6.4 The Infinitely Many Alleles Model
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1 6.4. THE INFINITELY MANY ALLELES MODEL 93 NE µ [F (X N 1 ) F (µ)] = + 1 i<j n ( f i f j, µp N f i, µp N f j, µp N ) l:l i,j n Af i, µ f j, µ f j, µp N + O(N 1 ) j:j<i = GF (µ) + o(1) uniformly in µ. The completes the verification of condition (6.7). j:j>i 6.4 The Infinitely Many Alleles Model f l, µp N This is a special case of the Fleming-Viot process which has played a crucial role in modern population biology. It has type space E = [, 1] and type-independent mutation operator with mutation source ν P([, 1]) Af(x) = θ( p(x, dy)f(y) f(x)) = θ( f(y)ν (dy) f(x)). Since A is a bounded operator we can take indicator functions of intervals in D(A). If we have a partition [, 1] = K j=1b j where the B j are intervals, consider the set D(G) of functions (6.1) F (µ) = f 1, µ... f n, µ with n 1 and where the functions f 1,..., f n are finite linear combinations of indicator functions of the intervals {A j }. Then the function GF (µ) can be written in the same form and we can prove that the -valued process {p t (A 1 ),..., p t (A K )} is a version of the K allele process with generator (6.11) G K f(p) = 1 2 i,j=1 p i (δ ij p j ) 2 f(p) + θ p i p j (ν (A i ) p i ) f(p) p i. We will next give an explicit construction of this process that allows us to derive a number of interesting properties of this important model Projective Limit Construction of the Infinitely Many Alleles Model Let µ, ν P(E), C = C [, ) ([, )). Let U denote the collection of finite partitions u = (A u 1,..., A u u ) of E into measurable subsets in B(E) and u denotes
2 94 CHAPTER 6. INFINITELY MANY TYPES MODELS the number of sets in the partition u. We place a partial ordering on U as follows: v u if v is a refinement of u. We can also identify partitions with the finite algebras of subsets of E they generate. Given a partition we define the probability measure, P u on C u as the law of the Wright-Fisher diffusion with generator G (K) f(p) = i,j=1 ν i := ν (A j ) p i (δ ij p j ) 2 f(p) p i p j θ(ν i p i ) f(p) p i and initial measure µ, that is, the law of (p t (A u 1),..., p t (A u u ) (and the additive extension of this to the algebra generated by u). Remark 6.6 Recall that the associated Markov transition function is determined by the joint moments as follows. Since the family of functions p k , p k belong to D(G(K) ) we can apply G (K) and obtain the following system of equations for the joint moments: (6.12) m k1,...,k (t) := E[p k 1 1 (t)... p k (t)], t m k 1,...,k (t) = 1 k i (k i 1)m k1,.,k 2 i 1,..k (t) i 1 k i k j m k1,...k 2 K (t) + θ 2 i j ν i k i m k1,.k j 1,.,k i +1,..k K (t) θ k i m k1,...k 2 (t) Since this system of linear equations is closed, there exists a unique solution which characterizes the K-allele Wright-Fisher diffusion. In a similar way we can apply this to the function corresponding to the coalescence of two partition elements f(p) = f( p) p = ( p 1,..., p ) = (p 1,..., p l 1, p l+1,..., p k 1, p k+1,..., p, (p l + p k ))
3 6.4. THE INFINITELY MANY ALLELES MODEL 95 G (K) f(p) = θ 2 K 2 i,j=1 p i (δ ij p j ) 2 f( p) p i p j K 2 ( ν i p i ) f( p) p i = G () f( p) In other words we have consistency under coalescence of the partition elements. Because of uniqueness this implies that the process p(t) = ( p 1 (t),..., p (t)) coincides with the (K 1) allele Wright-Fisher diffusion. We denote the canonical projections π u : C B(E) C u and π uv : C v C u if v u such that π u = π uv π v, v u. The family {P u } u U forms a projective system of probability laws, that is for every pair, (u, v), v u, {P u } then satisfies (6.13) π uv (P v ) = P u, P u (B) = P v (π 1 uv (B)). Therefore, by Theorem 17.6 (in Appendix I) there exists a projective limit measure, that is, a probability measure P on C B([,1]) such that for any u U, π u P = P u. For fixed t (or any finite set of times) we can identify the projective limit, (6.14) { p t (A) : A B([, 1])} with an element of X ([, 1]), the space of all finitely additive, non-negative, mass one measures on [, 1], equipped with the projective limit topology, i.e., the weakest topology such that for all Borel subset B of [, 1], µ(b) is continuous in µ. Under this topology, X ([, 1]) is Hausdorff. The σ-algebra B of the space X ([, 1]) is the smallest σ-algebra such that for all Borel subset B of [, 1], µ(b) is a measurable function of µ. For fixed t [, ), p t ( ) is a.s. a finitely additive measure, that is, a member of X [, 1] and satisfies the conditions of Theorem 17.8 in the Appendices (conditions 1,2 follow immediately from the construction, 3 follows since for any A B([, 1]) E(p t (A)) max(µ(a), ν (A)) and (4) is automatic since all measures are bounded by 1). Therefore for fixed t this determines a unique countably additive version p t ( ), that is, a random countable additive measure p t P([, 1]) a.s. Similarly, taking two times t 1, t 2 we obtain a the joint distribution of a pair (p t1, p t2 ) of random probability measures. We can then verify that t f(x)p t (dx) is a.s. continuous for a countable convergence determining class of functions so that there is an a.s. continuous version with respect to the topology of weak convergence.
4 96 CHAPTER 6. INFINITELY MANY TYPES MODELS Remark 6.7 We can carry out the same construction assuming that for each u U the Wright-Fisher diffusion starts with the stationary Dirichlet measure and obtain by the projective limit a probability measure on P(E) which for any partition has the associated Dirichlet distribution. 6.5 The Jirina Branching Process In 1964 Jirina [355] gave the first construction of a measure-valued branching process. The state space is the space of finite measures on [, 1], M f ([, 1]). ν M 1 ([, 1]). We will construct a version of this process with immigration by a projective limit construction. Given a partition (A 1,..., A K ) of [, 1] let {X t (A i ) : t, i = 1,..., K} satisfy the SDE (Feller CSB plus immigration): (6.15) dx t (A i ) = c(ν (A i ) X t (A i ))dt + 2γX t (A i )dw A i t X (A i ) = µ(a i ) where ν is in P([, 1]) and for each i, W A i t is a standard Brownian motion and for i j W A i t and W A j t are independent. We can then verify that the processes X t (A i ) : i = 1,..., K are independent and as t, X t (A i ) converges in distribution to a stationary measure X (A i ) with density which satisfies f i (x) = 1 Z xθ i 1 e θx, x > where θ = c γ, θ i = θν (A i ). This can be represented by X (A) = θ 1 G(θν (A)) where θ = c γ and L{(X (A 1 ),..., X (A K ))} = L{ 1 θ [G(θ 1), G(θ 1 + θ 2 ) G(θ 1 ),..., G(θ) G(θ θ K )]} where G(s) is the Moran subordinator - see subsection below. For u = (A u 1,..., A u u ) U (defined as in the last subsection) let {P u = L({(X t (A 1 ),..., X t (A u ) : t, A u})}. Then the collection {P u } u U forms a projective system and as in the previous section there exists a projective limit measure P on (C [, ) ([, ))) B([,1]). Moreover for fixed t [, ), X t ( ) is a.s. a finitely additive measure that is regular (on a countable generating subset of B([, 1])) we obtain a unique countably additive version (recall Theorem 17.8). Thus, {X t ( ) : t } is a measure-valued process and again we can obtain an a.s. continuous M F ([, 1])-valued version. This M F ([, 1])-valued process is called the Jirina process.
5 6.6. INVARIANT MEASURES OF THE IMA AND JIRINA PROCESSES 97 Corollary 6.8 The stationary measure for the Jirina process is given by the random measure (6.16) X (A) = 1 θ 1 1 A (x)dg(θs), A B([, 1]) where G( ) is the Moran gamma subordinator. 6.6 Invariant Measures of the IMA and Jirina Processes The Moran (Gamma) Subordinator We begin by recalling the the Gamma distribution with parameter α > given by the density function g α (u) = u α 1 e u /Γ(α) and Laplace transform of g α is g α (y)e λy dy = 1 (1 + λ) α, λ > 1, The Moran subordinator {G(α) : α } is an increasing process with stationary independent increments G(α 2 ) G(α 1 ), α 1 < α 2 given by g α2 α 1. Lévy representation Lemma 6.9 (6.17) E ( e λg(α)) = exp Proof. Note that λ ( α (1 e λz )z 1 e z dz = ) (1 e uλ ) e u u du. (1 e λz )z 1 e z dz = log(1 + λ) (e λz )e z dz = λ Hence we have the Lévy-Khinchin representation with Lévy measure e z z, z > (6.18) { 1 (1 + λ) = exp α α (1 e λz )z 1 e z dz }.
6 98 CHAPTER 6. INFINITELY MANY TYPES MODELS Poisson representation The Poisson random field with intensity measure µ is a random counting measure Π on a space S. Π(A i ), Π(A j ) are independent if i j and Π(A) is Poisson with parameter µ(a). Theorem 6.1 (Campbell s Theorem.) Let Π be a Poisson random field with intensity µ M(S) and f : S R, Σ = x Π f(x) = f(x)π(dx) converges a.s. if and only if min( f(x), 1)µ(dx) < S and then E(e s f(x)π(dx) ) = exp( (e sf(x) 1)µ(dx)), s R provided the integral on the right exists. Now consider the Poisson random measure on [, 1] (, ) (6.19) Ξ θ = δ {x,u} with intensity measure θν (dx) e u u du. Let X (A) := A uξ θ (dx, du). Then by Campbell s Theorem (6.2) E(e λ X (A) ) = E(e λ A uξ θ(dx,du) ) = e θν (A) (1 e λu ) e u u du. Hence we can represent equilibrium of the Jirina process by the random measure with Poisson representation {X (A) : A B([, 1])} by (6.21) X (A) = θ 1 u Ξ θ (dx, du) A and this can be obtained as the projective limit of the finite systems. If ν is Lebesgue measure on [, 1] then have that the {X ([, s)} s 1 = {G(s)} s 1 where G(s) is the Moran subordinator with with increments G(s 2 ) G(s 1 ) having the Gamma θ(s 2 s 1 ) distribution θ = c γ.
7 6.6. INVARIANT MEASURES OF THE IMA AND JIRINA PROCESSES Representation of the Infinitely Many Alleles Equilibrium Recall (Theorem 5.9) that the Dirichlet distribution Dirichlet(θ 1,..., θ n ) has the joint density on relative to (n 1)-dimensional Lebesgue measure on n 1 given by f(p 1,..., p n 1 ) = Γ(θ θ n ) Γ(θ 1 )... Γ(θ n ) pθ p θ p θn 1 n. Recall that if the θ are large the measure concentrates away from the boundary whereas is the θ are small things concentrate near the boundary corresponding to highly disparate p with a few large p j and the others small. For example if the θ j are small but equal there is a high probability that at least one of the p j is much greater than average; and which value or values of j have large p j is a matter of chance. Proposition 6.11 Let X denote the equilibrium random measure for the Jirina process and consider a partition [, 1] = n A i and define (6.22) Y (A i ) := X (A i ) X ([, 1]) = G(θ A i ). G(θ) Then the family (Y (A 1 ),..., Y (A K )) is independent of X ([, 1]) and has as distribution the Dirichlet(θ 1,... θ K ) where θ j = θν (A j ). Proof. Let Y be Gamma(θ) and (P 1,..., P K ) Dirichlet (θ 1,..., θ K ) with Y and (P 1,..., P K ) independent, and define (Y 1,..., Y K ) by (6.23) Y i := Y P i. We will verify that (Y 1,..., Y K ) has the joint probability density function (6.24) g(y 1,..., y K ) = K u θ i 1 i e u i /Γ(θ i ). Consider the 1-1 transformation (Y 1, Y 2,..., Y K ) (Y, P 2,..., P K ) with Jacobian { (6.25) J = x } 1,..., x K, x 1 = y, x 2 = p 2,..., x K = p K = 1 y 1,..., y K y. By independence of Y and (P 1,..., P K ), we obtain the joint density of (Y 1,..., Y K ) as
8 1 CHAPTER 6. INFINITELY MANY TYPES MODELS g(y 1,..., y K ) = f(p 1,..., p K Y )f Y (y) J 1 = f(p 1,..., p K ) Γ(θ) yθ 1 e y 1 y Γ(θ) = Γ(θ 1 )... Γ(θ K ) ( ) θ1 1 ( ) θk 1 y1 yk yi yi Γ(θ) ( y i ) (θ 1) e yi.( y i ) () K 1 = Γ(θ i ) yθ i 1 i e y i Note that this coincides with the Dirichlet(θ 1,..., θ K ) distribution. Corollary 6.12 The invariant measure of the infinitely many alleles model can be represented by the random probability measure (6.26) Y (A) = X (A),, A B([, 1]). X ([, 1]) where X ( ) is the equilibrium of the above Jirina process and Y ( ) and X ([, 1]) are independent. Reversibility Recall that the Dirichlet distribution is a reversible stationary measure for the K type Wright-Fisher model with house of cards mutation (Theorem 5.9). From this and the projective limit construction it can be verified that L(Y ( )) is a reversible stationary measure for the infinitely many alleles process. Note that reversibility actually characterizes the IMA model among neutral Fleming-Viot processes with mutation, that is, any mutation mechanism other than the typeindependent or house of cards mutation leads to a stationary measure that is not reversible (see Li-Shiga-Yau (1999) [431]) The Poisson-Dirichlet Distribution Without loss of generality we can assume that ν is Lebesgue measure on [, 1]. This implies that the IMA equilibrium is given by a random probability measure which is pure atomic (6.27) p = a i δ xi, a i = 1, x i [, 1]
9 Supplementary EXERCISES FOR LECTURE 9 1. Consider two Feller CSB with immigration as in Equation 6.15 in the notes (with Wt Ai, W Aj t independent Brownian motions, i j). Show that Y t := X t (A i ) + X t (A j ) is a Feller CSB with immigration. 2. Prove Campbell s Theorem. 3. Consider the Gamma subordinator {G(s) : s } and let a < b. (a) Prove that P (G(b) G(a) = ) = (b) Calcuate the probability that there is no jump in (a, b) larger than 1. 1
yk 1 y K 1 .( y i ) (K 1) Note that this coincides with the Dirichlet(θ 1,..., θ K ) distribution.
1 CHAPTER 6. INFINITELY MANY TYPES MODELS g(y 1,..., y K ) = f(p 1,..., p K Y )f Y (y) J 1 = f(p 1,..., p K ) Γ(θ) yθ 1 e y 1 y K 1 Γ(θ) = Γ(θ 1 )... Γ(θ K ) ( ) θ1 1 ( ) θk 1 y1 yk 1.... yi yi Γ(θ) (
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