13,166 research outputs found
Analytic measures and Bochner measurability
Let be a -algebra over , and let denote
the Banach space of complex measures. Consider a representation for
acting on . We show that under certain, very weak
hypotheses, that if for a given and all the
map is in , then it follows that the
map is Bochner measurable. The proof is based upon the idea
of the Analytic Radon Nikod\'ym Property.
Straightforward applications yield a new and simpler proof of Forelli's main
result concerning analytic measures ({\it Analytic and quasi-invariant
measures}, Acta Math., {\bf 118} (1967), 33--59)
Non-equilibrium theory of the allele frequency spectrum
A forward diffusion equation describing the evolution of the allele frequency
spectrum is presented. The influx of mutations is accounted for by imposing a
suitable boundary condition. For a Wright-Fisher diffusion with or without
selection and varying population size, the boundary condition is , where is the frequency
spectrum of derived alleles at independent loci at time and is
the relative population size at time . When population size and selection
intensity are independent of time, the forward equation is equivalent to the
backwards diffusion usually used to derive the frequency spectrum, but the
forward equation allows computation of the time dependence of the spectrum both
before an equilibrium is attained and when population size and selection
intensity vary with time. From the diffusion equation, we derive a set of
ordinary differential equations for the moments of and express the
expected spectrum of a finite sample in terms of those moments. We illustrate
the use of the forward equation by considering neutral and selected alleles in
a highly simplified model of human history. For example, we show that
approximately 30% of the expected heterozygosity of neutral loci is
attributable to mutations that arose since the onset of population growth in
roughly the last years.Comment: 24 pages, 7 figures, updated to accomodate referees' suggestions, to
appear in Theoretical Population Biolog
Bayesian inference of natural selection from allele frequency time series
The advent of accessible ancient DNA technology now allows the direct
ascertainment of allele frequencies in ancestral populations, thereby enabling
the use of allele frequency time series to detect and estimate natural
selection. Such direct observations of allele frequency dynamics are expected
to be more powerful than inferences made using patterns of linked neutral
variation obtained from modern individuals. We develop a Bayesian method to
make use of allele frequency time series data and infer the parameters of
general diploid selection, along with allele age, in non-equilibrium
populations. We introduce a novel path augmentation approach, in which we use
Markov chain Monte Carlo to integrate over the space of allele frequency
trajectories consistent with the observed data. Using simulations, we show that
this approach has good power to estimate selection coefficients and allele age.
Moreover, when applying our approach to data on horse coat color, we find that
ignoring a relevant demographic history can significantly bias the results of
inference. Our approach is made available in a C++ software package.Comment: 27 page
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