Topology and inference for Yule trees with multiple states

Abstract

We introduce two models for random trees with multiple states motivated by studies of trait dependence in the evolution of species. Our discrete time model, the multiple state ERM tree, is a generalization of Markov propagation models on a random tree generated by a binary search or ‘equal rates Markov’ mechanism. Our continuous time model, the multiple state Yule tree, is a generalization of the tree generated by a pure birth or Yule process to the tree generated by multi-type branching processes. We study state dependent topological properties of these two random tree models. We derive asymptotic results that allow one to infer model parameters from data on states at the leaves and at branch-points that are one step away from the leaves.

Appendix

Proof of Lemma 12

In the event that $\varvec{Z}(T)=0$ there is nothing to prove, so we consider $\varvec{W}$ on the event $\varvec{Z}(T)\ne 0\Leftrightarrow \varvec{W}(0)\ne 0$ (and $\varvec{W}(T)\ne 0$ as well).

For any $n\ge 1$ let $0\le t_0\le t_1\le \cdots \le t_n\le T$, we denote the joint distribution of $\varvec{W}$ at these times by

$$\begin{aligned} P_{t_0;t_1,\ldots ,t_n}(\varvec{z}_0;\varvec{w}_1,\ldots ,\varvec{w}_n)={\mathbb {P}}\left[ \varvec{W}({t_j})=\varvec{w}_j,\,1\le j\le n\,\big |\,\varvec{Z}(t_0)=\varvec{z}_0\right] . \end{aligned}$$

We first show, by induction, that $\forall n\ge 1$

$$\begin{aligned}&P_{t_0;t_1,\ldots ,t_n}(\varvec{z}_0;\varvec{w}_1,\ldots ,\varvec{w}_n)\nonumber \\&\quad = P_{t_0;t_1,\ldots ,t_{n-1}}(\varvec{z}_0;\varvec{w}_1,\ldots ,\varvec{w}_{n-1})\frac{ P_{t_0;t_{n-1},t_n}(\varvec{z}_0;\varvec{w}_{n-1},\varvec{w}_n) }{ P_{t_0;t_{n-1}}(\varvec{z}_0;\varvec{w}_{n-1}) }. \end{aligned}$$

(10)

This is evident for $n=2$. Assume the equation is true $\forall i\le n-1$ with $n>2$. Notice that

$$\begin{aligned}&P_{t_0;t_1,\ldots ,t_n}(\varvec{z}_0;\varvec{w}_1,\ldots ,\varvec{w}_n)\nonumber \\&\quad =\sum _{\varvec{z}_1\ge \varvec{w}_1}{\mathbb {P}}[\varvec{Z}({t_1})=\varvec{z}_1|\varvec{Z}({t_0})=\varvec{z}_0]P_{t_1;t_1,\ldots ,t_n}(\varvec{z}_1;\varvec{w}_1,\ldots ,\varvec{w}_n). \end{aligned}$$

(11)

The branching property of the birth-death process $\varvec{Z}$ guarantees independence of its subtrees originating from non-overlapping subsets of individuals present at any time $t_1$. Since all individuals surviving at time T must be descendants of the process $\varvec{W}$, we have

$$\begin{aligned} P_{t_1;t_1,\ldots ,t_n}(\varvec{z}_1;\varvec{w}_1,\ldots ,\varvec{w}_n)= & {} {\mathbb {P}}\left[ \varvec{W}({t_j})=\varvec{w}_j,\,1\le j\le n\,\big |\,\varvec{Z}({t_1})=\varvec{z}_1\right] \nonumber \\= & {} C_{\varvec{z}_1,\varvec{w}_1}{\mathbb {P}}\left[ \varvec{W}({t_j})\right. \nonumber \\= & {} \left. \varvec{w}_j,\,1\le j\le n\,\big |\,\varvec{Z}({t_1})=\varvec{w}_1\right] p_{\varvec{z}_1-\varvec{w}_1}^{\varvec{0}}(t_1,T)\nonumber \\= & {} C_{\varvec{z}_1,\varvec{w}_1}P_{t_1;t_1,\ldots ,t_n}(\varvec{w}_1;\varvec{w}_1,\ldots ,\varvec{w}_n)p_{\varvec{z}_1-\varvec{w}_1}^{\varvec{0}}(t_1,T)\qquad \end{aligned}$$

(12)

where $C_{\varvec{z}_1,\varvec{w}_1}$ denotes the combinatorial number of distinct ways of choosing $\varvec{w}_1$ out of $\varvec{z}_1$ individuals, and $p_{\varvec{z}}^{\varvec{0}}(t,T)={\mathbb {P}}[\varvec{Z}(T)=0|\varvec{Z}(t)=\varvec{z}]$ is the extinction probability by time T of the process $\varvec{Z}$ started at time t with $\varvec{Z}(t)=\varvec{z}$.

Given $\varvec{Z}({t_1})=\varvec{w}_1$, the process $(\varvec{Z}(t))_{t\ge t_1}$ is the sum of birth-death processes defined by subtrees $\{\mathcal T^{(i)}\}, i=1,\ldots ,|\varvec{w}_1|$, originated by one of each of the $|\varvec{w}_1|$ individuals at time $t_1$. We may assume that each ${\mathcal {T}}^{(i)}$ is started by an individual of state $\tau ^{(i)}$, where $\tau ^{(1)},\ldots ,\tau ^{(|\varvec{w}_1|)}$ is some ordering of the $|\varvec{w}_1|$ surviving originator states. Probability for the surviving lineages is

$$\begin{aligned}&P_{t_1;t_1,\ldots ,t_n}(\varvec{w}_1;\varvec{w}_1,\ldots ,\varvec{w}_n)\\&\quad ={\mathbb {P}}\left[ \varvec{W}({t_j})=\varvec{w}_j,\,1\le j\le n\,\big |\,\varvec{Z}({t_1})=\varvec{w}_1\right] \\&\quad ={\mathbb {P}}\left[ \varvec{W}({t_j})({\mathcal {T}}^{(i)})\ne 0\, \forall i,\;\, \sum _{i=1}^{|\varvec{w}_1|} \varvec{W}({t_j})({\mathcal {T}}^{(i)})=\varvec{w}_j, \,\forall 2\le j\le n\right] \end{aligned}$$

where $\varvec{W}(t)({\mathcal {T}}^{(i)})$ denotes the number of individuals of ${\mathcal {T}}^{(i)}$ at time t which have a surviving lineage at time T. Since the subtrees ${\mathcal {T}}^{(i)}$ are independent

$$\begin{aligned}&P_{t_1;t_1,\ldots ,t_n}\left( \varvec{w}_1;\varvec{w}_1,\ldots ,\varvec{w}_n\right) \nonumber \\&\quad = \displaystyle \sum _{\begin{array}{c} \forall 2\le j\le n,\; \left( \varvec{w}_j^{\left( i\right) }\right) _{1\le i\le |\varvec{w}_1|}:\\ \varvec{w}_j^{\left( i\right) }>0,\; \sum _{i=1}^{|\varvec{w}_1|}\varvec{w}_j^{\left( i\right) } = \varvec{w}_j \end{array}} \prod _{i=1}^{|\varvec{w}_1|}P_{t_1;t_2,\ldots ,t_n}\left( \varvec{e}_{\tau ^{\left( i\right) }};\varvec{w}^{\left( i\right) }_2,\ldots ,\varvec{w}^{\left( i\right) }_n\right) , \end{aligned}$$

(13)

where $\varvec{e}_i$ denotes the unit k-dimensional vector whose i-th coordinate is 1 and all other coordinates are 0, and the summation is over all possible decompositions of $\varvec{w}_j$ into vectors $(\varvec{w}_j^{(i)})_{i=1,\ldots ,|\varvec{w}_1|}$ with all nonzero coordinate values, for each $j=2,\dots , n$. By the inductive hypothesis (10) for $n-1$, the probabilities in the product on the right side are equal to

$$\begin{aligned}&P_{t_1;t_2,\ldots ,t_n}\left( \varvec{e}_{\tau ^{\left( i\right) }};\varvec{w}^{\left( i\right) }_2,\ldots ,\varvec{w}^{\left( i\right) }_n\right) \\&\quad =\displaystyle P_{t_1;t_2,\ldots ,t_{n-1}}\left( \varvec{e}_{\tau ^{\left( i\right) }};\varvec{w}^{\left( i\right) }_2,\ldots ,\varvec{w}^{\left( i\right) }_{n-1}\right) \frac{ P_{t_1;t_{n-1},t_n}\left( \varvec{e}_{\tau ^{\left( i\right) }};\varvec{w}^{\left( i\right) }_{n-1},\varvec{w}^{\left( i\right) }_n\right) }{ P_{t_1;t_{n-1}}\left( \varvec{e}_{\tau ^{\left( i\right) }};\varvec{w}^{\left( i\right) }_{n-1}\right) }\\&\quad = P_{t_1;t_2,\ldots ,t_{n-1}}\left( \varvec{e}_{\tau ^{\left( i\right) }};\varvec{w}^{\left( i\right) }_2,\ldots ,\varvec{w}^{\left( i\right) }_{n-1}\right) {\mathbb {P}}\left[ \varvec{W}\left( {t_n}\right) =\varvec{w}_n^{\left( i\right) }|\varvec{W}\left( {t_{n-1}}\right) \right. \\&\quad \left. =\varvec{Z}\left( {t_{n-1}}\right) =\varvec{w}_{n-1}^{\left( i\right) }\right] \end{aligned}$$

where the last equality follows from (11) and (12) since

$$\begin{aligned}&P_{t_1;t_{n-1},t_n}\left( \varvec{e}_{\tau ^{\left( i\right) }};\varvec{w}^{\left( i\right) }_{n-1},\varvec{w}^{\left( i\right) }_n\right) \\&\quad =\,{\mathbb {P}}\left[ \varvec{W}\left( {t_{n-1}}\right) =\varvec{w}_{n-1}^{\left( i\right) }, \varvec{W}\left( {t_n}\right) =\varvec{w}_n^{\left( i\right) }|\varvec{Z}\left( {t_1}\right) =\varvec{e}_{\tau ^{\left( i\right) }}\right] \\&\quad =\sum _{\varvec{z}_{n-1}\ge \varvec{w}_{n-1}} {\mathbb {P}}\left[ Z\left( {t_n-1}\right) =\varvec{z}_{n-1}|\varvec{Z}\left( {t_1}\right) =\varvec{e}_{\tau ^{\left( i\right) }}\right] C_{\varvec{z}_{n-1};\varvec{w}_{n-1}^{\left( i\right) }} p^{\varvec{0}}_{\varvec{z}_{n-1}-\varvec{w}_{n-1}}\left( t_{n-1},T\right) \\&\qquad \times {\mathbb {P}}\left[ \varvec{W}\left( {t_n}\right) =\varvec{w}_n^{\left( i\right) }|\varvec{W}\left( {t_{n-1}}\right) =\varvec{Z}\left( {t_{n-1}}\right) =\varvec{w}_{n-1}^{\left( i\right) }\right] \\&\quad =\,{\mathbb {P}}\left[ \varvec{W}\left( {t_n}\right) =\varvec{w}_n^{\left( i\right) }|\varvec{W}\left( {t_{n-1}}\right) =\varvec{Z}\left( {t_{n-1}}\right) =\varvec{w}_{n-1}^{\left( i\right) }\right] P_{t_1;t_{n-1}}\left( \varvec{e}_{\tau ^{\left( i\right) }};\varvec{w}^{\left( i\right) }_{n-1}\right) . \end{aligned}$$

As the first factor on the right side above does not depend on $(\varvec{w}_n^{(i)})_{i=1,\ldots ,|\varvec{w}_1|}$ the sum in (13) may be split into outer sums, over $2\le j\le n-1$, and an inner sum, over $j=n$ that is equal to

$$\begin{aligned} \sum _{\begin{array}{c} \left( \varvec{w}_n\right) :\varvec{w}_n^{\left( i\right) }>0,\\ \sum _{i=1}^{|\varvec{w}_1|}\varvec{w}_n^{\left( i\right) } = \varvec{w}_n \end{array}} \prod _{i=1}^{|\varvec{w}_1|}{\mathbb {P}}\left[ \varvec{W}\left( {t_n}\right) =\varvec{w}_n^{\left( i\right) }|\varvec{W}\left( {t_{n-1}}\right) =\varvec{Z}\left( {t_{n-1}}\right) =\varvec{w}_{n-1}^{\left( i\right) }\right] . \end{aligned}$$

By the same argument using splitting over independent subtrees, but this time splitting the individuals at time $t_{n-1}$ into subsets of sizes $(\varvec{w}_{n-1}^{(i)})_{i=1,\ldots ,|\varvec{w}_1|}$, we can show that this sum contributes to the outer sums a factor of

$$\begin{aligned} {\mathbb {P}}\left[ \varvec{W}\left( {t_n}\right) =\varvec{w}_n|\varvec{W}\left( {t_{n-1}}\right) =\varvec{Z}\left( {t_{n-1}}\right) =\varvec{w}_{n-1}\right] = \frac{ P_{t_0;t_{n-1},t_n}\left( \varvec{z}_0;\varvec{w}_{n-1},\varvec{w}_n\right) }{ P_{t_0;t_{n-1}}\left( \varvec{z}_0;\varvec{w}_{n-1}\right) }, \end{aligned}$$

where the last equality follows again from Eqs. (11) and (12), and combining with the outer sums in (13) implies

$$\begin{aligned}&P_{t_1;t_1,\ldots ,t_n}\left( \varvec{w}_1;\varvec{w}_1,\ldots ,\varvec{w}_n\right) \\&\quad = P_{t_1;t_1,\ldots ,t_{n-1}}\left( \varvec{w}_1;\varvec{w}_1,\ldots ,\varvec{w}_{n-1}\right) \frac{ P_{t_0;t_{n-1},t_n}\left( \varvec{z}_0;\varvec{w}_{n-1},\varvec{w}_n\right) }{ P_{t_0;t_{n-1}}\left( \varvec{z}_0;\varvec{w}_{n-1}\right) }, \end{aligned}$$

as wanted. By using once again Eqs. (11) and (12), this becomes Eq. (10) for step n. Equation (10) may be written in terms of conditional probabilities as

$$\begin{aligned}&{\mathbb {P}}\left[ \varvec{W}\left( {t_n}\right) =\varvec{w}_n\,\big |\,\varvec{W}\left( {t_j}\right) =\varvec{w}_j,\,1\le j\le n-1,\,\varvec{Z}\left( {t_0}\right) =\varvec{z}_0\right] \\&\quad ={\mathbb {P}}\left[ \varvec{W}\left( {t_n}\right) =\varvec{w}_n\,\big |\,\varvec{W}\left( {t_{n-1}}\right) =\varvec{w}_{n-1},\,\varvec{Z}\left( {t_0}\right) =\varvec{z}_0\right] \end{aligned}$$

which implies the Markov property for $(\varvec{W}(t))_{t\ge 0}$. $\square$