373 research outputs found
Effect of selection on ancestry: an exactly soluble case and its phenomenological generalization
We consider a family of models describing the evolution under selection of a
population whose dynamics can be related to the propagation of noisy traveling
waves. For one particular model, that we shall call the exponential model, the
properties of the traveling wave front can be calculated exactly, as well as
the statistics of the genealogy of the population. One striking result is that,
for this particular model, the genealogical trees have the same statistics as
the trees of replicas in the Parisi mean-field theory of spin glasses. We also
find that in the exponential model, the coalescence times along these trees
grow like the logarithm of the population size. A phenomenological picture of
the propagation of wave fronts that we introduced in a previous work, as well
as our numerical data, suggest that these statistics remain valid for a larger
class of models, while the coalescence times grow like the cube of the
logarithm of the population size.Comment: 26 page
The Time Machine: A Simulation Approach for Stochastic Trees
In the following paper we consider a simulation technique for stochastic
trees. One of the most important areas in computational genetics is the
calculation and subsequent maximization of the likelihood function associated
to such models. This typically consists of using importance sampling (IS) and
sequential Monte Carlo (SMC) techniques. The approach proceeds by simulating
the tree, backward in time from observed data, to a most recent common ancestor
(MRCA). However, in many cases, the computational time and variance of
estimators are often too high to make standard approaches useful. In this paper
we propose to stop the simulation, subsequently yielding biased estimates of
the likelihood surface. The bias is investigated from a theoretical point of
view. Results from simulation studies are also given to investigate the balance
between loss of accuracy, saving in computing time and variance reduction.Comment: 22 Pages, 5 Figure
Evolution of the most recent common ancestor of a population with no selection
We consider the evolution of a population of fixed size with no selection.
The number of generations to reach the first common ancestor evolves in
time. This evolution can be described by a simple Markov process which allows
one to calculate several characteristics of the time dependence of . We also
study how is correlated to the genetic diversity.Comment: 21 pages, 10 figures, uses RevTex4 and feynmf.sty Corrections :
introduction and conclusion rewritten, references adde
Noisy traveling waves: effect of selection on genealogies
For a family of models of evolving population under selection, which can be
described by noisy traveling wave equations, the coalescence times along the
genealogical tree scale like , where is the size of the
population, in contrast with neutral models for which they scale like . An
argument relating this time scale to the diffusion constant of the noisy
traveling wave leads to a prediction for which agrees with our
simulations. An exactly soluble case gives trees with statistics identical to
those predicted for mean-field spin glasses in Parisi's theory.Comment: 4 pages, 2 figures New version includes more numerical simulations
and some rewriting of the text presenting our result
Approximate Bayesian Computation: a nonparametric perspective
Approximate Bayesian Computation is a family of likelihood-free inference
techniques that are well-suited to models defined in terms of a stochastic
generating mechanism. In a nutshell, Approximate Bayesian Computation proceeds
by computing summary statistics s_obs from the data and simulating summary
statistics for different values of the parameter theta. The posterior
distribution is then approximated by an estimator of the conditional density
g(theta|s_obs). In this paper, we derive the asymptotic bias and variance of
the standard estimators of the posterior distribution which are based on
rejection sampling and linear adjustment. Additionally, we introduce an
original estimator of the posterior distribution based on quadratic adjustment
and we show that its bias contains a fewer number of terms than the estimator
with linear adjustment. Although we find that the estimators with adjustment
are not universally superior to the estimator based on rejection sampling, we
find that they can achieve better performance when there is a nearly
homoscedastic relationship between the summary statistics and the parameter of
interest. To make this relationship as homoscedastic as possible, we propose to
use transformations of the summary statistics. In different examples borrowed
from the population genetics and epidemiological literature, we show the
potential of the methods with adjustment and of the transformations of the
summary statistics. Supplemental materials containing the details of the proofs
are available online
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