## Advances in Applied Probability

- Adv. in Appl. Probab.
- Volume 36, Number 2 (2004), 434-454.

### Importance sampling on coalescent histories. II: Subdivided population models

Maria De Iorio and Robert C. Griffiths

#### Abstract

De Iorio and Griffiths (2004) developed a new method of constructing sequential importance-sampling proposal distributions on coalescent histories of a sample of genes for computing the likelihood of a type configuration of genes in the sample by simulation. The method is based on approximating the diffusion-process generator describing the distribution of population gene frequencies, leading to an approximate sample distribution and finally to importance-sampling proposal distributions. This paper applies that method to construct an importance-sampling algorithm for computing the likelihood of samples of genes in subdivided population models. The importance-sampling technique of Stephens and Donnelly (2000) is thus extended to models with a Markov chain mutation mechanism between gene types and migration of genes between subpopulations. An algorithm for computing the likelihood of a sample configuration of genes from a subdivided population in an infinitely-many-alleles model of mutation is derived, extending Ewens's (1972) sampling formula in a single population. Likelihood calculation and ancestral inference in gene trees constructed from DNA sequences under the infinitely-many-sites model are also studied. The Griffiths-Tavaré method of likelihood calculation in gene trees of Bahlo and Griffiths (2000) is improved for subdivided populations.

#### Article information

**Source**

Adv. in Appl. Probab. Volume 36, Number 2 (2004), 434-454.

**Dates**

First available in Project Euclid: 11 June 2004

**Permanent link to this document**

http://projecteuclid.org/euclid.aap/1086957580

**Digital Object Identifier**

doi:10.1239/aap/1086957580

**Mathematical Reviews number (MathSciNet)**

MR2058144

**Zentralblatt MATH identifier**

1124.62317

**Subjects**

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60]

Secondary: 93E25: Other computational methods 92D25: Population dynamics (general)

**Keywords**

Coalescent process diffusion process importance sampling migration subdivided population

#### Citation

De Iorio, Maria; Griffiths, Robert C. Importance sampling on coalescent histories. II: Subdivided population models. Adv. in Appl. Probab. 36 (2004), no. 2, 434--454. doi:10.1239/aap/1086957580. http://projecteuclid.org/euclid.aap/1086957580.