Phase transition in the sample complexity of likelihood-based phylogeny inference

Abstract

Reconstructing evolutionary trees from molecular sequence data is a fundamental problem in computational biology. Stochastic models of sequence evolution are closely related to spin systems that have been extensively studied in statistical physics and that connection has led to important insights on the theoretical properties of phylogenetic reconstruction algorithms as well as the development of new inference methods. Here, we study maximum likelihood, a classical statistical technique which is perhaps the most widely used in phylogenetic practice because of its superior empirical accuracy. At the theoretical level, except for its consistency, that is, the guarantee of eventual correct reconstruction as the size of the input data grows, much remains to be understood about the statistical properties of maximum likelihood in this context. In particular, the best bounds on the sample complexity or sequence-length requirement of maximum likelihood, that is, the amount of data required for correct reconstruction, are exponential in the number, n, of tips—far from known lower bounds based on information-theoretic arguments. Here we close the gap by proving a new upper bound on the sequence-length requirement of maximum likelihood that matches up to constants the known lower bound for some standard models of evolution. More specifically, for the r-state symmetric model of sequence evolution on a binary phylogeny with bounded edge lengths, we show that the sequence-length requirement behaves logarithmically in n when the expected amount of mutation per edge is below what is known as the Kesten-Stigum threshold. In general, the sequence-length requirement is polynomial in n. Our results imply moreover that the maximum likelihood estimator can be computed efficiently on randomly generated data provided sequences are as above. Our main technical contribution, which may be of independent interest, relates the total variation distance between the leaf state distributions of two trees with a notion of combinatorial distance between the trees. In words we show in a precise quantitative manner that the more different two evolutionary trees are, the easier it is to distinguish their output.

Preliminary lemmas

In this section, we collect a few useful lemmas.

1.1 Ancestral reconstruction

An important part of our construction involves reconstructing ancestral states. We will use the following lemma from [14] which we typically apply to a rooted subtree. Let $T = (V,E;\phi ;w) \in \mathbb {Y}$ rooted at $\rho$. Let $e = (x,y) \in E$ and assume that x is closest to $\rho$ (in topological distance). We define $\mathrm {P}(\rho ,e) = \mathrm {P}(\rho ,y)$, $|e|_\rho = |\mathrm {P}(\rho ,e)|$, and

$$\begin{aligned} R_\rho (e) = \left( 1 - \theta _e^2\right) \Theta _{\rho ,y}^{-2}, \end{aligned}$$

(60)

where $\Theta _{\rho ,y} = e^{-\mathrm {d}_T(\rho ,y)}$ and $\theta _e = e^{-w_e}$.

Lemma 2

(Ancestral reconstruction [14]) For any unit flow $\Psi$ from $\rho$ to [n],

$$\begin{aligned} \mathbb {E}_T\left| \mathbb {P}_T[\sigma _\rho = +1|\sigma _X] - \mathbb {P}_T[\sigma _\rho = -1|\sigma _X] \right| \ge \frac{1}{1+ \sum _{e \in E} R_\rho (e) \Psi (e)^2}, \end{aligned}$$

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where the LHS is the difference between the probability of correct and incorrect reconstruction using MLE. (See [14, Equation (14), Lemma 5.1 and Theorem 1.2’].)

1.2 Random cluster representation

We use a convenient percolation-based representation of the CFN model known as the random cluster model (see e.g. [23]). Let $T = (V,E;\phi ;w) \in \mathbb {Y}$ with corresponding $(\delta _e)_{e\in E}$.

Lemma 3

(Random cluster representation) Run a percolation process on T where edge e is open with probability $1 - 2 \delta _e$. Then associate to each open connected component a state according to the uniform distribution on $\{+1,-1\}$. The state vector on the vertices so obtained $(\sigma _v)_{v\in V}$ has the same distribution as the corresponding CFN model.

1.3 Concentration inequalities

Recall the following standard concentration inequality (see e.g. [38]):

Lemma 4

(Azuma-Hoeffding Inequality) Suppose $\mathbf{Z}=(Z_1,\ldots ,Z_m)$ are independent random variables taking values in a set S, and $h:S^m \rightarrow \mathbb {R}$ is any t-Lipschitz function: $|h(\mathbf{z}) - h(\mathbf{z'})|\le t$ whenever $\mathbf{z}, \mathbf{z'} \in S^m$ differ at just one coordinate. Then, $\forall \zeta > 0$,

$$\begin{aligned} \mathbb {P}\left[ |h(\mathbf{Z}) - \mathbb {E}[h(\mathbf{Z})]| \ge \zeta \right] \le 2\exp \left( -\frac{\zeta ^2 }{2 t^2 m}\right) . \end{aligned}$$