Kingman, J. F. C. (1982). Exchangeability and the evolution of large of populations. In Exchangeability in Provability and Statistics. ed. G. Koch and F. Spizzichino. North Holland, Amsterdam pp. 97--112.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....still leads to useful (and strong) information on the location of the gene, provided that the size of the diseased population is considerably larger than the sample size. Indeed, in such cases, the sample of haplotypes, H 1 ; H n , can be regarded as a set of exchangeable random variables (Kingman 1982). Thus, by de Finetti s theorem, H 1 ; H n can be considered as i.i.d. samples conditional on an ancestral measure P, where P is the probability distribution of H j in a hypothetical in nite population. The star genealogy assumption, then, should be regarded as a mathematical description ....

Kingman (1982). Exchangeability and the evolution of large populations. In Exchangeability in Probability and Statistics, 97-112.

....is the appropriate tool to analyse the ancestral structure of a sample of n genes (or individuals) if the total number of genes in the population is sufficiently large. A corresponding convergence theorem for a large class of exchangeable population models was first proved by Kingman ( 6] [7], 8] During the last years a variety of publications appeared in order to extend the coalescent theory to more general and more complicated models, for example for models with underlying mutation, selection or recombination, for models with variable population size or for nonexchangeable models. ....

....and infinitesimal generator G = lim t 0 Pi(t) Gamma Pi(0 ) t = 2 Gamma s)PBP = 0 0 0 0 1 Gamma1 0 q Gammaq 0 1 A : Call this process the 2 coalescent with selfing probability s. The state 3 is instantaneous. Eliminating this state leads to the usual 2 coalescent (see Kingman [6] [7], 8] 5 Remarks. 1. The effective population size is (approximately) given by N e = N(2 Gamma s) 2 (Wright [16] Pollak [14] Thus it is reasonable to measure the time in units of 2N e = 2 Gamma s)N generations. 2. In applications it is often assumed that the selfing probability is not a ....

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Kingman, J.F.C.: Exchangeability and the Evolution of Large Populations. in: Koch, G. and Spizzichino, F.: Exchangeability in Probability and Statistics, North--Holland Publishing Company, pp. 97--112 (1982).

....stochastic process, called the coalescent, is of fundamental interest. Under certain conditions this process can be used to approximate the ancestral structure of the model, if the total population size is sufficiently large. A corresponding convergence theorem was first established by Kingman [8, 9, 10] for a class of exchangeable models including the classical WrightFisher model. Convergence results for larger classes of models followed shortly later. See for example Donnelly and Tavar e [5] or Tavar e [12] and references therein. Unfortunately most of these results do only focus on the ....

Kingman, J.F.C.: Exchangeability and the Evolution of Large Populations. in: Koch, G. and Spizzichino, F.: Exchangeability in Probability and Statistics, North--Holland Publishing Company, pp. 97--112 (1982).

....process is the appropriate tool to analyse the ancestral structure of a sample of n genes (or individuals) if the total number of genes in the population is sufficiently large. A corresponding convergence theorem for a large class of exchangeable population models was first proved by Kingman ( 6] [7], 8] During the last years the coalescent theory has been 1 Postal addresses: The University of Chicago, Department of Statistics, 5734 University Avenue, Chicago Delta Illinois 60637, USA and Johannes Gutenberg Universitat Mainz, Fachbereich 17 Mathematik, Saarstra e 21, 55099 Mainz, ....

....: and infinitesimal generator G = lim t 0 Pi(t) Gamma Pi(0 ) t = 2 Gamma s)PBP = 0 0 0 0 1 Gamma1 0 q Gammaq 0 1 A : Call this process the 2 coalescent with selfing probability s. The state 3 is instantaneous. Eliminating this state leads to the usual 2 coalescent (see Kingman [6] [7], 8] Remarks. 1. The effective population size is (approximately) given by N e = N(2 Gamma s) 2 (Pollak [13] Wright [15] Thus it is reasonable to measure the time in units of 2N e = 2 Gamma s)N generations. 2. In applications it is often assumed that the selfing probability is not a ....

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Kingman, J.F.C.: Exchangeability and the Evolution of Large Populations. in: Koch, G. and Spizzichino, F.: Exchangeability in Probability and Statistics, North--Holland Publishing Company, pp. 97--112 (1982).

.... As the population size is assumed to be fixed it follows for each r 2 IN that the offspring variables (r) i , i 2 f1; Ng have to satisfy the condition N X i=1 (r) i = N: 1) The behavior of the ancestral structure of such models has been well studied (first by Kingman [14] [15], 16] later by many other authors; see for example [5] 9] 19] for the class of exchangeable neutral models. For a finite sequence of random variables 1 ; N the exchangeability is defined by the property that the joint distribution of these variables is invariant under permutation, ....

....of all equivalence relations on f1; ng and the initial value is R 0 = Delta : f(i; i) j i 2 f1; ngg. For ; j 2 En let p j : P (R r = j j R r Gamma1 = denote the transition probabilities of the ancestral process. Obviously p j = 0 for 6 j. Assume now that j. As in [15] let C 1 ; C a denote the equivalence classes of j and let C fffi , ff 2 f1; ag, fi 2 f1; b ff g denote the equivalence classes of such that C ff = S b ff fi=1 C fffi . From the second condition it follows by a combinatorial putting balls into boxes argument (see for ....

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Kingman, J.F.C.: Exchangeability and the Evolution of Large Populations. in: Koch, G. and Spizzichino, F.: Exchangeability in Probability and Statistics, North--Holland Publishing Company, pp. 97--112 (1982).

....process with the k particles born at time 0, with time running from 0 to 1 and with each pair of clusters merging at rate 1. We later need the easy fact that, in Kingman s coalescent, the number N k (t) of clusters at time t 0 satisfies N k (t) N1 (t) 1 a.s. 5) As noted by Kingman [11] the non homogeneous case is just a deterministic time change of the homogeneous case. While many variations have been considered in population genetics [15] our random birth times setting has no visible biological interpretation and so has apparently not been studied explicitly. b) In our ....

J.F. C. Kingman. Exchangeability and the evolution of large populations. In G. Koch and F. Spizzichino, editors, Exchangeability in Probability and Statistics, pages 97--112. NorthHolland Publishing Company, 1982.

....c 0, corresponding to 1 with a stable distribution of index ff, then V has PD(ff; 0) distribution. The PD(0; distribution has well known applications in population genetics, number theory, and combinatorics, as reviewed in [17, 6] Formula (15) in this case is a variation due to Kingman [26] of the Ewens sampling formula [16] 20, Ch. 41] See [30, 37] for interpretations of PD(ff; 0) in terms of excursions of a Markov process such as a Brownian motion or a recurrent Bessel process whose zero set is the closed range of a stable subordinator of index ff, and [39, 10, 11, 9] for ....

J.F. C. Kingman. Exchangeability and the evolution of large populations. In G. Koch and F. Spizzichino, editors, Exchangeability in Probability and Statistics. North-Holland Publishing Company, 1982.

.... ) with Sigma m i = k, and Sigma im i = n, p (ff; m) n i Q k Gamma1 =1 ( ff) j [ 1] n Gamma1 n Y i=1 [1 Gamma ff] i Gamma1 i m i 1 m i (42) For ff = 0; 0 this is the Ewens sampling formula [23] which has found extensive applications in population genetics [47, 24, 25]. Antoniak [3] showed that the distribution of the partition of n generated by sampling from a dirichlet( prior for a diffuse is governed by this formula. The case ff = Gamma 0 and = m gives the distribution of the partition of n generated by Fisher s m species model, corresponding to ....

J.F. C. Kingman. Exchangeability and the evolution of large populations. In G. Koch and F. Spizzichino, editors, Exchangeability in Probability and Statistics. North-Holland Publishing Company, 1982.

....backwards in time and let (r) i , i 2 f1; M r g, denote the number of descendants of the i th individual alive in the r th generation. Obviously Mr X i=1 (r) i = M r Gamma1 : 1) The behaviour of the ancestral structure of such models has been well studied (first by Kingman [7] [8], 9] later by many other authors; see for example [2] 6] 11] for the class of exchangeable neutral models, where the exchangeability is mostly used in an assumption that the individuals present at a given time have the same propensity to reproduce. More precisely, for a finite sequence of ....

....is En , the set of all equivalence relations on f1; ng and the initial value is R 0 = Delta : f(i; i) j i 2 f1; ngg. Obviously for ; j 2 En the transition probability p j (r) P (R r = j j R r Gamma1 = is equal to zero if 6 j. Assume now j. In analogy to Kingman [8] let C 1 ; C a denote the equivalence classes of j and let C fffi , ff 2 f1; ag, fi 2 f1; b ff g denote the equivalence classes of such that C ff = S b ff fi=1 C fffi . Because all assignments of offsprings to parents are equally likely (conditional on (r) 1 ; ....

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Kingman, J.F.C.: Exchangeability and the Evolution of Large Populations. in: Koch, G. and Spizzichino, F.: Exchangeability in Probability and Statistics, North-Holland Publishing Company, pp. 97 -- 112 (1982).

....Saarstra e 21, 55099 Mainz, Germany; e mail: moehle mathematik.uni mainz.de 2 Postal address: Chalmers University of Technology, Department of Mathematical Statistics, 41296 Goteborg, Sweden; e mail: serik math.chalmers.se equal to zero for 6 j. Assume now that j. In analogy to Kingman [4, 5, 6] let C 1 ; C a denote the equivalence classes of j and let C fffi , ff 2 f1; ag, fi 2 f1; b ff g denote the equivalence classes of such that C ff = S b ff fi=1 C fffi . The transition probability is given by p j = 1 (N ) b N X i 1 ; i a=1 all distinct E( i 1 ) ....

.... process looking forwards in time (see [1] Note that for N 2 the correlation coefficient between 1 and 2 is given by ( 1 ; 2 ) E( 1 2 ) Gamma 1 Var( 1 ) Gamma 1 N Gamma 1 0: 4) For a large class of models, for example for the Moran model and the WrightFisher model (Kingman [5]) it is well known that the finite dimensional distributions of the time scaled ancestral process (R [t=cN ] t0 converge to those of the (classical) n coalescent. The n coalescent is a time continuous Markov process with state space En , initial state Delta and infinitesimal generator Q = q ....

Kingman, J.F.C.: Exchangeability and the Evolution of Large Populations. in: Koch, G. and Spizzichino, F.: Exchangeability in Probability and Statistics, North--Holland Publishing Company, pp. 97--112 (1982).

.... is beta(1, #) follows immediately from (20) It follows easily from any one of the previous general formulas (27) 35) 36) or (40) that the EPPF of a pd(#) partition # = # n ) is given by the formula p # (n 1 , #(# n) n i 1) n = n i ) 49) This is a known equivalent [32, 43] of the Ewens sampling formula [18, 17] for the joint distribution of the number of blocks of # n of various sizes. It is also known [41, 49] that the following conditions on # are equivalent: i) # is of the form (47) for some b 0, # 0; ii) pk(# t) pk(#) for all t 0; iii) pk(#) ....

J. F. C. Kingman. Exchangeability and the evolution of large populations. In G. Koch and F. Spizzichino, editors, Exchangeability in probability and statistics (Rome, 1981.

.... one of the previous general formulas (23) 31) 32) or (36) that the EPPF of a pd( partition Pi = Pi n ) is given by the formula p (n 1 ; Delta Delta Delta ; n k ) k Gamma( Gamma( n) k Y i=1 (n i Gamma 1) n = k X i=1 n i ) 45) This is a known equivalent [21, 30] of Ewens sampling formula [12, 11] for the joint distribution of the number of components of Pi n of various sizes. It is known [29, 35] that the following conditions on ae are equivalent: i) ae is of the form (43) for some b 0; 0; 13 (ii) pk(ae j t) pk(ae) for all t 0; iii) ....

J. F. C. Kingman. Exchangeability and the evolution of large populations. In G. Koch and F. Spizzichino, editors, Exchangeability in probability and statistics (Rome,

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Kingman, J. F. C. (1982). Exchangeability and the evolution of large of populations. In Exchangeability in Provability and Statistics. ed. G. Koch and F. Spizzichino. North Holland, Amsterdam pp. 97--112.

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Kingman, J.F.C. (1982b) Exchangeability and the Evolution of Large Populations. in: Koch, G. and Spizzichino, F.: Exchangeability in Probability and Statistics, North{Holland Publishing Company, pp. 97-112.

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Kingman, J.F.C. (1982b). Exchangeability and the Evolution of Large Populations. in: Koch, G. and Spizzichino, F.: Exchangeability in Probability and Statistics, North--Holland Publishing Company, pp. 97--112.

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Kingman, J. F. C. (1982) Exchangeability and the evolution of large populations. Pp. 97-- 112 in Exchangeability in Probability and Statistics , G. Koch and F. Spizzichino, Eds., North-Holland Publishing Co., New York.

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Kingman, J.F.C. (1982b) Exchangeability and the Evolution of Large Populations. in: Koch, G. and Spizzichino, F.: Exchangeability in Probability and Statistics, North--Holland Publishing Company, pp. 97 -- 112.

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