A
likelihood function for organisms with discrete
generations is developed, based on the  Wright-Fisher model and a hidden
Markov chain formulation.  Importance sampling methods for computing this
likelihood are presented and applied to published data on
{\em Drosophila}.  Some modeling
assumptions implicit in the use of the Wright-Fisher model are detailed,
and a new model for genetic inheritance, based on a P\'{o}lya urn scheme,
is presented and characterized.  Methods are developed using this model
for the MCMC estimation of the likelihood or posterior probability for the
effective size of a population or for the ratio
$\lambda$ of the per-generation number of effective breeders to census
breeders in a population.  The P\'{o}lya
urn model forms the basis for a probability model
of allele frequency change conditional on
$\lambda$ in salmon populations. 
Specifying this model in terms of a large  directed
graph simplifies the application of a single-component
Metropolis-Hastings algorithm for computing the posterior distribution of
$\lambda$.  The method is applied to genetic data
simulated upon the census counts of a threatened salmon population in
Idaho, demonstrating that the method allows precise estimates
of $\lambda$ with such data.  

The second half of the thesis focuses on approaches to inference within
populations of recently-hybridized populations.  First, I extend the
methods of \citeN{Pritchardetal2000} to allow inference of pure and
admixed categories of individuals in structured populations.   Finally, I
develop methods based on explicit modeling of recent hybridization and
evaluate the potential for using genetic data to distinguish
$\F_1$,
$\F_2$ and backcrossed hybrid categories among sympatric, hybridizing
populations. 


}



%% TOC's and Lists of Figures, etc
%
\tableofcontents
\listoffigures
\listoftables  % I have no tables




\acknowledgments{I owe thanks to a great many people who helped me in
the preparation of this dissertation.  I must first acknowledge the
support, encouragement and criticism of my supervisory committee, starting
with the mentoring given me by my committee chair,
Elizabeth Thompson.  I will be forever grateful that Elizabeth was
willing to take me on as a student five years ago, despite my relative
lack, at the time, of formal, college, mathematical training.  In
addition to profiting from Elizabeth's habit of being always
available to her students, I also benefited tremendously from her
mentoring style,  knowledge of statistical genetics, and innate
curiosity of statistical and genetic matters of all variety.  I also owe
special thanks to Joe Felsenstein.  Were it not for his generosity in
offering me an independent study in population genetics six years ago, I would  never
have found my way to
this field that I find so rewarding.  Robin Waples has been  helpful throughout the dissertation
process.  In the early stages he suggested fruitful lines of
research, in the middle stages he kindly provided me with useful
datasets, and in the final stages he contributed many insightful
comments and criticisms of the work in these pages.  I met Matthew
Stephens while he was a postdoctoral researcher at the University of
Oxford.  My visit to Oxford in October of 1999 inspired two of the
chapters in this dissertation.  Matthew, who has provided a great amount
of technical help, is also an inspiring statistician to me, as
well as a valued friend.  Finally, Richard Fenske has been a model
Graduate-School Representative.  He was flexible in scheduling
meetings, and it was a pleasure to get to know him during my years at the
University. 

I thank the QERM program for its sturdy and resilient existence through
some of the trying times in the last four years.  I have enjoyed the
contact I have had with fellow QERM students.  I particularly acknowledge
my discussions with David Caccia on stochastic processes,
Markov chain Monte Carlo, and perfect sampling methods, and I give thanks for the typically animated
comments that Professor David Ford always had for me and others when presenting at the QERM seminar
series.  

Elizabeth Thompson's students and postdocs in Statistics and
Biostatistics were an invaluable resource for me, and I particularly
thank Nicky Chapman (with whom I shared dissertation-completion-neurosis
during spring quarter) and my esteemed office mate, Andrew George. 
I have had the pleasure of occupying an office in the Department of
Statistics for the last three years.  My exposure to the
distinguished faculty and the dedicated students there was a
wonderful experience.  The Department never ceases to impress me, and
I am  honored to have been  a part of it. 

I regularly attended  several seminar groups that enriched and broadened
my experience at 
the University.  I particularly thank the
MathBio Czar, Professor Thomas Daniel in the Department of Zoology
and Professors Garry Odell and Elizabeth Thompson for organizing and
orchestrating the Mathematical Biology seminar in
Zoology.  I have also enjoyed the long-runnning Statistical Genetics
seminar group in Statistics and Biostatistics, and, in my last year
here, have had the pleasure of attending Adrian Raftery's Model-Based
Clustering Working Group in Statistics.    

I thank my collaborators on talks and papers in the last four years.  I
collaborated with Ellen Williamson at University of California, Berkeley on a paper that formed the
material for Chapter~\ref{chap:ForBack} of this dissertation.  In connection with that collaboration, I
acknowledge the hospitality of Montgomery Slatkin and his lab group during my visits to
Berkeley.  Jonathan Pritchard was a collaborator for a talk I presented at the International Biometrics
Conference in 2000.  This work, which eventually grew into Chapter~\ref{chap:MixAdmix}, started from
several conversations during a brief visit to University of Oxford, where I was hosted by the lab of Peter
Donnelly in the  Department of Statistics.  Most recently I have had the pleasure of teaming up with Paul
Scheet for a short article in {\em Genetics}.  Mark Beaumont and David Teel provided datasets that I
use in Chapters~\ref{chap:MixAdmix} and~\ref{chap:Hybrids}.

During my years working on this dissertation I have been supported by National Science Foundation grant
BIR--9807747 to Elizabeth Thompson, the National Science Foundation Mathematical Biology Training Grant
\#BIR--9256532 to Thomas Daniel and Garry Odell,  the Burroughs Wellcome Fund, Program in Mathematics and
Molecular Biology, and a  QERM Student Training Grant.

While in Seattle, outside the academic sphere, I have received support from a network of
friends too numerous to  thank individually.  Through it all, always, I thank my mother, father, and sister
for their unflagging love and support.

Finally, as my undergraduate training was in biology, with few mathematics courses, I have
had to rely on the math background I gained as a high school student.  The Mathematics Department at The
Thacher School, and particularly Kurt Meyer, deserve special recognition.  And I thank my dear friend and
mentor of math and mountains, John Rosendahl.    }



%%Mark Riegner at Prescott College---my introduction to genetics which got
%%me interested in biology.  My, what a circuitous route I have taken. 
%%
%%
%% Chapter 2.
%%This study was supported by National Science Foundation grant BIR--9807747 to E.T\@. 
%%Additionally, E.A. was supported by National Science Foundation
%%grant BIR--9256537, the University of Washington QERM program, and a Burroughs Wellcome Fund
%%PMMB training fellowship.  E.W. was supported by National Institutes of Health grant GM40282 to M.
%%Slatkin.
%%
%%
%%
%%
%%
%%Chapter 5.
%%This work developed out of conversations during a brief visit
%%to University of Oxford, where the author was hosted by the lab of Peter Donnelly in the 
%%Department of Statistics. Partial support came from
%%National Science Foundation Grant BIR--9807747.  The author thanks Jonathan Pritchard, Elizabeth Thompson, and
%%Matthew Stephens for helpful discussion and comments on earlier drafts. Special thanks to Mark Beaumont for providing
%%the data on Scottish wildcats.
%%



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%%%\dedication{Not currently dedicated to anyone or anything}
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