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Importance sampling on coalescent histories. I. (English) Zbl 1045.62111
Summary: M. Stephens and P. Donnelly [J. R. Stat. Soc., Ser. B, Stat. Methodol. 62, 605–655 (2000; Zbl 0962.62107)] 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.

MSC:
62P10 Applications of statistics to biology and medical sciences; meta analysis
92D10 Genetics and epigenetics
60J70 Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)
62L99 Sequential statistical methods
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[1] Bahlo, M. and Griffiths, R. C. (2000). Inference from gene trees in a subdivided population. Theoret. Pop. Biol. 57, 79–95. · Zbl 0984.92020
[2] Beaumont, M. (1999). Detecting population expansion and decline using microsatellites. Genetics 153, 2013–2029.
[3] Beaumont, M. (2001). Conservation genetics. In Handbook of Statistical Genetics , eds D. J. Balding, M. Bishop and C. Cannings, John Wiley, Chichester, pp. 779–809.
[4] 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.
[5] Carbone, I. and Kohn, M. (2001). A microbial population–species interface: nested cladistic and coalescent inference with multilocus data. Molecular Ecology 10, 947–964.
[6] De Iorio, M. and Griffiths, R. C. (2004). Importance sampling on coalescent histories. II: Subdivided population models. Adv. Appl. Prob. 36, 434–454. · Zbl 1124.62317
[7] 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. · Zbl 1101.62105
[8] Ethier, S. N. and Griffiths, R. C. (1987). The infinitely-many-sites model as a measure-valued diffusion. Ann. Prob. 15, 515–545. JSTOR: · Zbl 0634.92007
[9] Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes. Characterization and Convergence . John Wiley, New York. · Zbl 0592.60049
[10] Fearnhead, P. and Donnelly, P. (2001). Estimating recombination rates from population genetics data. Genetics 159, 1299–1318.
[11] 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.
[12] Griffiths, R. C. (1989). Genealogical-tree probabilities in the infinitely-many-sites model. J. Math. Biol. 27, 667–680. · Zbl 0716.92012
[13] 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 P. Donnelly and R. Foley, IOS Press, Amsterdam, pp. 137–172.
[14] Griffiths, R. C. and Marjoram, P. (1996). Ancestral inference from samples of DNA sequences with recombination. J. Comput. Biol. 3, 479–502.
[15] Griffiths, R. C. and Tavaré, S. (1994a). Ancestral inference in population genetics. Statist. Sci. 9, 307–319. JSTOR: · Zbl 0955.62644
[16] Griffiths, R. C. and Tavaré, S. (1994b). Sampling theory for neutral alleles in a varying environment. Proc. R. Soc. London B 344, 403–410.
[17] Griffiths, R. C. and Tavaré, S. (1994c). Simulating probability distributions in the coalescent. Theoret. Pop. Biol. 46, 131–159. · Zbl 0807.92015
[18] Griffiths, R. C. and Tavaré, S. (1996). Markov chain inference methods in population genetics. Math. Comput. Modelling 23, 141–158. · Zbl 0853.92014
[19] 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 P. Donnelly and S. Tavaré, Springer, Berlin, pp. 165–182. · Zbl 0893.92021
[20] Griffiths, R. C. and Tavaré, S. (1999). The ages of mutations in gene trees. Ann. Appl. Prob. 9, 567–590. · Zbl 0948.92016
[21] Harding, R. M. \et (1997). Archaic African and Asian lineages in the genetic ancestry of modern humans. Amer. J. Human Genet. 60, 772–789. · Zbl 0883.26001
[22] Kingman, J. F. C. (1982). The coalescent. Stoch. Process. Appl. 13, 235–248. · Zbl 0491.60076
[23] 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.
[24] 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 P. Donnelly and S. Tavaré, Springer, Berlin, pp. 257–270. · Zbl 0892.92013
[25] Liu, J. S. (2001). Monte Carlo Strategies in Scientific Computing . Springer, New York. · Zbl 0991.65001
[26] Markovtsova, L., Marjoram, P. and Tavaré, S. (2000a). The age of a unique event polymorphism. Genetics 156, 401–409.
[27] Markovtsova, L., Marjoram, P. and Tavaré, S. (2000b). The effects of rate variation on ancestral inference in the coalescent. Genetics 156, 1427–1436.
[28] Nath, M. and Griffiths, R. C. (1996). Estimation in an island model using simulation. Theoret. Pop. Biol. 3, 227–253. · Zbl 0871.62095
[29] Nielsen, R. (1997). A likelihood approach to population samples of microsatellite alleles. Genetics 146, 711–716.
[30] Slade, P. (2000a). Simulation of selected genealogies. Theoret. Pop. Biol. 57, 35–49. · Zbl 1035.92036
[31] Slade, P. (2000b). Most recent common ancestor probability distribution in gene genealogies under selection. Theoret. Pop. Biol. 58, 291–305. · Zbl 1036.92022
[32] Stephens, M. (2001). Inference under the coalescent. In Handbook of Statistical Genetics , eds D. J. Balding, M. Bishop and C. Cannings, John Wiley, Chichester, pp. 213–238.
[33] Stephens, M. and Donnelly, P. (2000). Inference in molecular population genetics. J. R. Statist. Soc. B 62, 605–655. · Zbl 0962.62107
[34] 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.
[35] Wilson, I. J. and Balding, D. J. (1998). Genealogical inference from microsatellite data. Genetics 150, 499–510. · Zbl 0902.62037
[36] 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.
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