Hostname: page-component-848d4c4894-2pzkn Total loading time: 0 Render date: 2024-06-02T22:01:13.969Z Has data issue: false hasContentIssue false
  • English
  • Français

Importance sampling on coalescent histories. I

Part of: Stochastic processes Stochastic systems and control

Published online by Cambridge University Press:  01 July 2016

Maria De Iorio  and
Robert C. Griffiths
Maria De Iorio*
Affiliation:
Imperial College London
Robert C. Griffiths*
Affiliation:
University of Oxford
*
∗ Postal address: Department of Mathematics, Imperial College London, 180 Queen's Gate, London SW7 2BZ, UK
∗∗ Postal address: Department of Statistics, University of Oxford, 1 South Parks Road, Oxford OX1 3TG, UK. Email address: griff@stats.ox.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

Stephens and Donnelly (2000) constructed an efficient sequential importance-sampling proposal distribution on coalescent histories of a sample of genes for computing the likelihood of a type configuration of genes in the sample. In the current paper a characterization of their importance-sampling proposal distribution is given in terms of the diffusion-process generator describing the distribution of the population gene frequencies. This characterization leads to a new technique for constructing importance-sampling algorithms in a much more general framework when the distribution of population gene frequencies follows a diffusion process, by approximating the generator of the process.

Keywords

Coalescent processdiffusion processimportance sampling

MSC classification

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory
Secondary: 93E25: Other computational methods 92D25: Population dynamics (general)
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 36 , Issue 2 , June 2004 , pp. 417 - 433
Copyright
Copyright © Applied Probability Trust 2004 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

Supported by BBSRC Bioinformatics grant 43/BIO14435.

References

Bahlo, M. and Griffiths, R. C. (2000). Inference from gene trees in a subdivided population. Theoret. Pop. Biol. 57, 79–95.Google Scholar
Beaumont, M. (1999). Detecting population expansion and decline using microsatellites. Genetics 153, 2013–2029.Google Scholar
Beaumont, M. (2001). Conservation genetics. In Handbook of Statistical Genetics, eds Balding, D. J., Bishop, M. and Cannings, C., John Wiley, Chichester, pp. 779–809.Google Scholar
Beerli, P. and Felsenstein, J. (1999). Maximum likelihood estimation of migration rates and effective population numbers in two populations using a coalescent approach. Genetics 152, 763–773.Google Scholar
Carbone, I. and Kohn, M. (2001). A microbial population–species interface: nested cladistic and coalescent inference with multilocus data. Molecular Ecology 10, 947–964.Google Scholar
De Iorio, M. and Griffiths, R. C. (2004). Importance sampling on coalescent histories. II: Subdivided population models. Adv. Appl. Prob. 36, 434–454.Google Scholar
De Iorio, M., Griffiths, R. C., Leblois, R. and Rousset, F. (2004). Stepwise mutation likelihood computation by sequential importance sampling in subdivided population models. Tech. Rep., Oxford University.Google Scholar
Ethier, S. N. and Griffiths, R. C. (1987). The infinitely-many-sites model as a measure-valued diffusion. Ann. Prob. 15, 515–545.Google Scholar
Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes. Characterization and Convergence. John Wiley, New York.Google Scholar
Fearnhead, P. and Donnelly, P. (2001). Estimating recombination rates from population genetics data. Genetics 159, 1299–1318.Google Scholar
Felsenstein, J., Kuhner, M. K., Yamato, J. and Beerli, P. (1999). Likelihoods on coalescents: a Monte Carlo sampling approach to inferring parameters from population samples of molecular data. In Statistics in Molecular Biology and Genetics (IMS Lecture Notes Monogr. Ser. 33), Institute of Mathematical Statistics, Hayward, CA, pp. 163–185.Google Scholar
Griffiths, R. C. (1989). Genealogical-tree probabilities in the infinitely-many-sites model. J. Math. Biol. 27, 667–680.Google Scholar
Griffiths, R. C. (2001). Ancestral inference from gene trees. In Genes, Fossils, and Behaviour: An Integrated Approach to Human Evolution (NATO Sci. Ser. A Life Sci. 310), eds Donnelly, P. and Foley, R., IOS Press, Amsterdam, pp. 137–172.Google Scholar
Griffiths, R. C. and Marjoram, P. (1996). Ancestral inference from samples of DNA sequences with recombination. J. Comput. Biol. 3, 479–502.Google Scholar
Griffiths, R. C. and Tavaré, S. (1994a). Ancestral inference in population genetics. Statist. Sci. 9, 307–319.Google Scholar
Griffiths, R. C. and Tavaré, S. (1994b). Sampling theory for neutral alleles in a varying environment. Proc. R. Soc. London B 344, 403–410.Google Scholar
Griffiths, R. C. and Tavaré, S. (1994c). Simulating probability distributions in the coalescent. Theoret. Pop. Biol. 46, 131–159.Google Scholar
Griffiths, R. C. and Tavaré, S. (1996). Markov chain inference methods in population genetics. Math. Comput. Modelling 23, 141–158.Google Scholar
Griffiths, R. C. and Tavaré, S. (1997). Computational methods for the coalescent. In Progress in Population Genetics and Human Evolution (IMA Vols Math. Appl. 87), eds Donnelly, P. and Tavaré, S., Springer, Berlin, pp. 165–182.CrossRefGoogle Scholar
Griffiths, R. C. and Tavaré, S. (1999). The ages of mutations in gene trees. Ann. Appl. Prob. 9, 567–590.Google Scholar
Harding, R. M. et al. (1997). Archaic African and Asian lineages in the genetic ancestry of modern humans. Amer. J. Human Genet. 60, 772–789.Google Scholar
Kingman, J. F. C. (1982). The coalescent. Stoch. Process. Appl. 13, 235–248.Google Scholar
Kuhner, M. K., Yamato, J. and Felsenstein, J. (1995). Estimating effective population size and mutation rate from sequence data using Metropolis–Hastings sampling. Genetics 140, 1421–1430.CrossRefGoogle ScholarPubMed
Kuhner, M. K., Yamato, J. and Felsenstein, J. (1997). Appliecations of Metropolis–Hastings genealogy sampling. In Progress in Population Genetics and Human Evolution (IMA Vols Math. Appl. 87), eds Donnelly, P. and Tavaré, S., Springer, Berlin, pp. 257–270.Google Scholar
Liu, J. S. (2001). Monte Carlo Strategies in Scientific Computing. Springer, New York.Google Scholar
Markovtsova, L., Marjoram, P. and Tavaré, S. (2000a). The age of a unique event polymorphism. Genetics 156, 401–409.Google Scholar
Markovtsova, L., Marjoram, P. and Tavaré, S. (2000b). The effects of rate variation on ancestral inference in the coalescent. Genetics 156, 1427–1436.Google Scholar
Nath, M. and Griffiths, R. C. (1996). Estimation in an island model using simulation. Theoret. Pop. Biol. 3, 227–253.Google Scholar
Nielsen, R. (1997). A likelihood approach to population samples of microsatellite alleles. Genetics 146, 711–716.Google Scholar
Slade, P. (2000a). Simulation of selected genealogies. Theoret. Pop. Biol. 57, 35–49.Google Scholar
Slade, P. (2000b). Most recent common ancestor probability distribution in gene genealogies under selection. Theoret. Pop. Biol. 58, 291–305.Google Scholar
Stephens, M. (2001). Inference under the coalescent. In Handbook of Statistical Genetics, eds Balding, D. J., Bishop, M. and Cannings, C., John Wiley, Chichester, pp. 213–238.Google Scholar
Stephens, M. and Donnelly, P. (2000). Inference in molecular population genetics. J. R. Statist. Soc. B 62, 605–655.Google Scholar
Wakeley, J., Nielsen, R., Liu-Cordero, S. and Ardlie, K. (2001). The discovery of single nucleotide polymorphisms, and inferences about human demographic history. Amer. J. Human Genet. 69, 1332–1347.Google Scholar
Wilson, I. J. and Balding, D. J. (1998). Genealogical inference from microsatellite data. Genetics 150, 499–510.Google Scholar
Wilson, I. J., Weale, M. E. and Balding, D. J. (2003). Inferences from DNA data: population histories, evolutionary processes and forensic match probabilities. J. R. Statist. Soc. A 166, 155–201.Google Scholar