APPROXIMATE SAMPLING FORMULAS FOR GENERAL FINITE-ALLELES MODELS OF MUTATION

Abstract

Many applications in genetic analyses utilize sampling distributions, which describe the probability of observing a sample of DNA sequences randomly drawn from a population. In the one-locus case with special models of mutation such as the infinite-alleles model or the finite-alleles parent-independent mutation model, closed-form sampling distributions under the coalescent have been known for many decades. However, no exact formula is currently known for more general models of mutation that are of biological interest. In this paper, models with finitely-many alleles are considered, and an urn construction related to the coalescent is used to derive approximate closed-form sampling formulas for an arbitrary irreducible recurrent mutation model or for a reversible recurrent mutation model, depending on whether the number of distinct observed allele types is at most three or four, respectively. It is demonstrated empirically that the formulas derived here are highly accurate when the per-base mutation rate is low, which holds for many biological organisms.

1. Introduction

An important problem in genetic analyses concerns computing the probability of observing a randomly drawn sample of chromosomes under a given model of evolution. Popular applications of this probability computation include maximum likelihood estimation of model parameters and ancestral inference (see [19] for a nice introduction). The coalescent [14, 15] is a useful mathematical framework for performing model-based full-likelihood analyses, but in most cases it is intractable to obtain a closed-form formula for the probability of a given dataset. A well-known exception to this complication is the celebrated Ewens sampling formula (ESF) [3], which describes the stationary probability distribution of a sample configuration under the one-locus infinite-alleles model in the coalescent or the diffusion limit. A Pólyalike urn model interpretation [9] of the formula has been known for some time, and recently a new combinatorial proof of the ESF has been provided [6]. Furthermore, the ESF also arises in several interesting contexts outside biology, including random partition structures; the ESF is a special case of the two-parameter sampling formula [17, 18] for exchangeable random partitions. See [1] for examples of other interesting combinatorial connections.

In the case of finitely-many alleles, a closed-form sampling formula is known [20] only for the parent-independent mutation (PIM) model, in which the probability of mutating from allele j to allele i depends only on the child allele i. For a general non-PIM mutation model, finding an exact, closed-form sampling formula has remained a challenging open problem.

In this paper, we make progress on this problem by deriving approximate, closed-form sampling formulas that are highly accurate when the mutation rate is low. More precisely, given a sample configuration n and the model parameters (mutation rate θ and transition matrix P), we consider the Taylor expansion of the sampling probability q(n | θ, P) about θ = 0. As discussed later, if P is irreducible when restricted to the observed alleles in the sample, then the leading order term in the expansion is proportional to θ^|𝒪_n|−1, where |𝒪_n| is the number of distinct observed alleles in the sample configuration n. Hence,

$$q(\mathit{n}|\theta ,\mathit{P})={\theta }^{|{𝒪}_{\mathit{n}}|-1}Q(\mathit{n}|\mathit{P})+O({\theta }^{|{𝒪}_{\mathit{n}}|}),$$ (1)

where Q(n | P) is the leading order coefficient that depends on the mutation transition matrix P but not on the mutation rate θ. In this paper, we consider the problem of obtaining exact closed-form formulas for Q(n | P). As many organisms typically have small per-base mutation rates, our results are of biological interest.

By restricting the set of events in the coalescent genealogy for a given sample, Jenkins and Song [12] provided closed-form formulas for Q(n | P) for an arbitrary transition matrix P when |𝒪_n| ≤ 3. In this paper, we provide new proofs of those results, and extend them by supplying a closed-form formula for Q(n | P) when |𝒪_n| = 4 and the transition matrix P is reversible restricted to the observed alleles. We prove our results using martingale arguments and use an urn construction related to the coalescent to develop a recursion for the approximate sampling probability, which can then be solved in closed-form using combinatorial techniques. As a corollary of our results, it can be seen that the simple general formula in [12, Theorem 6.3] for Q(n | P) when P is parent-independent restricted to the observed alleles also holds when P is reversible restricted to the observed alleles, provided that |𝒪_n| ≤ 3. That formula fails to hold when |𝒪_n| = 4 and P is not parent-independent restricted to the observed alleles.

As there are four distinct DNA bases, our extension to the |𝒪_n| = 4 case seems natural. A more interesting reason is as follows: In multi-locus models with finite recombination rates, no closed-form sampling formula is known, even for the simplest case of two loci with either infinite-alleles or finite-alleles PIM models. However, recently a new framework based on asymptotic series has been developed [2, 10, 11, 13] to derive useful closed-form results when the recombination rate is moderate to large. The main idea behind that research is to perform an asymptotic expansion of the sampling probability in inverse powers of the recombination rate. We note that our one-locus sampling formula for the |𝒪_n| = 4 case provides an accurate approximation of the sampling probability for a completely linked (i.e., with zero recombination rate) pair of loci with two observed alleles at each locus (as is typical in single-nucleotide polymorphism data). Hence, our work serves as a starting point for finding approximate two-locus sampling formulas when the recombination rate is small, complementary to the earlier work [2, 10, 11, 13] for large recombination rates. We leave this problem for future research.

We remark that, for a given sample configuration n and fixed parameters θ and P, the exact sampling probability q(n | θ, P) can be found numerically by solving a system of coupled linear equations in O(|n|^K) variables, where |n| denotes the total sample size and K denotes the number of allele types in the assumed model. One of the main motivations of our work is to remedy this high computational complexity. Evaluating our closed-form approximations is much more efficient, in both time and space complexity.

The rest of this paper is structured as follows. In Section 2, we lay out the model and notation used throughout the paper. In Section 3, we summarize our main closed-form sampling formulas, which we prove in Section 4 using martingale arguments and an urn construction. Numerical experiments demonstrating the usefulness of our approximate sampling formulas are provided in Section 5.

2. Model and notation

We consider Kingman’s coalescent with a K-allelic recurrent mutation model specified by the population-scaled mutation rate θ/2 and ergodic transition matrix P, where P_ji denotes the probability of allele j mutating to allele i forward in time given that a mutation occurs. The stationary distribution of P is denoted by π = (π₁, …, π_K).

The following definitions will be used throughout:

Definition 1. (n, sample configuration.) A sample of individuals is denoted by n = (n_i)_i∈[K], where n_i ∈ ℤ_≥0 denotes the number of individuals in the sample with allele i. The size |n| of the sample n is denoted by the same letter in non-bold-face, n. For notational convenience, we use e_i to denote the sample configuration with a single individual of type i and write n = n₁e₁ + ⋯ + n_Ke_K. For a subset S ⊆ [K], we define n_S = ∑_i∈S n_ie_i and n_S = |n_S|.

Definition 2. (𝒪_n, observed allele types.) Given a sample n, let 𝒪_n ⊆ [K] denote the set of observed allele types; i.e., 𝒪_n = {i ∈ [K] | n_i > 0}. The number of observed allele types is denoted by |𝒪_n|.

When the indices h, i, j, k and l are used in indefinite summations or products, they are assumed to range over 𝒪_n, unless stated otherwise.

By exchangeability, the probability of any ordered sample with configuration n is invariant under all permutations of the sampling order. We use q(n | θ, P) to denote the stationary sampling probability of any particular ordered sample with configuration n. From the standard coalescent arguments [7, 8], it can be deduced that q(n | θ, P) is the unique solution to the recursion

$$n(n-1+\theta )q(\mathit{n}|\theta ,\mathit{P})={\displaystyle \sum _i}{n}_i(n_i-1)q(\mathit{n}-{\mathit{e}}_i|\theta ,\mathit{P})+\theta {\displaystyle \sum _{i,j}}{P}_{\mathit{\text{ji}}}{n}_{i}q(\mathit{n}-{\mathit{e}}_i+{\mathit{e}}_j|\theta ,\mathit{P}),$$ (2)

with boundary conditions

$$q({\mathit{e}}_i|\theta ,\mathit{P})={\pi }_i,\text{for all }i\in [K].$$ (3)

If P is irreducible when restricted to the observed alleles 𝒪_n, then by unwinding recursion (2), it can be seen that |𝒪_n| − 1 is the smallest power of θ with a non-vanishing coefficient in the Taylor series expansion of q(n | θ, P) about θ = 0. Intuitively, for a sample with m distinct observed alleles, the coefficient of θ^m−1 in the Taylor expansion corresponds to the total probability of coalescent genealogies with the most parsimonious number (i.e., m − 1) of mutations. That P is irreducible when restricted to 𝒪_n is a sufficient (but not necessary) condition for the existence of such a parsimonious genealogy for sample n.

Letting Q(n | P) denote the coefficient of θ^|𝒪_n|−1 in the Taylor expansion, q(n | θ, P) can be written as in (1). For simplicity, in what follows we simply write q(n) and Q(n) instead of q(n | θ, P) and Q(n | P), respectively.

We now introduce some notation used throughout the paper. For a sample configuration n, we define the combinatorial quantity Λ(n) as

$$\mathrm{\Lambda }(\mathit{n})=\frac{{\displaystyle {\prod }_{i\in {𝒪}_{\mathit{n}}}}(n_i-1)!}{(n-1)!}.$$ (4)

For k ∈ ℤ_≥0, the kth falling factorial of x (denoted (x)_k↓) and the kth rising factorial of x (denoted (x)_k↑) are defined as

$${(x)}_{k\downarrow }=x(x-1)\cdots (x-k+1),$$

$${(x)}_{k\uparrow }=x(x+1)\cdots (x+k-1),$$

with (x)_0↓ = (x)_0↑ = 1. The kth harmonic number H_k is defined as

$$H_k=1+\frac{1}{2}+\cdots +\frac{1}{k},$$

with H₀ = 0. Given a sample configuration n = (n₁, …, n_K), a K-tuple m = (m₁, …, m_K) satisfying 0 ⪯ m ≺ n means 0 ≤ m_i < n_i for all i ∈ 𝒪_n and m_i = 0 for all i ∉ 𝒪_n, while 0 ≺ m ⪯ n means 0 < m_i ≤ n_i for all i ∈ 𝒪_n and m_i = 0 for all i ∉ 𝒪_n. Also, 0 ⪯ m ⪯ n denotes 0 ≤ m_i ≤ n_i for all i ∈ [K].

3. A summary of closed-form results for Q(n)

In the case of |𝒪_n| = 1, it is easy to see that Q(n) = π_i for n = ne_i. In this paper, we derive closed-form expressions for the leading order coefficient Q(n) when |𝒪_n| ≤ 3 and P is an arbitrary mutation transition matrix that is irreducible when restricted to the observed alleles 𝒪_n; and also when |𝒪_n| = 4, and P is irreducible and reversible when restricted to 𝒪_n (i.e., π_iP_ij = π_jP_ji for all i, j ∈ 𝒪_n). These closed-form results are summarized below.

Theorem 1. For |𝒪_n| = 2 and P an arbitrary mutation transition matrix that is irreducible when restricted to 𝒪_n, Q(n) is given by

$$Q(\mathit{n})=\mathrm{\Lambda }(\mathit{n}){\displaystyle \sum _{i,j\in {𝒪}_{\mathit{n}}:i\ne j}}\frac{n_j}{n}{\pi }_j{P}_{\mathit{\text{ji}}}.$$

Theorem 2. For |𝒪_n| = 3 and P an arbitrary mutation transition matrix that is irreducible when restricted to 𝒪_n, Q(n) is given by

$$Q(\mathit{n})=\mathrm{\Lambda }(\mathit{n}){\displaystyle \sum _{\text{distinct }i,j,k\in {𝒪}_{\mathit{n}}}}\left\{{\pi }_j{P}_{\mathit{\text{ji}}}{P}_{\mathit{\text{jk}}}\right[\frac{{(n_j)}_{2\downarrow }}{n(n_j+n_k-1)}-\frac{n_i{n}_j}{n(n_i+n_k)}-2\frac{n_i{n}_j{n}_k}{n{(n_j+n_k)}_{2\downarrow }}+2\frac{n_i{n}_j{n}_k}{{(n_j+n_k+1)}_{3\downarrow }}(H_n-H_{n_i-1})]+{\pi }_k{P}_{\mathit{\text{kj}}}{P}_{\mathit{\text{ji}}}[\frac{n_j{n}_k}{n(n_j+n_k-1)}+2\frac{n_i{n}_j{n}_k}{n{(n_j+n_k)}_{2\downarrow }}-2\frac{n_i{n}_j{n}_k}{{(n_j+n_k+1)}_{3\downarrow }}(H_n-H_{n_i-1})\left]\right\}.$$

Corollary 3. Suppose |𝒪_n| = 3 with sample configuration n = n_ae_a + n_be_b + n_ce_c, where a, b, c are distinct alleles in [K]. If the mutation transition matrix P is reversible and irreducible when restricted to the observed alleles 𝒪_n, Q(n) is given by

$$Q(\mathit{n})=\mathrm{\Lambda }(\mathit{n})(\frac{n_a}{n}{\pi }_a{P}_{\mathit{\text{ab}}}{P}_{\mathit{\text{ac}}}+\frac{n_b}{n}{\pi }_b{P}_{\mathit{\text{ba}}}{P}_{\mathit{\text{bc}}}+\frac{n_c}{n}{\pi }_c{P}_{\mathit{\text{ca}}}{P}_{\mathit{\text{cb}}}).$$

Theorem 4. For |𝒪_n| = 4, if the mutation transition matrix P is reversible and irreducible when restricted to the observed alleles 𝒪_n, then Q(n) is given by

$$Q(\mathit{n})=\mathrm{\Lambda }(\mathit{n}){\displaystyle \sum _{\text{distinct }i,j,k,l\in {𝒪}_{\mathit{n}}}}[{\pi }_i{P}_{\mathit{\text{ij}}}{P}_{\mathit{\text{ik}}}{P}_{\mathit{\text{il}}}\gamma (\mathit{n},i,j,k,l)+{\pi }_i{P}_{\mathit{\text{ij}}}{P}_{\mathit{\text{ik}}}{P}_{\mathit{\text{jl}}}\delta (\mathit{n},i,j,k,l)],$$

where

$$\gamma (\mathit{n},i,j,k,l)=\frac{n_i}{n}\left\{\right[\frac{n_i-1}{2(n_i+n_j+n_k-1)}-\frac{2n_j{n}_l}{{(n_i+n_j+n_k)}_{2\downarrow }}]+\frac{n_l}{2(n_j+n_k+n_l)}-[\frac{n_l(n_i-1)}{(n_k+n_l)(n_i+n_j-1)}-\frac{2n_j{n}_l}{{(n_i+n_j)}_{2\downarrow }}\left]\right\}+\frac{2n_i{n}_j{n}_l}{{(n_i+n_j+n_k+1)}_{3\downarrow }}(H_n-H_{n_l-1})-\frac{2n_i{n}_j{n}_l}{{(n_i+n_j+1)}_{3\downarrow }}(H_n-H_{n_k+n_l-1}),$$

and

$$\delta (\mathit{n},i,j,k,l)=\frac{n_i}{n}\left\{\right[\frac{n_j}{n_i+n_j+n_k-1}+\frac{2n_j{n}_l}{{(n_i+n_j+n_k)}_{2\downarrow }}]-[\frac{n_j{n}_l}{(n_k+n_l)(n_i+n_j-1)}+\frac{2n_j{n}_l}{{(n_i+n_j)}_{2\downarrow }}\left]\right\}-\frac{2n_i{n}_j{n}_l}{{(n_i+n_j+n_k+1)}_{3\downarrow }}(H_n-H_{n_l-1})+\frac{2n_i{n}_j{n}_l}{{(n_i+n_j+1)}_{3\downarrow }}(H_n-H_{n_k+n_l-1}).$$

4. Proofs of the main results

In this section, we construct an urn process to derive the closed-form formulas for Q(n) mentioned in the previous section. We use the urn process to decompose Q(n) into a sum-product of two vectors, one which depends only on the sample configuration n and the other which depends only on the mutation transition matrix P. Using this decomposition, we show that Q(n) corresponds to the probability of a certain event in the urn process.

Throughout, we use R(n) to denote the following rescaled version of Q(n):

$$R(\mathit{n})=\frac{Q(\mathit{n})}{\mathrm{\Lambda }(\mathit{n})},$$ (5)

where Λ (n) is the combinatorial coefficient defined in (4).

4.1. Description of the urn process

Let n be the sample configuration of interest. We have an urn with n balls, n_i of which have color i. We remove balls one at a time uniformly at random until there are no more balls in the urn. However, whenever we “kill” a color (i.e., remove the last ball of that color), we add back a ball of a different color. We do this by picking another ball from the urn, copying it, and returning both copies to the urn. Note that when we kill the last color, we do not add any balls back, since there are no more colors to choose from.

Suppose that when we kill color i, we add back a ball of color j. We then call j the parent of i, and call the last surviving color the root. This generates a rooted tree whose vertices consist of the |𝒪_n| observed colors (alleles).

Let T be any rooted tree on 𝒪_n. We denote the probability of generating T under the above process as ℙ_n(T). Let E(T) be the edge set of T, and let ρ(T) denote the root vertex of T. By convention, we draw edges as pointing away from the root, so the edge (j → i) indicates that j is the parent of i.

The main idea of this section is that to compute Q(n), it is enough to compute ℙ_n(T) for each T. In particular, we prove the following theorem in Section 4.2:

Theorem 5. Recall that for a transition matrix P that is irreducible when restricted to 𝒪_n, Q(n) denotes the first nonzero coefficient in the Taylor expansion (1) of q (n) about θ = 0. Given a rooted tree T described above, define f_P (T) as

$$f_{\mathit{P}}(T)={\pi }_{\rho (T)}{\displaystyle \prod _{(j\to i)\in E(T)}}{P}_{\mathit{\text{ji}}}.$$

Then, the quantity R(n) = Q(n)/Λ(n) is given by

$$R(\mathit{n})={\displaystyle \sum _T}{\mathbb{P}}_{\mathit{n}}(T)f_{\mathit{P}}(T)={𝔼}_{\mathit{n}}[f_{\mathit{P}}(T)],$$ (6)

where the sum is taken over all rooted trees T with |𝒪_n| vertices bijectively labeled by 𝒪_n. That is, R(n) is the expectation of f_P (T) under the above process.

Note that we can view f_P (T) as a probability as well. In particular, suppose we relabel the vertices of T as follows: we assign a new label from [K] to ρ(T) according to the stationary distribution π, and for each edge in T, we assign a new label to the child according to the new label of its parent and the transition matrix P. Then f_P (T) is the probability that we assign the original labels to all the vertices, given that we drew T. That is, if 𝒞_𝒪n is the event that we assign the original labels to all vertices, then

$$f_{\mathit{P}}(T)=\mathbb{P}({𝒞}_{{𝒪}_{\mathit{n}}}|T){\pi }_{\rho (T)}{\displaystyle \prod _{(j\to i)\in E(T)}}{P}_{\mathit{\text{ji}}}.$$

This immediately leads to the following interpretation:

$$R(\mathit{n})={\displaystyle \sum _T}\mathbb{P}({𝒞}_{{𝒪}_{\mathit{n}}}|T){\mathbb{P}}_{\mathit{n}}(T)={\mathbb{P}}_{\mathit{n}}({𝒞}_{{𝒪}_{\mathit{n}}}).$$ (7)

That is, R(n) is the unconditional probability that we correctly label all the alleles, if we use the urn process to generate a tree on the alleles and then use the tree to assign labels.

4.2. An inductive proof of Theorem 5

In this subsection, we provide an inductive proof of Theorem 5. In Section 4.3, we provide an alternative proof based on a modified coalescent process which provides a more intuitive explanation for why the urn process works.

Proof of Theorem 5. Recall the recursion in (2):

$$n(n-1+\theta )q(\mathit{n})={\displaystyle \sum _i{n}_i(n_i-1)q(\mathit{n}-{\mathit{e}}_i)+\theta {\displaystyle \sum _{i,j}}}{P}_{\mathit{\text{ji}}}{n}_{i}q(\mathit{n}-{\mathit{e}}_i+{\mathit{e}}_j).$$

Recall also that if P is irreducible when restricted to 𝒪_n, q(n) has leading order power θ^|𝒪_n|−1 in its Taylor series. Hence we get the following recursion for Q(n):

$$n(n-1)Q(\mathit{n})={\displaystyle \sum _{i:n_i>1}}{n}_i(n_i-1)Q(\mathit{n}-{\mathit{e}}_i)+{\displaystyle \sum _{i:n_i=1}}{\displaystyle \sum _{j:j\ne i}}{P}_{\mathit{\text{ji}}}{n}_{i}Q(\mathit{n}-{\mathit{e}}_i+{\mathit{e}}_j).$$

Plugging in Q(n) = Λ(n)R(n) and simplifying gives us the following recursion for R(n):

$$n(n-1)R(\mathit{n})={\displaystyle \sum _{i:n_i>1}}{n}_i(n-1)R(\mathit{n}-{\mathit{e}}_i)+{\displaystyle \sum _{i:n_i=1}}{\displaystyle \sum _{j:j\ne i}}{P}_{\mathit{\text{ji}}}{n}_{j}R(\mathit{n}-{\mathit{e}}_i+{\mathit{e}}_j).$$ (8)

A simple induction over |𝒪_n| and n shows that this recursion has a unique solution given the boundary conditions R(e_i). So if we can show (6) when |𝒪_n| = n = 1, and then show that ∑_T ℙ (𝒞_𝒪n | T)ℙ_n(T) satisfies the recursion (8), then we will be done. The base case is trivial: when 𝒪_n = {a}, there is only one possible tree, T = {a}, with ℙ_n(T) = 1 and ℙ(𝒞_𝒪n | T) = π_a = lim_θ→0 q(n) = Q(n) = Λ(n)R(n) = R(n).

To show ∑_T ℙ(𝒞_𝒪n | T)ℙn(T) satisfies (8), we start by giving recursions for ℙ_n(T) and ℙ(𝒞_𝒪n | T). Let z(i) be the parent of i in T, and let L(T) be the set of leafs of T (where the root is not considered a leaf). Conditioning on the first event in the urn process gives us

$${\mathbb{P}}_{\mathit{n}}(T)={\displaystyle \sum _{i:n_i>1}}\frac{n_i}{n}{\mathbb{P}}_{\mathit{n}-{\mathit{e}}_i}(T)+{\displaystyle \sum _{i\in L(T):n_i=1}}\frac{n_{z(i)}}{n(n-1)}{\mathbb{P}}_{\mathit{n}-{\mathit{e}}_i+{\mathit{e}}_{z(i)}}(T\backslash \{i\}).$$ (9)

Furthermore, if i ∈ L(T), we have

$$\mathbb{P}({𝒞}_{{𝒪}_{\mathit{n}}}|T)=P_{z(i),i}\mathbb{P}({𝒞}_{{𝒪}_{\mathit{n}}\backslash \{i\}}|T\backslash \{i\}).$$ (10)

Using (9) and (10), and collecting terms, we arrive at

$$n(n-1){\displaystyle \sum _T}\mathbb{P}({𝒞}_{{𝒪}_{\mathit{n}}}|T){\mathbb{P}}_{\mathit{n}}(T)={\displaystyle \sum _T}\mathbb{P}({𝒞}_{{𝒪}_{\mathit{n}}}|T)[{\displaystyle \sum _{i:n_i>1}}{n}_i(n-1){\mathbb{P}}_{\mathit{n}-{\mathit{e}}_i}(T)+{\displaystyle \sum _{i\in L(T):n_i=1}}{n}_{z(i)}{\mathbb{P}}_{\mathit{n}-{\mathit{e}}_i+{\mathit{e}}_{z(i)}}(T\backslash \{i\})]={\displaystyle \sum _{i:n_i>1}}{n}_i(n-1){\displaystyle \sum _T}{\mathbb{P}}_{\mathit{n}-{\mathit{e}}_i}(T)\mathbb{P}({𝒞}_{{𝒪}_{\mathit{n}}}|T)+{\displaystyle \sum _{i:n_i=1}}{\displaystyle \sum _{j:j\ne i}}{P}_{\mathit{\text{ji}}}{n}_j{\displaystyle \sum _{T\prime }}{\mathbb{P}}_{\mathit{n}-{\mathit{e}}_i+{\mathit{e}}_j}(T\prime )\mathbb{P}({𝒞}_{{𝒪}_{\mathit{n}}\backslash \{i\}}|T\prime ),$$

where the sum over T′ is taken over all rooted trees with vertex set 𝒪_n \ {i}. Hence, ∑_T ℙ(𝒞_𝒪n | T)ℙn(T) satisfies (8).

4.3. Connection to the coalescent

In this subsection, we motivate our urn process by drawing a connection to the coalescent. We then use this connection with the coalescent to provide an alternate proof of Theorem 5.

Let ℋ be a history of mutation and coalescence events on n labeled individuals, and let q(ℋ) be the probability of ℋ. Then we have

$$q(\mathit{n})={\displaystyle \sum _{\mathscr{H}\text{ consistent with }\mathit{n}}}q(\mathscr{H}).$$ (11)

It turns out that only histories with exactly |𝒪_n| − 1 mutations contribute to the leading order term of q(n); this is the observation also utilized in [12]. Furthermore, each history of choices in our urn process corresponds with a genealogical history of |𝒪_n| − 1 mutations. This provides the basic intuition for why the urn sampling scheme works.

We start by providing a modified coalescent that generates a history ℋ that is consistent with n and has exactly |𝒪n| − 1 mutations. We then show that this modified coalescent is equivalent to our urn sampling process. Finally, we prove Theorem 5 by relating the modified coalescent with Kingman’s coalescent.

Consider the following modified coalescent process on our sample:

1. Select allele i with probability m_i/m, where m is our current configuration of alleles.

2. If m_i > 1, choose a random pair in allele i to coalesce (so m is replaced with m − e_i).

3. If m_i = 1, have the last individual of allele i mutate to allele j with probability m_j/(m − 1) (so m is replaced with m − e_i + e_j).

4. Repeat steps 1 to 3 until all individuals have coalesced.

It should be clear that the modified coalescent only generates histories with exactly |𝒪_n| − 1 mutations, since each mutation kills an allele permanently.

If we take an unordered view of our sample, then the modified coalescent is equivalent to the urn process, for they have the same initial configuration and transition probabilities between configurations. In particular, when m_i > 1 we move from m to m − e_i with probability m_i/m, and when m_i = 1 we move from m to m − e_i + e_j with probability m_j/(m)_2↓. We generate trees on 𝒪_n by drawing an edge (j → i) whenever we make a transition from m to m − e_i + e_j, i.e. whenever there is a mutation from i to j.

We now give a proof of Theorem 5, using the modified coalescent in place of the urn process:

Alternative proof of Theorem 5. Let ℋ be a coalescent history with exactly M mutations. Running time backwards from the present, we suppose that the ith mutation was from allele u_i to allele υ_i, and that the most recent common ancestor has allele ρ. We further suppose that J_i is the total number of lineages at the time of the ith mutation. Then we have that

$$q(\mathscr{H})={\pi }_{\rho }\left({\displaystyle \prod _{i=1}^M}{P}_{{\upsilon }_i{u}_i}\right)\frac{{\theta }^M}{{\displaystyle {\prod }_{i=1}^M}{J}_i(\theta +J_i-1)}\frac{2^{n-1}}{n!{(\theta +n-1)}_{(n-1)\downarrow }},$$

since the ith coalescence contributes probability $\frac{n-i}{n-i+\theta }{\left(\begin{array}{c}\hfill n-i+1\hfill \\ \hfill 2\hfill \end{array}\right)}^{-1}=\frac{2}{(n-i+1)(n-i+\theta )}$, and the ith mutation contributes probability $\frac{\theta P_{{\upsilon }_i{u}_i}}{J_i(J_i-1+\theta )}$.

Now, observe that

$$Q(\mathscr{H})\equiv \underset{\theta \to 0}{\text{lim}}\frac{q(\mathscr{H})}{{\theta }^M}={\pi }_{\rho }\left({\displaystyle \prod _{i=1}^M}{P}_{{\upsilon }_i{u}_i}\right)\frac{2^{n-1}}{n!(n-1)!{\displaystyle {\prod }_{i=1}^M}{J}_i(J_i-1)}.$$ (12)

Therefore, the Taylor series for q(ℋ) has leading power θ^M, with coefficient Q(ℋ).

Hence by (11), the Taylor series for q(n) has leading power θ^|𝒪_n|−1, and its leading coefficient is given by the sum of all Q(ℋ) such that ℋ is consistent with n and has |𝒪_n| − 1 mutations.

For such an ℋ, let ℙ_n(ℋ) be the probability of generating ℋ under our modified coalescent. Then we have that

$${\mathbb{P}}_{\mathit{n}}(\mathscr{H})=\frac{2^{n-1}}{n!{\displaystyle {\prod }_{k=1}^{|{𝒪}_{\mathit{n}}|}}(n_k-1)!{\displaystyle {\prod }_{i=1}^{|{𝒪}_{\mathit{n}}|-1}}{J}_i(J_i-1)}.$$ (13)

To see this, note that if our current sample is m, the probability that the next event is a coalescence on allele i with m_i > 1 is

$$\frac{m_i}{m}\frac{2}{m_i(m_i-1)}=\frac{2}{m(m_i-1)},$$

and if m_i = 1, the probability that the next event is a mutation from allele i to allele j (where j ≠ i) is

$$\frac{m_j}{m(m-1)}.$$

Multiplying the probabilities of the mutation and coalescence events in ℋ, and noting that the numerator of each mutation term cancels with the denominator of a future coalescence term, yields the equation (13).

Combining (12) with (13) yields

$$Q(\mathscr{H})=\mathrm{\Lambda }(\mathit{n}){\pi }_{\rho }\left({\displaystyle \prod _{i=1}^{|{𝒪}_{\mathit{n}}|-1}}{P}_{{\upsilon }_i{u}_i}\right){\mathbb{P}}_{\mathit{n}}(\mathscr{H})$$

Now let 𝒯 (ℋ) be the resulting tree on 𝒪_n if we draw an edge (j → i) when allele i mutates to allele j. Then we have

$$Q(\mathit{n})={\displaystyle \sum _{\stackrel{\mathscr{H}\text{consistent with }\mathit{n}}{\mathscr{H}\text{ has }|{𝒪}_{\mathit{n}}|-1\text{ mutations}}}}Q(\mathscr{H})=\mathrm{\Lambda }(\mathit{n}){\displaystyle \sum _T}{\pi }_{\rho (T)}\left({\displaystyle \prod _{(j\to i)\in T}}{P}_{\mathit{\text{ji}}}\right)\left({\displaystyle \prod _{\mathscr{H}:𝒯(\mathscr{H})=T}}{\mathbb{P}}_{\mathit{n}}(\mathscr{H})\right)=\mathrm{\Lambda }(\mathit{n}){\displaystyle \sum _T}{\pi }_{\rho (T)}\left({\displaystyle \prod _{(j\to i)\in T}}{P}_{\mathit{\text{ji}}}\right){\mathbb{P}}_{\mathit{n}}(T)=\mathrm{\Lambda }(\mathit{n}){\displaystyle \sum _T}{f}_{\mathit{P}}(T){\mathbb{P}}_{\mathit{n}}(T),$$

and hence

$$R(\mathit{n})={\displaystyle \sum _T}{f}_{\mathit{P}}(T){\mathbb{P}}_{\mathit{n}}(T),$$

as needed.

4.4. A martingale property

Here, we prove Theorem 1 and Corollary 3 by using martingales to compute ℙ_n(T) for 𝒪_n = {a, b}, and for 𝒪_n = {a, b, c} when P is reversible when restricted to 𝒪_n. We run time as follows: whenever we remove a ball in the urn process, count this as one time step. If in doing so, we kill a color, count the adding of another ball as a separate time step.

Let ℱ_t be the σ-algebra generated by all sequences of choices up to time t. Let X_t be the proportion of balls that have color a at time t; so X₀ = n_a/n. It is easy to check that {X_t} is a martingale with respect to {ℱ_t}: Suppose that m is the remaining sample after time t − 1, and we are deleting a ball at time t. Then,

$$𝔼[X_t|{ℱ}_{t-1}]=\frac{m_a}{m}\frac{m_a-1}{m-1}+{\displaystyle \sum _{i\ne a}}\frac{m_i}{m}\frac{m_a}{m-1}=\frac{m_a}{m}=X_{t-1}.$$

On the other hand, if we are adding a ball at time t, then

$$𝔼[X_t|{ℱ}_{t-1}]=\frac{m_a}{m}\frac{m_a+1}{m+1}+{\displaystyle \sum _{i\ne a}}\frac{m_i}{m}\frac{m_a}{m+1}=\frac{m_a}{m}=X_{t-1}.$$

So, {(X_t, ℱ_t), t ≥ 0} is a martingale.

Proof of Theorem 1. Suppose 𝒪_n = {a, b}. Let T be the tree whose vertex set is 𝒪_n, with a being the root. Let τ be the the first time we kill a color. Noting that τ is a stopping time, we obtain

$${\mathbb{P}}_{\mathit{n}}(T)=𝔼[{\mathbb{P}}_{\mathit{n}}(T|{ℱ}_{\tau })]=𝔼[𝕀(\text{Color }a\text{ is the last remaining at time}\tau )]=𝔼[X_{\tau }]=𝔼[X_0]=\frac{n_a}{n}.$$

Therefore, by Theorem 5,

$$Q(\mathit{n})=\mathrm{\Lambda }(\mathit{n})\left(\frac{n_a}{n}{\pi }_a{P}_{\mathit{\text{ab}}}+\frac{n_b}{n}{\pi }_b{P}_{\mathit{\text{ba}}}\right).$$

Proof of Corollary 3. Suppose 𝒪_n = {a, b, c} and P is reversible when restricted to 𝒪_n. Note that ℙ (𝒞_𝒪n | T) does not depend on how T is rooted, for by reversibility we can move the root around by

$$\begin{array}{cc}\hfill {\pi }_{\rho }{P}_{\rho k}={\pi }_k{P}_{k\rho },\hfill & \hfill \forall k\in {𝒪}_{\mathit{n}},k\ne \rho .\hfill \end{array}$$

Therefore, we redefine ℙ_n(T) to be the probability of drawing the undirected tree T. We still have R(n) = ∑_T ℙ (𝒞_𝒪n | T)ℙ_n(T), but now the sum is taken over undirected T. Now let T be the tree on {a, b, c} whose interior vertex is a. We draw T if and only if a is chosen as the parent of the first color that we kill. So, letting τ be the first killing time and noting X_τ = ℙ_n(T | ℱ_τ), we have

$${\mathbb{P}}_{\mathit{n}}(T)=𝔼[{\mathbb{P}}_{\mathit{n}}(T|{ℱ}_{\tau })]=𝔼[X_{\tau }]=𝔼[X_0]=\frac{n_a}{n}.$$

Therefore, by Theorem 5,

$$Q(\mathit{n})=\mathrm{\Lambda }(\mathit{n})\left(\frac{n_a}{n}{\pi }_a{P}_{\mathit{\text{ab}}}{P}_{\mathit{\text{ac}}}+\frac{n_b}{n}{\pi }_b{P}_{\mathit{\text{ba}}}{P}_{\mathit{\text{bc}}}+\frac{n_c}{n}{\pi }_c{P}_{\mathit{\text{ca}}}{P}_{\mathit{\text{cb}}}\right).$$

4.5. A recursion for R(n)

In this section, we derive a recursion for R(n) which will be useful for deriving closed-form formulas for Q(n) when |𝒪_n| = 3, 4. Given a sample configuration n and a subsample m, define the expression $\left(\begin{array}{c}\hfill \mathit{n}\hfill \\ \hfill \mathit{m}\hfill \end{array}\right)$ as

$$\left(\begin{array}{c}\hfill \mathit{n}\hfill \\ \hfill \mathit{m}\hfill \end{array}\right)={\displaystyle \prod _{i\in {𝒪}_{\mathit{n}}}}\left(\begin{array}{c}\hfill n_i\hfill \\ \hfill m_i\hfill \end{array}\right).$$

The following proposition provides a recursion relating R(n) to R(m) where |𝒪_m| = |𝒪_n| − 1.

Proposition 6. Suppose P is irreducible when restricted to 𝒪_n and let θ^|𝒪_n|−1 Q(n) = θ^|𝒪_n|−1 Λ(n)R(n) denote the leading order term in the Taylor expansion (1) of q(n) about θ = 0. Then, R(n) for |𝒪_n| > 1 satisfies the recursion

$$R(\mathit{n})={\displaystyle \sum _{i,j\in {𝒪}_{\mathit{n}}:i\ne j}}{P}_{\mathit{\text{ji}}}{\displaystyle \sum _{\stackrel{\mathbf{0}\prec \mathit{m}⪯\mathit{n}:}{m_i=1}}}\frac{\left(\begin{array}{c}\hfill \mathit{n}\hfill \\ \hfill \mathit{m}\hfill \end{array}\right)}{\left(\begin{array}{c}\hfill n\hfill \\ \hfill m\hfill \end{array}\right)}\frac{m_{j}R(\mathit{m}-{\mathit{e}}_i+{\mathit{e}}_j)}{m(m-1)},$$ (14)

with boundary conditions

$$R(\mathit{n})={\pi }_a,$$ (15)

for all sample configurations n = n_ae_a, where a ∈ [K].

Proof of Proposition 6. We can derive this recursion from the urn process as follows. Let D_ij(m) be the event where the first killing replaces a ball of color i with a ball of color j, and where m is the (unordered) configuration immediately before this killing. Then for any event A,

$${\mathbb{P}}_{\mathit{n}}(A)={\displaystyle \sum _{i,j\ne i}}{\displaystyle \sum _{\stackrel{\mathbf{0}\prec \mathit{m}⪯\mathit{n}:}{m_i=1}}}{\mathbb{P}}_{\mathit{n}}(D_{\mathit{\text{ij}}}(\mathit{m})){\mathbb{P}}_{\mathit{n}}(A|D_{\mathit{\text{ij}}}(\mathit{m}))$$ (16)

where we use the fact that ℙ_n(D_ij(m)) = 0 if m_i ≠ 1 or m_j = 0 for any j ∈ 𝒪_n.

We compute ℙ_n(D_ij(m)) when m ≻ 0 and m_i = 1. The probability that m is the remaining configuration after n − m draws is

$$\frac{(n-m)!}{{\displaystyle {\prod }_k}(n_k-m_k)!}\frac{{\displaystyle {\prod }_k}{(n_k)}_{n_k-m_k\downarrow }}{{(n)}_{n-m\downarrow }}=\frac{\left(\begin{array}{c}\hfill \mathit{n}\hfill \\ \hfill \mathit{m}\hfill \end{array}\right)}{\left(\begin{array}{c}\hfill n\hfill \\ \hfill m\hfill \end{array}\right)}.$$

To see this, note that the first term is the number of ways we can make n − m draws that result in the configuration m, and the second term is the probability of each such sequence of draws.

When our current configuration is m with m_i = 1, the probability that on the next draw we replace the last ball of color i with a ball of color j is m_j/(m)_2↓. Hence we get that

$${\mathbb{P}}_{\mathit{n}}(D_{\mathit{\text{ij}}}(\mathit{m}))=\frac{\left(\begin{array}{c}\hfill \mathit{n}\hfill \\ \hfill \mathit{m}\hfill \end{array}\right)}{\left(\begin{array}{c}\hfill n\hfill \\ \hfill m\hfill \end{array}\right)}\frac{m_j}{m(m-1)}.$$

when m ≻ 0 and m_i = 1.

Plugging this into (16) yields

$${\mathbb{P}}_{\mathit{n}}(A)={\displaystyle \sum _{i,j\ne i}}{\displaystyle \sum _{\stackrel{\mathbf{0}\prec \mathit{m}⪯\mathit{n}:}{m_i=1}}}\frac{\left(\begin{array}{c}\hfill \mathit{n}\hfill \\ \hfill \mathit{m}\hfill \end{array}\right)}{\left(\begin{array}{c}\hfill n\hfill \\ \hfill m\hfill \end{array}\right)}\frac{m_j}{m(m-1)}{\mathbb{P}}_{\mathit{n}}(A|D_{\mathit{\text{ij}}}(\mathit{m})).$$ (17)

Now recall from (7) that R(n) = ℙ_n(𝒞_𝒪n). That is, R(n) is the probability that we assign the original labels to all alleles, if we use the urn process to generate a tree on 𝒪_n and then use the tree to assign new labels to the alleles. Note that

$$\mathbb{P}({𝒞}_{{𝒪}_{\mathit{n}}}|D_{\mathit{\text{ij}}}(\mathit{m}))=P_{\mathit{\text{ji}}}{\mathbb{P}}_{\mathit{m}-{\mathit{e}}_i+{\mathit{e}}_j}({𝒞}_{{𝒪}_{\mathit{n}}\backslash \{i\}})=P_{\mathit{\text{ji}}}R(\mathit{m}-{\mathit{e}}_i+{\mathit{e}}_j),$$

since we need to use the urn process with sample m − e_i+e_j to correctly relabel 𝒪_n\{i}, and then assign the correct label to {i} with probability P_ji. Plugging this into (17) with A = 𝒞_𝒪n yields the desired recursion,

$$R(\mathit{n})={\displaystyle \sum _{i,j\ne i}}{P}_{\mathit{\text{ji}}}{\displaystyle \sum _{\stackrel{\mathbf{0}\prec \mathit{m}⪯\mathit{n}:}{m_i=1}}}\frac{\left(\begin{array}{c}\hfill \mathit{n}\hfill \\ \hfill \mathit{m}\hfill \end{array}\right)}{\left(\begin{array}{c}\hfill n\hfill \\ \hfill m\hfill \end{array}\right)}\frac{m_{j}R(\mathit{m}-{\mathit{e}}_i+{\mathit{e}}_j)}{m(m-1)}.$$

In the next two subsections, we use the recursion in Proposition 6 to provide proofs of Theorem 2 and Theorem 4.

4.6. Proof of Theorem 2 (|𝒪_n| = 3)

For |𝒪_n| = 3, the following expression for R(n) can be derived using Proposition 6:

$$R(\mathit{n})={\displaystyle \sum _{i,j\ne i}}{P}_{\mathit{\text{ji}}}{\displaystyle \sum _{\stackrel{\mathbf{0}\prec \mathit{m}⪯\mathit{n}:}{m_i=1}}}\frac{\left(\begin{array}{c}\hfill \mathit{n}\hfill \\ \hfill \mathit{m}\hfill \end{array}\right)}{\left(\begin{array}{c}\hfill n\hfill \\ \hfill m\hfill \end{array}\right)}\frac{m_{j}R(\mathit{m}-{\mathit{e}}_i+{\mathit{e}}_j)}{m(m-1)}={\displaystyle \sum _{i,j\ne i}}{P}_{\mathit{\text{ji}}}{\displaystyle \sum _{\stackrel{\mathbf{0}\prec \mathit{m}⪯\mathit{n}:}{m_i=1}}}\frac{\left(\begin{array}{c}\hfill \mathit{n}\hfill \\ \hfill \mathit{m}\hfill \end{array}\right)}{\left(\begin{array}{c}\hfill n\hfill \\ \hfill m\hfill \end{array}\right)}\frac{m_j}{m(m-1)}{\displaystyle \sum _{\stackrel{k,l:}{l\ne k\text{ and }k,l\ne i}}}\frac{m_k+{\delta }_{j,k}}{m}{\pi }_k{P}_{\mathit{\text{kl}}}={\displaystyle \sum _{i,j\ne i}}{P}_{\mathit{\text{ji}}}{\displaystyle \sum _{\stackrel{\mathbf{0}\prec \mathit{m}⪯\mathit{n}:}{m_i=1}}}\{\frac{\left(\begin{array}{c}\hfill \mathit{n}\hfill \\ \hfill \mathit{m}\hfill \end{array}\right)}{\left(\begin{array}{c}\hfill n\hfill \\ \hfill m\hfill \end{array}\right)}\frac{1}{m^2(m-1)}\times [{\displaystyle \sum _{l:l\ne i,j}}{m}_j(m_j+1){\pi }_j{P}_{\mathit{\text{jl}}}+{\displaystyle \sum _{k:k\ne i,j}}{m}_j{m}_k{\pi }_k{P}_{\mathit{\text{kj}}}\left]\right\}={\displaystyle \sum _{i,j\ne i}}{P}_{\mathit{\text{ji}}}{\displaystyle \sum _{m=3}^n}{\displaystyle \sum _{\stackrel{\mathbf{0}\prec \mathit{m}⪯\mathit{n}:}{m_i=1,|\mathit{m}|=m}}}\{\frac{\left(\begin{array}{c}\hfill \mathit{n}\hfill \\ \hfill \mathit{m}\hfill \end{array}\right)}{\left(\begin{array}{c}\hfill n\hfill \\ \hfill m\hfill \end{array}\right)}\frac{1}{m^2(m-1)}\times [{\displaystyle \sum _{k:k\ne i,j}}{m}_j(m_j+1){\pi }_j{P}_{\mathit{\text{jk}}}+{\displaystyle \sum _{k:k\ne i,j}}{m}_j{m}_k{\pi }_k{P}_{\mathit{\text{kj}}}\left]\right\}={\displaystyle \sum _{i,j,k\text{ distinct}}}{\displaystyle \sum _{m=3}^n}{\displaystyle \sum _{\stackrel{\mathbf{0}\prec \mathit{m}⪯\mathit{n}:}{m_i=1,|\mathit{m}|=m}}}\frac{\left(\begin{array}{c}\hfill \mathit{n}\hfill \\ \hfill \mathit{m}\hfill \end{array}\right)}{\left(\begin{array}{c}\hfill n\hfill \\ \hfill m\hfill \end{array}\right)}\frac{{\pi }_j{P}_{\mathit{\text{ji}}}{P}_{\mathit{\text{jk}}}{m}_j(m_j+1)+{\pi }_k{P}_{\mathit{\text{kj}}}{P}_{\mathit{\text{ji}}}{m}_j{m}_k}{m^2(m-1)},$$ (18)

where in the second equality, the formula from Theorem 1 is used, noting that |𝒪_m−e_i+e_j| = 2. If we define the quantities α(n, i, j, k) and β(n, i, j, k) as

$$\alpha (\mathit{n},i,j,k)={\displaystyle \sum _{m=3}^n}\frac{1}{m^2(m-1)}{\displaystyle \sum _{\stackrel{\mathbf{0}\prec \mathit{m}⪯\mathit{n}:}{m_i=1,|\mathit{m}|=m}}}\frac{\left(\begin{array}{c}\hfill \mathit{n}\hfill \\ \hfill \mathit{m}\hfill \end{array}\right)}{\left(\begin{array}{c}\hfill n\hfill \\ \hfill m\hfill \end{array}\right)}{m}_j(m_j+1),$$ (19)

and

$$\beta (\mathit{n},i,j,k)={\displaystyle \sum _{m=3}^n}\frac{1}{m^2(m-1)}{\displaystyle \sum _{\stackrel{\mathbf{0}\prec \mathit{m}⪯\mathit{n}:}{m_i=1,|\mathit{m}|=m}}}\frac{\left(\begin{array}{c}\hfill \mathit{n}\hfill \\ \hfill \mathit{m}\hfill \end{array}\right)}{\left(\begin{array}{c}\hfill n\hfill \\ \hfill m\hfill \end{array}\right)}{m}_j{m}_k,$$

then (18) can be rewritten as

$$R(\mathit{n})={\displaystyle \sum _{i,j,k\text{ distinct}}}{\pi }_j{P}_{\mathit{\text{ji}}}{P}_{\mathit{\text{jk}}}\alpha (\mathit{n},i,j,k)+{\displaystyle \sum _{i,j,k\text{ distinct}}}{\pi }_k{P}_{\mathit{\text{kj}}}{P}_{\mathit{\text{ji}}}\beta (\mathit{n},i,j,k).$$ (20)

Now consider α(n, i, j, k) defined by (19). We can remove the restriction in the inner sum that m_i = 1 by defining m′ = m − e_i, and so |m′| = m − 1. Also, since j ≠ i in (20), $m_j^{\prime }=m_j$. Making this change of variables from m to m′ in the inner sum of (19), we get

$${\displaystyle \sum _{\stackrel{\mathbf{0}\prec \mathit{m}⪯\mathit{n}:}{m_i=1,|\mathit{m}|=m}}}\frac{\left(\begin{array}{c}\hfill \mathit{n}\hfill \\ \hfill \mathit{m}\hfill \end{array}\right)}{\left(\begin{array}{c}\hfill n\hfill \\ \hfill m\hfill \end{array}\right)}{m}_j(m_j+1)=\frac{\left(\begin{array}{c}\hfill n-n_i\hfill \\ \hfill m-1\hfill \end{array}\right)}{\left(\begin{array}{c}\hfill n\hfill \\ \hfill m\hfill \end{array}\right)}{n}_i{\displaystyle \sum _{\stackrel{\mathbf{0}\prec \mathit{m}\prime ⪯\mathit{n}-n_i{\mathit{e}}_i:}{|\mathit{m}\prime |=m-1}}}\frac{\left(\begin{array}{c}\hfill \mathit{n}-n_i{\mathit{e}}_i\hfill \\ \hfill \mathit{m}\prime \hfill \end{array}\right)}{\left(\begin{array}{c}\hfill n-n_i\hfill \\ \hfill m-1\hfill \end{array}\right)}{m}_j^{\prime }(m_j^{\prime }+1).$$ (21)

Using identity (34) in Fact 5 of the Appendix, the summation over m′ in (21) can be written as

$${\displaystyle \sum _{\stackrel{\mathbf{0}\prec \mathit{m}\prime ⪯\mathit{n}-n_i{\mathit{e}}_i:}{|\mathit{m}\prime |=m-1}}}\frac{\left(\begin{array}{c}\hfill \mathit{n}-n_i{\mathit{e}}_i\hfill \\ \hfill \mathit{m}\prime \hfill \end{array}\right)}{\left(\begin{array}{c}\hfill n-n_i\hfill \\ \hfill m-1\hfill \end{array}\right)}{m}_j^{\prime }(m_j^{\prime }+1)={\displaystyle \sum _{\stackrel{T\subseteq [L]:}{i,j\notin T}}}{(-1)}^{|T|}\left[\frac{{(n_j)}_{2\downarrow }{(m-1)}_{2\downarrow }}{{(n-n_i-n_T)}_{2\downarrow }}+\frac{2n_j(m-1)}{n-n_i-n_T}\right]\frac{\left(\begin{array}{c}\hfill n-n_i-n_T\hfill \\ \hfill m-1\hfill \end{array}\right)}{\left(\begin{array}{c}\hfill n-n_i\hfill \\ \hfill m-1\hfill \end{array}\right)}$$ (22)

The only sets T satisfying the conditions in the summation in (22) are T = ∅ and T = {k}. Hence, substituting (21) and (22) in (19), we have

$$\alpha (\mathit{n},i,j,k)={\displaystyle \sum _{m=3}^n}\frac{1}{m^2(m-1)}{\displaystyle \sum _{\stackrel{\mathbf{0}\prec \mathit{m}⪯\mathit{n}:}{m_i=1,|\mathit{m}|=m}}}\frac{\left(\begin{array}{c}\hfill \mathit{n}\hfill \\ \hfill \mathit{m}\hfill \end{array}\right)}{\left(\begin{array}{c}\hfill n\hfill \\ \hfill m\hfill \end{array}\right)}{m}_j(m_j+1)={\displaystyle \sum _{m=3}^n}\frac{1}{m^2(m-1)}\frac{\left(\begin{array}{c}\hfill n-n_i\hfill \\ \hfill m-1\hfill \end{array}\right)}{\left(\begin{array}{c}\hfill n\hfill \\ \hfill m\hfill \end{array}\right)}{n}_i{\displaystyle \sum _{\stackrel{\mathbf{0}\prec \mathit{m}\prime ⪯\mathit{n}-n_i{\mathit{e}}_i:}{|\mathit{m}\prime |=m-1}}}\frac{\left(\begin{array}{c}\hfill \mathit{n}-n_i{\mathit{e}}_i\hfill \\ \hfill \mathit{m}\prime \hfill \end{array}\right)}{\left(\begin{array}{c}\hfill n-n_i\hfill \\ \hfill m-1\hfill \end{array}\right)}{m}_j^{\prime }(m_j^{\prime }+1)={\displaystyle \sum _{m=3}^n}\frac{n_i\left(\begin{array}{c}\hfill n-n_i\hfill \\ \hfill m-1\hfill \end{array}\right)}{m^2(m-1)\left(\begin{array}{c}\hfill n\hfill \\ \hfill m\hfill \end{array}\right)}\left\{\frac{\left(\begin{array}{c}\hfill n_j+n_k\hfill \\ \hfill m-1\hfill \end{array}\right)}{\left(\begin{array}{c}\hfill n-n_i\hfill \\ \hfill m-1\hfill \end{array}\right)}\left[\frac{{(n_j)}_{2\downarrow }}{{(n_j+n_k)}_{2\downarrow }}{(m-1)}_{2\downarrow }+2\frac{n_j(m-1)}{n_j+n_k}\right]-\frac{\left(\begin{array}{c}\hfill n_j\hfill \\ \hfill m-1\hfill \end{array}\right)}{\left(\begin{array}{c}\hfill n-n_i\hfill \\ \hfill m-1\hfill \end{array}\right)}\left[\frac{{(n_j)}_{2\downarrow }}{{(n_j)}_{2\downarrow }}{(m-1)}_{2\downarrow }+2\frac{n_j}{n_j}(m-1)\right]\right\}={\displaystyle \sum _{m=3}^n}\frac{n_i}{m^2(m-1)}\left\{\frac{\left(\begin{array}{c}\hfill n_j+n_k\hfill \\ \hfill m-1\hfill \end{array}\right)}{\left(\begin{array}{c}\hfill n\hfill \\ \hfill m\hfill \end{array}\right)}\left[\frac{{(n_j)}_{2\downarrow }}{{(n_j+n_k)}_{2\downarrow }}{(m-1)}_{2\downarrow }+2\frac{n_j(m-1)}{n_j+n_k}\right]-\frac{\left(\begin{array}{c}\hfill n_j\hfill \\ \hfill m-1\hfill \end{array}\right)}{\left(\begin{array}{c}\hfill n\hfill \\ \hfill m\hfill \end{array}\right)}m(m-1)\right\}={\displaystyle \sum _{m=1}^n}\frac{n_i}{n}\left\{\frac{\left(\begin{array}{c}\hfill n_j+n_k\hfill \\ \hfill m\hfill \end{array}\right)}{\left(\begin{array}{c}\hfill n-1\hfill \\ \hfill m\hfill \end{array}\right)}\left[\frac{{(n_j)}_{2\downarrow }}{{(n_j+n_k)}_{2\downarrow }}\frac{m-1}{m+1}+2\frac{n_j}{n_j+n_k}\frac{1}{m+1}\right]-\frac{\left(\begin{array}{c}\hfill n_j\hfill \\ \hfill m\hfill \end{array}\right)}{\left(\begin{array}{c}\hfill n-1\hfill \\ \hfill m\hfill \end{array}\right)}\right\}.$$ (23)

Applying Facts 1 and 3 in the Appendix to (23) yields

$$\alpha (\mathit{n},i,j,k)={\displaystyle \sum _{m=1}^n}\frac{n_i}{n}\left[\frac{{(n_j)}_{2\downarrow }}{{(n_j+n_k)}_{2\downarrow }}\frac{\left(\begin{array}{c}\hfill n_j+n_k\hfill \\ \hfill m\hfill \end{array}\right)}{\left(\begin{array}{c}\hfill n-1\hfill \\ \hfill m\hfill \end{array}\right)}-\frac{\left(\begin{array}{c}\hfill n_j\hfill \\ \hfill m\hfill \end{array}\right)}{\left(\begin{array}{c}\hfill n-1\hfill \\ \hfill m\hfill \end{array}\right)}+\frac{2n_j{n}_k}{{(n_j+n_k)}_{2\downarrow }}\frac{\left(\begin{array}{c}\hfill n_j+n_k\hfill \\ \hfill m\hfill \end{array}\right)}{\left(\begin{array}{c}\hfill n-1\hfill \\ \hfill m\hfill \end{array}\right)}\frac{1}{m+1}\right]=\frac{n_i}{n}\left\{\frac{{(n_j)}_{2\downarrow }}{{(n_j+n_k)}_{2\downarrow }}\frac{n_j+n_k}{n_i}-\frac{n_j}{n_i+n_k}+2\frac{n_j{n}_k}{{(n_i+n_k)}_{2\downarrow }}\left[\frac{n}{n_j+n_k+1}(H_n-H_{n_i-1})-1\right]\right\}=\frac{{(n_j)}_{2\downarrow }}{n(n_j+n_k-1)}-\frac{n_i{n}_j}{n(n_i+n_k)}-2\frac{n_i{n}_j{n}_k}{n{(n_j+n_k)}_{2\downarrow }}+2\frac{n_i{n}_j{n}_k}{{(n_j+n_k+1)}_{3\downarrow }}(H_n-H_{n_i-1}).$$ (24)

Following a similar line of computation as above, we can find a closed-form expression for β(n, i, j, k) as follows:

$$\beta (\mathit{n},i,j,k)={\displaystyle \sum _{m=3}^n}\frac{1}{m^2(m-1)}{\displaystyle \sum _{\stackrel{\mathbf{0}\prec \mathit{m}⪯\mathit{n}:}{m_i=1,|\mathit{m}|=m}}}\frac{\left(\begin{array}{c}\hfill \mathit{n}\hfill \\ \hfill \mathit{m}\hfill \end{array}\right)}{\left(\begin{array}{c}\hfill n\hfill \\ \hfill m\hfill \end{array}\right)}{m}_j{m}_k={\displaystyle \sum _{m=3}^n}\frac{1}{m^2(m-1)}\frac{\left(\begin{array}{c}\hfill n-n_i\hfill \\ \hfill m-1\hfill \end{array}\right)}{\left(\begin{array}{c}\hfill n\hfill \\ \hfill m\hfill \end{array}\right)}{n}_i{\displaystyle \sum _{\stackrel{\mathbf{0}\prec \mathit{m}\prime ⪯\mathit{n}-n_i{\mathit{e}}_i:}{|\mathit{m}\prime |=m-1}}}\frac{\left(\begin{array}{c}\hfill \mathit{n}-n_i{\mathit{e}}_i\hfill \\ \hfill \mathit{m}\prime \hfill \end{array}\right)}{\left(\begin{array}{c}\hfill n-n_i\hfill \\ \hfill m-1\hfill \end{array}\right)}{m}_j^{\prime }{m}_k^{\prime }={\displaystyle \sum _{m=3}^n}\frac{1}{m^2(m-1)}\frac{\left(\begin{array}{c}\hfill n_j+n_k\hfill \\ \hfill m-1\hfill \end{array}\right)}{\left(\begin{array}{c}\hfill n\hfill \\ \hfill m\hfill \end{array}\right)}{n}_i\frac{n_j{n}_k}{{(\mathit{\text{nj}}+n_k)}_{2\downarrow }}{(m-1)}_{2\downarrow }={\displaystyle \sum _{m=1}^n}\frac{n_i}{n}\frac{n_j{n}_k}{{(\mathit{\text{nj}}+n_k)}_{2\downarrow }}\frac{\left(\begin{array}{c}\hfill n_j+n_k\hfill \\ \hfill m\hfill \end{array}\right)}{\left(\begin{array}{c}\hfill n\hfill \\ \hfill m\hfill \end{array}\right)}\left(1-\frac{2}{m+1}\right)=\frac{n_i}{n}\frac{n_j{n}_k}{{(\mathit{\text{nj}}+n_k)}_{2\downarrow }}\left\{\frac{n_j+n_k}{n_i}-2\left[\frac{n}{n_j+n_k+1}(H_n-H_{n_i-1})-1\right]\right\}=\frac{n_j{n}_k}{n(n_j+n_k-1)}+2\frac{n_i{n}_j{n}_k}{n{(n_j+n_k)}_{2\downarrow }}-2\frac{n_i{n}_j{n}_k}{{(n_j+n_k+1)}_{3\downarrow }}(H_n-H_{n_i-1}),$$ (25)

where the second equality above is the same change of variables from m to m′ = m − e_i as in the α(n, i, j, k) term. The third equality follows from identity (35) in Fact 5, and the second to last equality follows from Facts 1 and 3. Substituting (24) and (25) into (20), and using (5) gives

$$Q(\mathit{n})=\mathrm{\Lambda }(\mathit{n}){\displaystyle \sum _{i,j,k\text{ distinct}}}[{\pi }_i{P}_{\mathit{\text{ji}}}{P}_{\mathit{\text{jk}}}\alpha (\mathit{n},i,j,k)+{\pi }_k{P}_{\mathit{\text{kj}}}{P}_{\mathit{\text{ji}}}\beta (\mathit{n},i,j,k)]=\mathrm{\Lambda }(\mathit{n}){\displaystyle \sum _{i,j,k\text{ distinct}}}\{{\pi }_i{P}_{\mathit{\text{ji}}}{P}_{\mathit{\text{jk }}}[\frac{{(n_j)}_{2\downarrow }}{n(n_j+n_k-1)}-\frac{n_i{n}_j}{n(n_i+n_k)}-2\frac{n_i{n}_j{n}_k}{n{(n_j+n_k)}_{2\downarrow }}+2\frac{n_i{n}_j{n}_k}{{(n_j+n_k+1)}_{3\downarrow }}(H_n-H_{n_i-1})]+{\pi }_k{P}_{\mathit{\text{kj}}}{P}_{\mathit{\text{ji }}}[\frac{n_j{n}_k}{n(n_j+n_k-1)}+2\frac{n_i{n}_j{n}_k}{n{(n_j+n_k)}_{2\downarrow }}-2\frac{n_i{n}_j{n}_k}{{(n_j+n_k+1)}_{3\downarrow }}(H_n-H_{n_i-1})]\}.$$ (26)

Note that if P is reversible when restricted to the observed alleles 𝒪_n, then (26) simplifies to the expression given in Corollary 3.

4.7. Proof of Theorem 4 (|𝒪_n| = 4)

Using Corollary 3, we first note the following alternate expression for R(n) when |𝒪_n| = 3 and P is reversible restricted to the observed alleles:

$$R(\mathit{n})={\displaystyle \sum _{i,j,k\text{ distinct}}}\frac{n_i}{n}{\pi }_i\frac{P_{\mathit{\text{ij}}}{P}_{\mathit{\text{ik}}}}{2}.$$ (27)

Suppose |𝒪_n| = 4 and assume that P is reversible restricted to the observed alleles 𝒪_n. Then using Proposition 6, we obtain

$$R(\mathit{n})={\displaystyle \sum _{l,h\ne l}}{P}_{\mathit{\text{hl}}}{\displaystyle \sum _{\stackrel{\mathbf{0}\prec \mathit{m}⪯\mathit{n}:}{m_l=1}}}\frac{\left(\begin{array}{c}\hfill \mathit{n}\hfill \\ \hfill \mathit{m}\hfill \end{array}\right)}{\left(\begin{array}{c}\hfill n\hfill \\ \hfill m\hfill \end{array}\right)}\frac{m_{h}R(\mathit{m}-{\mathit{e}}_l+{\mathit{e}}_h)}{m(m-1)}={\displaystyle \sum _{l,h\ne l}}{P}_{\mathit{\text{hl}}}{\displaystyle \sum _{\stackrel{\mathbf{0}\prec \mathit{m}⪯\mathit{n}:}{m_l=1}}}\frac{\left(\begin{array}{c}\hfill \mathit{n}\hfill \\ \hfill \mathit{m}\hfill \end{array}\right)}{\left(\begin{array}{c}\hfill n\hfill \\ \hfill m\hfill \end{array}\right)}\frac{m_h}{m(m-1)}{\displaystyle \sum _{\stackrel{i,j,k\text{ distinct},}{i,j,k\ne l}}}\frac{m_i+{\delta }_{i,h}}{m}{\pi }_i\frac{P_{\mathit{\text{ij}}}{P}_{\mathit{\text{ik}}}}{2}={\displaystyle \sum _{i,j,k,l\text{ distinct}}}\frac{1}{2}{\pi }_i{P}_{\mathit{\text{ij}}}{P}_{\mathit{\text{ik}}}{P}_{\mathit{\text{il}}}{\displaystyle \sum _{\stackrel{\mathbf{0}\prec \mathit{m}⪯\mathit{n}}{m_l=1}}}\frac{\left(\begin{array}{c}\hfill \mathit{n}\hfill \\ \hfill \mathit{m}\hfill \end{array}\right)}{\left(\begin{array}{c}\hfill n\hfill \\ \hfill m\hfill \end{array}\right)}\frac{m_i(m_i+1)}{m^2(m-1)}+{\displaystyle \sum _{i,j,k,l\text{ distinct}}}{\pi }_i{P}_{\mathit{\text{ij}}}{P}_{\mathit{\text{ik}}}{P}_{\mathit{\text{jl}}}{\displaystyle \sum _{\underset{m_l=1}{\mathbf{0}\prec \mathit{m}⪯\mathit{n}}}}\frac{\left(\begin{array}{c}\hfill \mathit{n}\hfill \\ \hfill \mathit{m}\hfill \end{array}\right)}{\left(\begin{array}{c}\hfill n\hfill \\ \hfill m\hfill \end{array}\right)}\frac{m_i{m}_j}{m^2(m-1)},$$ (28)

where the second equality follows from using (27) since P is reversible when restricted to the alleles {i, j, k} ⊂ 𝒪_n. Similar to the proof in Section 4.6, if we define quantities ζ(n, i, j, k, l) and δ(n, i, j, k, l) as

$$\zeta (\mathit{n},i,j,k,l)={\displaystyle \sum _{m=4}^n}\frac{1}{m^2(m-1)}{\displaystyle \sum _{\underset{m_l=1,|\mathit{m}|=m}{\mathbf{0}\prec \mathit{m}⪯\mathit{n}:}}}\frac{\left(\begin{array}{c}\hfill \mathit{n}\hfill \\ \hfill \mathit{m}\hfill \end{array}\right)}{(\begin{array}{c}\hfill n\hfill \\ \hfill m\hfill \end{array})}{m}_i(m_i+1),$$

and

$$\delta (\mathit{n},i,j,k,l)={\displaystyle \sum _{m=4}^n}\frac{1}{m^2(m-1)}{\displaystyle \sum _{\underset{m_l=1,|\mathit{m}|=m}{\mathbf{0}\prec \mathit{m}⪯\mathit{n}:}}}\frac{\left(\begin{array}{c}\hfill \mathit{n}\hfill \\ \hfill \mathit{m}\hfill \end{array}\right)}{(\begin{array}{c}\hfill n\hfill \\ \hfill m\hfill \end{array})}{m}_i{m}_j,$$

then, using (5) and (28), we obtain the following expression for Q(n) = Λ(n)R(n):

$$Q(\mathit{n})=\mathrm{\Lambda }(\mathit{n}){\displaystyle \sum _{i,j,k,l\text{ distinct}}}[{\pi }_i{P}_{\mathit{\text{ij}}}{P}_{\mathit{\text{ik}}}{P}_{\mathit{\text{il}}}\frac{\zeta (\mathit{n},i,j,k,l)}{2}+{\pi }_i{P}_{\mathit{\text{ij}}}{P}_{\mathit{\text{ik}}}{P}_{\mathit{\text{jl}}}\delta (\mathit{n},i,j,k,l)].$$ (29)

By a very similar calculation to that in Section 4.6, using Facts 1 and 3, and identities (34) and (35) in Fact 5 of the Appendix, we obtain the following closed-form expressions for ζ(n, i, j, k, l) and δ(n, i, j, k, l):

$$\zeta (\mathit{n},i,j,k,l)=\frac{n_l}{n}\{\frac{n_i+n_j+n_k}{n_l}\frac{{(n_i)}_{2\downarrow }}{{(n_i+n_j+n_k)}_{2\downarrow }}+\frac{n_i}{n_j+n_k+n_l}+\frac{2n_i(n_j+n_k)}{{(n_i+n_j+n_k)}_{2\downarrow }}(\frac{n}{n_i+n_j+n_k+1}(H_n-H_{n_l-1})-1)-[\frac{n_i+n_j}{n_k+n_l}\frac{{(n_i)}_{2\downarrow }}{{(n_i+n_j)}_{2\downarrow }}+\frac{2n_i{n}_j}{{(n_i+n_j)}_{2\downarrow }}(\frac{n}{n_i+n_j+1}(H_n-H_{n_k+n_l-1})-1\left)\right]-[\frac{n_i+n_k}{n_j+n_l}\frac{{(n_i)}_{2\downarrow }}{{(n_i+n_k)}_{2\downarrow }}+\frac{2n_i{n}_k}{{(n_i+n_k)}_{2\downarrow }}(\frac{n}{n_i+n_k+1}(H_n-H_{n_j+n_l-1})-1\left)\right]\}.$$

and

$$\delta (\mathit{n},i,j,k,l)=\frac{n_l}{n}\left\{\frac{n_i+n_j+n_k}{n_l}\frac{n_i{n}_j}{{(n_i+n_j+n_k)}_{2\downarrow }}-\frac{2n_i{n}_j}{{(n_i+n_j+n_k)}_{2\downarrow }}(\frac{n}{n_i+n_j+n_k+1}(H_n-H_{n_l-1})-1)-\left[\frac{n_i+n_j}{n_k+n_l}\frac{n_i{n}_j}{{(n_i+n_j)}_{2\downarrow }}-\frac{2n_i{n}_j}{{(n_i+n_j)}_{2\downarrow }}(\frac{n}{n_i+n_j+1}(H_n-H_{n_k+n_l-1})-1)\right]\right\}.$$

Simplifying the expression for δ(n, i, j, k, l), we get the expression stated in Theorem 4. Observing that ζ(n, i, j, k, l) is symmetric in j and k, we see that for all i, j, k, and l distinct in 𝒪_n,

$$\frac{\zeta (\mathit{n},i,j,k,l)+\zeta (\mathit{n},i,k,j,l)}{2}=\gamma (\mathit{n},i,j,k,l)+\gamma (\mathit{n},i,k,j,l),$$

where γ(n, i, j, k, l) is given by:

$$\gamma (\mathit{n},i,j,k,l)=\frac{n_i}{n}\left\{\right[\frac{n_i-1}{2(n_i+n_j+n_k-1)}-\frac{2n_j{n}_l}{{(n_i+n_j+n_k)}_{2\downarrow }}]+\frac{n_l}{2(n_j+n_k+n_l)}-[\frac{n_l(n_i-1)}{(n_k+n_l)(n_i+n_j-1)}-\frac{2n_j{n}_l}{{(n_i+n_j)}_{2\downarrow }}]\}+\frac{2n_i{n}_j{n}_l}{{(n_i+n_j+n_k+1)}_{3\downarrow }}(H_n-H_{n_l-1})-\frac{2n_i{n}_j{n}_l}{{(n_i+n_j+1)}_{3\downarrow }}(H_n-H_{n_k+n_l-1}).$$

Using the fact that π_iP_ijP_ikP_il is also symmetric in j and k, we can then rewrite (29) as

$$Q(\mathit{n})=\mathrm{\Lambda }(\mathit{n}){\displaystyle \sum _{i,j,k,l\text{ distinct}}}[{\pi }_i{P}_{\mathit{\text{ij}}}{P}_{\mathit{\text{ik}}}{P}_{\mathit{\text{il}}}\gamma (\mathit{n},i,j,k,l)+{\pi }_i{P}_{\mathit{\text{ij}}}{P}_{\mathit{\text{ik}}}{P}_{\mathit{\text{jl}}}\delta (\mathit{n},i,j,k,l)].$$

5. Empirical study of accuracy

Here, we investigate the accuracy of approximating the sampling probability q(n) by using only the leading order term θ^|𝒪_n|−1 Q(n). In this study, we solve the recursion (2) numerically to obtain the true sampling probability q(n) for moderate sample sizes.

For a given sample n, define the approximate sampling probability, q_approx(n), by

$$q_{\text{approx}}(\mathit{n})={\theta }^{|{𝒪}_{\mathit{n}}|-1}Q(\mathit{n}).$$

We can then define the relative error, Err(n), of the approximation q_approx(n) from the true sampling probability q(n) as

$$\text{Err}(\mathit{n})=\frac{|q(\mathit{n})-q_{\text{approx}}(\mathit{n})|}{q(\mathit{n})}.$$

For a given sample size n, another natural measure of the approximation quality is the expected relative error under the distribution arising from the coalescent on samples of size n. Since q(n) is the probability of a particular ordered sample consistent with n, the probability p(n) of the unordered sample n, when sampling order is ignored, is given by

$$p(\mathit{n})=\left(\begin{array}{c}\hfill n\hfill \\ \hfill n_1,\dots ,n_K\hfill \end{array}\right)q(\mathit{n}).$$

We can then define the expected relative error for a sample size n by AvgErr(n), given by

$$\text{AvgErr}(n)={\displaystyle \sum _{\mathit{n}:|\mathit{n}|=n}}p(\mathit{n})\text{ Err}(\mathit{n})={\displaystyle \sum _{\mathit{n}:|\mathit{n}|=n}}\left(\begin{array}{c}\hfill n\hfill \\ \hfill n_1,\dots ,n_K\hfill \end{array}\right)|q(\mathit{n})-q_{\text{approx}}(\mathit{n})|.$$

We also define the worst-case relative error, WorstErr(n), for a given sample size n as the worse relative error among all samples of size n. Specifically,

$$\text{WorstErr}(n)=\underset{\mathit{n}:|\mathit{n}|=n}{\text{max}}\text{ Err}(\mathit{n})=\underset{\mathit{n}:|\mathit{n}|=n}{\text{max}}\frac{|q(\mathit{n})-q_{\text{approx}}(\mathit{n})|}{q(\mathit{n})}.$$

To study the accuracy of approximating q(n) by q_approx(n), we examine the behavior of AvgErr(n) and WorstErr(n) for a transition matrix estimated from real biological data. Specifically, we use the reversible phylogenetic mutation rate matrix estimated in [21, Table 1, matrix (1)] for the ψη-globin pseudogenes of six primate species. Since their estimated matrix is a matrix of nucleotide substitution rates used for phylogenetic analysis, we rescale it by the minimum amount that can make it a valid Markov transition matrix. This rescaled matrix, denoted by P̂, is given below to three digits of precision, and is used in our numerical experiments with different values of the mutation parameter θ:

$$\hat{\mathit{P}}=\left(\begin{array}{cccc}\hfill 0.433\hfill & \hfill 0.398\hfill & \hfill 0.074\hfill & \hfill 0.095\hfill \\ \hfill 0.665\hfill & \hfill 0.000\hfill & \hfill 0.164\hfill & \hfill 0.171\hfill \\ \hfill 0.074\hfill & \hfill 0.098\hfill & \hfill 0.394\hfill & \hfill 0.434\hfill \\ \hfill 0.147\hfill & \hfill 0.159\hfill & \hfill 0.674\hfill & \hfill 0.020\hfill \end{array}\right),$$ (30)

in the (T, C, A, G) basis. The stationary distribution corresponding to this transition matrix is π̂ = (0.308, 0.185, 0.308, 0.199) to three digits of precision.

For many neutral regions of the human genome, typical mutation rates per base are in the range 10⁻³ ≤ θ ≤ 10⁻² [16], and we consider θ ∈ {10⁻³, 5 × 10⁻³, 10⁻²} in our study. For the transition matrix in (30), the expected relative error AvgErr(n) and the worst-case relative error WorstErr(n) are plotted in Figures 1(a) and 1(b), respectively, as functions of the sample size n. As can be seen from the plots, both the expected relative error and the worst-case relative error grow very slowly with the sample size n. Further, the ratio of WorstErr(n) to AvgErr(n) is a small number between 1.3 and 2.1 for all n ≤ 360, and is decreasing in n. Hence, it appears that the approximation quality of q_approx(n) is uniformly good over all samples n for any given size n.

Figure 1.

Error plots as a function of the sample size n, for the transition matrix P̂ in (30) and mutation rate θ ∈ {10⁻³, 5 × 10⁻³, 10⁻²}. (a) The expected relative error, AvgErr(n). (b) The worst-case relative error, WorstErr(n).

Appendix

Here, we provide some general combinatorial identities which are used several times for proving the main results in this paper.

Fact 1. For any positive integers x, y, a and b where b ≤ a and x ≤ y,

$${\displaystyle \sum _{m=x}^y}\frac{\left(\begin{array}{c}\hfill b\hfill \\ \hfill m\hfill \end{array}\right)}{\left(\begin{array}{c}\hfill a\hfill \\ \hfill m\hfill \end{array}\right)}=\frac{\left(\begin{array}{c}\hfill a+1-x\hfill \\ \hfill a+1-b\hfill \end{array}\right)-\left(\begin{array}{c}\hfill a-y\hfill \\ \hfill a+1-b\hfill \end{array}\right)}{\left(\begin{array}{c}\hfill a\hfill \\ \hfill b\hfill \end{array}\right)}.$$ (31)

Proof. Starting from the left hand side of (31), we have:

$${\displaystyle \sum _{m=x}^y}\frac{\left(\begin{array}{c}\hfill b\hfill \\ \hfill m\hfill \end{array}\right)}{\left(\begin{array}{c}\hfill a\hfill \\ \hfill m\hfill \end{array}\right)}=\frac{b!(a-b)!}{a!}{\displaystyle \sum _{m=x}^y}\left(\begin{array}{c}\hfill a-m\hfill \\ \hfill a-b\hfill \end{array}\right)=\frac{\left(\begin{array}{c}\hfill a+1-x\hfill \\ \hfill a+1-b\hfill \end{array}\right)-\left(\begin{array}{c}\hfill a-y\hfill \\ \hfill a+1-b\hfill \end{array}\right)}{\left(\begin{array}{c}\hfill a\hfill \\ \hfill b\hfill \end{array}\right)},$$

where the last equality follows from the standard combinatorial identity that for all positive integers a, n, and k,

$${\displaystyle \sum _{i=a}^n}\left(\begin{array}{c}\hfill n-i\hfill \\ \hfill k\hfill \end{array}\right)=\left(\begin{array}{c}\hfill n-a+1\hfill \\ \hfill k+1\hfill \end{array}\right).$$

Fact 2. For positive integers a and b,

$${\displaystyle \sum _{m=1}^a}\frac{1}{m}\left(\begin{array}{c}\hfill a-m\hfill \\ \hfill b\hfill \end{array}\right)=\left(\begin{array}{c}\hfill a\hfill \\ \hfill b\hfill \end{array}\right)(H_a-H_b).$$

Fact 2 can be verified by induction [4] or by the method of Wilf-Zeilberger pairs [5].

Fact 3. For positive integers a and b where b ≤ a,

$${\displaystyle \sum _{m=1}^b}\frac{\left(\begin{array}{c}\hfill b\hfill \\ \hfill m\hfill \end{array}\right)}{\left(\begin{array}{c}\hfill a\hfill \\ \hfill m\hfill \end{array}\right)}\frac{1}{m+1}=\frac{a+1}{b+1}(H_{a+1}-H_{a-b})-1.$$ (32)

Proof. Starting from the left hand side of (32), we have:

$${\displaystyle \sum _{m=1}^b}\frac{\left(\begin{array}{c}\hfill b\hfill \\ \hfill m\hfill \end{array}\right)}{\left(\begin{array}{c}\hfill a\hfill \\ \hfill m\hfill \end{array}\right)}\frac{1}{m+1}=\frac{b!(a-b)!}{a!}{\displaystyle \sum _{m=1}^b}\left(\begin{array}{c}\hfill a-m\hfill \\ \hfill a-b\hfill \end{array}\right)\frac{1}{m+1}=\frac{1}{\left(\begin{array}{c}\hfill a\hfill \\ \hfill b\hfill \end{array}\right)}{\displaystyle \sum _{m=2}^{b+1}}\left(\begin{array}{c}\hfill a+1-m\hfill \\ \hfill a-b\hfill \end{array}\right)\frac{1}{m}=\frac{1}{\left(\begin{array}{c}\hfill a\hfill \\ \hfill b\hfill \end{array}\right)}\left[{\displaystyle \sum _{m=1}^{b+1}}\left(\begin{array}{c}\hfill a+1-m\hfill \\ \hfill a-b\hfill \end{array}\right)\frac{1}{m}-\left(\begin{array}{c}\hfill a\hfill \\ \hfill b\hfill \end{array}\right)\right]=\frac{1}{\left(\begin{array}{c}\hfill a\hfill \\ \hfill b\hfill \end{array}\right)}\left[\left(\begin{array}{c}\hfill a+1\hfill \\ \hfill b+1\hfill \end{array}\right)(H_{a+1}-H_{a-b})-\left(\begin{array}{c}\hfill a\hfill \\ \hfill b\hfill \end{array}\right)\right]=\frac{a+1}{b+1}(H_{a+1}-H_{a-b})-1,$$

where the fourth equality follows from using Fact 2.

We also list some facts about the moments of a hypergeometric distribution which are appealed to several times in the paper.

Fact 4. If a multivariate hypergeometric distribution is parameterized by n = (n₁, n₂, …, n_L), where n = |n|, and a sample of size m, m = (m₁, m₂, …, m_L), is drawn from it, then for any t = (t₁, t₂, …, t_L) where t_i ≥ 0 for all i, t = |t| and t ≤ n,

$$𝔼\left[{\displaystyle \prod _{i=1}^L}{(m_i)}_{t_{i\downarrow }}\right]={\displaystyle \sum _{\stackrel{\mathbf{0}⪯\mathit{m}⪯\mathit{n}:}{|\mathit{m}|=m}}}\frac{\left(\begin{array}{c}\hfill \mathit{n}\hfill \\ \hfill \mathit{m}\hfill \end{array}\right)}{\left(\begin{array}{c}\hfill n\hfill \\ \hfill m\hfill \end{array}\right)}{\displaystyle \prod _{i=1}^L}{(m_i)}_{t_{i\downarrow }}=\frac{{\displaystyle {\prod }_{i=1}^L}(n_i)t_{i\downarrow }}{{(n)}_{t\downarrow }}{(m)}_{t\downarrow }$$ (33)

Proof. Starting from the middle term in (33), we get:

$${\displaystyle \sum _{\stackrel{\mathbf{0}⪯\mathit{m}⪯\mathit{n}:}{|\mathit{m}|=m}}}\frac{\left(\begin{array}{c}\hfill \mathit{n}\hfill \\ \hfill \mathit{m}\hfill \end{array}\right)}{\left(\begin{array}{c}\hfill n\hfill \\ \hfill m\hfill \end{array}\right)}{\displaystyle \prod _{i=1}^L}{(m_i)}_{t_i\downarrow }={\displaystyle \sum _{\stackrel{\mathbf{0}⪯\mathit{m}⪯\mathit{n}:}{|\mathit{m}|=m}}}\frac{{\displaystyle {\prod }_{i=1}^L}{(n_i)}_{t_i\downarrow }}{{(n)}_{t\downarrow }}{(m)}_{t\downarrow }\frac{\left(\begin{array}{c}\hfill \mathit{n}-\mathit{t}\hfill \\ \hfill \mathit{m}-\mathit{t}\hfill \end{array}\right)}{\left(\begin{array}{c}\hfill n-t\hfill \\ \hfill m-t\hfill \end{array}\right)}=\frac{{\displaystyle {\prod }_{i=1}^L}{(n_i)}_{t_i\downarrow }}{{(n)}_{t\downarrow }}{(m)}_{t\downarrow }{\displaystyle \sum _{\stackrel{\mathbf{0}⪯\mathit{m}⪯\mathit{n}-\mathit{t}:}{|\mathit{m}|=m-t}}}\frac{\left(\begin{array}{c}\hfill \mathit{n}-\mathit{t}\hfill \\ \hfill \mathit{m}\hfill \end{array}\right)}{\left(\begin{array}{c}\hfill n-t\hfill \\ \hfill m\hfill \end{array}\right)}=\frac{{\displaystyle {\prod }_{i=1}^L}{(n_i)}_{t_i\downarrow }}{{(n)}_{t\downarrow }}{(m)}_{t\downarrow },$$

where the last equality follows because the term being summed is the probability mass function of a multivariate hypergeometric distribution parameterized by n − t, and the summation is over the entire domain of the distribution, and hence is 1.

In the following fact, we compute some second moments of the hypergeometric distribution parameterized by n when restricted to those samples m which are non-zero at all types.

Fact 5. If n = (n₁, n₂, …, n_L), where n = |n|, and 1 ≤ j ≠ k ≤ L, then we have the following identities:

$${\displaystyle \sum _{\stackrel{\mathbf{0}\prec \mathit{m}⪯\mathit{n}:}{|\mathit{m}|=m}}}\frac{\left(\begin{array}{c}\hfill \mathit{n}\hfill \\ \hfill \mathit{m}\hfill \end{array}\right)}{\left(\begin{array}{c}\hfill n\hfill \\ \hfill m\hfill \end{array}\right)}{m}_j(m_j+1)={\displaystyle \sum _{\stackrel{T\subseteq [L]:}{j\notin T}}}{(-1)}^{|T|}[\frac{{(n_j)}_{2\downarrow }{(m)}_{2\downarrow }}{{(n-n_T)}_{2\downarrow }}+\frac{2n_{j}m}{n-n_T}]\frac{\left(\begin{array}{c}\hfill n-n_T\hfill \\ \hfill m\hfill \end{array}\right)}{\left(\begin{array}{c}\hfill n\hfill \\ \hfill m\hfill \end{array}\right)}$$ (34)

$${\displaystyle \sum _{\stackrel{\mathbf{0}\prec \mathit{m}⪯\mathit{n}:}{|\mathit{m}|=m}}}\frac{\left(\begin{array}{c}\hfill \mathit{n}\hfill \\ \hfill \mathit{m}\hfill \end{array}\right)}{\left(\begin{array}{c}\hfill n\hfill \\ \hfill m\hfill \end{array}\right)}{m}_j{m}_k={\displaystyle \sum _{\stackrel{T\subseteq [L]:}{j\notin T}}}{(-1)}^{|T|}\frac{m_j{m}_k{(m)}_{2\downarrow }}{{(n-n_T)}_{2\downarrow }}\frac{\left(\begin{array}{c}\hfill n-n_T\hfill \\ \hfill m\hfill \end{array}\right)}{\left(\begin{array}{c}\hfill n\hfill \\ \hfill m\hfill \end{array}\right)}$$ (35)

Proof. Applying the inclusion-exclusion principle and using Fact 4, the identity in (34) can be obtained as

$${\displaystyle \sum _{\stackrel{\mathbf{0}\prec \mathit{m}⪯\mathit{n}:}{|\mathit{m}|=m}}}\frac{\left(\begin{array}{c}\hfill \mathit{n}\hfill \\ \hfill \mathit{m}\hfill \end{array}\right)}{\left(\begin{array}{c}\hfill n\hfill \\ \hfill m\hfill \end{array}\right)}{m}_j(m_j+1)={\displaystyle \sum _{\stackrel{T\subseteq [L]:}{j\notin T}}}{(-1)}^{|T|}[{\displaystyle \sum _{\stackrel{\mathbf{0}⪯\mathit{m}⪯\mathit{n}-{\mathit{n}}_T:}{|\mathit{m}|=m}}}\frac{\left(\begin{array}{c}\hfill \mathit{n}-{\mathit{n}}_T\hfill \\ \hfill \mathit{m}\hfill \end{array}\right)}{\left(\begin{array}{c}\hfill n-n_T\hfill \\ \hfill m\hfill \end{array}\right)}({(m_j)}_{2\downarrow }+2m_j)]\frac{\left(\begin{array}{c}\hfill n-n_T\hfill \\ \hfill m\hfill \end{array}\right)}{\left(\begin{array}{c}\hfill n\hfill \\ \hfill m\hfill \end{array}\right)}={\displaystyle \sum _{\stackrel{T\subseteq [L]:}{j\notin T}}}{(-1)}^{|T|}[\frac{{(n_j)}_{2\downarrow }{(m)}_{2\downarrow }}{{(n-n_T)}_{2\downarrow }}+\frac{2n_{j}m}{n-n_T}]\frac{\left(\begin{array}{c}\hfill n-n_T\hfill \\ \hfill m\hfill \end{array}\right)}{\left(\begin{array}{c}\hfill n\hfill \\ \hfill m\hfill \end{array}\right)}.$$

Similarly for (35), we have

$${\displaystyle \sum _{\stackrel{\mathbf{0}\prec \mathit{m}⪯\mathit{n}:}{|\mathit{m}|=m}}}\frac{\left(\begin{array}{c}\hfill \mathit{n}\hfill \\ \hfill \mathit{m}\hfill \end{array}\right)}{\left(\begin{array}{c}\hfill n\hfill \\ \hfill m\hfill \end{array}\right)}{m}_j{m}_k={\displaystyle \sum _{\stackrel{T\subseteq [L]:}{j,k\notin T}}}{(-1)}^{|T|}\left[{\displaystyle \sum _{\stackrel{\mathbf{0}⪯\mathit{m}⪯\mathit{n}-{\mathit{n}}_T:}{|\mathit{m}|=m}}}\frac{\left(\begin{array}{c}\hfill \mathit{n}-{\mathit{n}}_T\hfill \\ \hfill \mathit{m}\hfill \end{array}\right)}{\left(\begin{array}{c}\hfill n-n_T\hfill \\ \hfill m\hfill \end{array}\right)}{m}_j{m}_k\right]\frac{\left(\begin{array}{c}\hfill n-n_T\hfill \\ \hfill m\hfill \end{array}\right)}{\left(\begin{array}{c}\hfill n\hfill \\ \hfill m\hfill \end{array}\right)}={\displaystyle \sum _{\stackrel{T\subseteq [L]:}{j,k\notin T}}}{(-1)}^{|T|}\frac{m_j{m}_k{(m)}_{2\downarrow }}{{(n-n_T)}_{2\downarrow }}\frac{\left(\begin{array}{c}\hfill n-n_T\hfill \\ \hfill m\hfill \end{array}\right)}{\left(\begin{array}{c}\hfill n\hfill \\ \hfill m\hfill \end{array}\right)}.$$