\chapter{Conclusions}
Each of the preceding five chapters presents a Monte Carlo method for
computing the likelihood function or the posterior probability in an
inference problem in population genetics.  The Monte Carlo method
is a computational tool for approximating expectations, and, therefore,
many of the advances presented in this dissertation are concerned
with methods for computing particular quantities.   However, the novel
developments herein are not to be found solely in the computational
methods---each chapter includes some degree of stochastic modeling
necessary to actually define the quantities (likelihoods, posterior
probabilities, \etc) that are to be computed.  
While some chapters have a 
computational focus, others deal more directly with the underlying
modeling issues. However, in each case, the computational and modeling
elements are complementary to one another in a synergistic
fashion---with the new computational methods it is possible to do
inference using stochastic models that are more 
faithful representations of the actual
population genetic processes than the models previously used.  Likewise,
judicious modeling choices may simplify the computational challenges that
must be confronted.   Here I summarize the work in each of
Chapters~\ref{chap:ForBack} to~\ref{chap:Hybrids} with particular
attention to the distinction between the stochastic modeling and the
computational component of each.  

Chapter~\ref{chap:ForBack} focuses heavily on a computational
method---importance sampling---to compute the likelihood for the
effective size $N_e$ of a population with discrete generations.  The
stochastic modeling involved is standard, with the genetic samples
taken from the population being easily recognized as observations of a
hidden  Markov chain.  This recognition, however, makes possible the
importance sampling method which employs forward-backward methods for
hidden Markov chains \cite{Baumetal1970} and a variate transformation
previously used in theoretical genetics \cite{Cav&Edw1967}.

Chapter~\ref{chap:Urns}, on the other hand, deals primarily with issues
of the stochastic modeling of genetic inheritance in populations.  It
details some of the inaccuracies of the Wright-Fisher model as
regards fixation probabilities when the census size of the population is
known to be larger than the variance effective size of the population. 
It then describes a different model for genetic inheritance.  This model
is based on a P\'{o}lya urn scheme, and is a special case of the
conditional branching process models investigated by
\citeN{Kar&McG1965}.  This urn scheme provides a better
model of genetic inheritance in populations of known census size, and is
used to define a stochastic model that depends on the parameter
$\lambda$, the ratio of effective to census breeders in a population.  At
the end of Chapter~\ref{chap:Urns}, it is shown that
implementing Markov chain Monte Carlo (MCMC) using the urn model
is simpler than trying to implement MCMC while adhering strictly to the
Wright-Fisher model.  This is thus a case in which the more accurate
stochastic model also carries certain computational advantages.  

The main modeling development in Chapter~\ref{chap:overlap} is a
stochastic model for genetic transmission that faithfully represents the
life history features of salmon populations with semelparous individuals
maturing at different ages.  This model uses the urn model to represent
genetic inheritance between different components of the salmon population
(\eg juveniles and adults of different ages).   The connections between
these different population components are summarized by a directed graph
between them.  The first half of the chapter provides a clear example of
how graphical modeling can assist in the development, portrayal, and
understanding of probabilistic dependence in complex biological systems. 
The second half of the chapter shows how, with the
neighborhood structure easily inferred from the graph, an
MCMC algorithm may be implemented to estimate $\lambda$ in salmon
populations.    

Chapters~\ref{chap:MixAdmix} and~\ref{chap:Hybrids} both deal with
genetic admixture. Chapter~\ref{chap:MixAdmix} is more concerned with
computational issues than modeling.  There are a few very slight modeling
concerns in extending the model of
\citeN{Pritchardetal2000} to include a purebred category, however, the
primary advances in the chapter are 1) the application of a
forward-backward algorithm to compute the probability of an individual's
multilocus genotype, and 2) the development of a reversible jump MCMC
scheme that allows computation of the Bayes factor for comparing the
model with purebred categories to the model without those categories.
Chapter~\ref{chap:Hybrids} employs standard MCMC
techniques for the Bayesian analysis of mixtures, but it does so within a
different modeling framework than that pursued in
Chapter~\ref{chap:MixAdmix}.  While the model adopted in
Chapter~\ref{chap:MixAdmix} derives from an heuristic model for structured
populations, proposed by \citeN{Pritchardetal2000} to encompass a wide
range of population scenarios, the efforts of Chapter~\ref{chap:Hybrids}
are directed toward explicitly modeling the hybridization process between
different species.  This allows a more natural and interpretable analysis
of genetic variation in sympatric populations of occasionally hybridizing
species.  


In conclusion, I have presented five applications of Monte Carlo
methods to inference from population genetics data.  
These methods allow inferences to be made based on the likelihood
function in the frequentist setting, or on the posterior probability
function in the Bayesian setting.  These Monte Carlo
methods allow the computation of likelihoods or posterior probabilities
from complex models, and this, in turn, permits a greater degree of
biological reality in the stochastic models used. 



There is considerable room
for future work in the field of inference from population genetics data. 
In some of the chapters, I have indicated extensions that could be made.
There are also certainly some estimation problems  in
conservation genetics that would require either custom-tailoring of the
models used here or completely new models altogether.  The treatment of
inference problems in this dissertation provides several
examples of the use of Monte Carlo, likelihood and Bayesian analysis,
and graphical modeling, and I hope that these examples will illuminate
and stimulate the future application of these useful tools to problems
in conservation genetics.