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Zig-zag sampling for discrete structures and non-reversible phylogenetic MCMC

Jere Koskela

TL;DR

A zig-zag process targeting a posterior distribution defined on a hybrid state space consisting of both discrete and continuous variables that can lead to efficiency gains over classical Metropolis–Hastings algorithms, and that it is well suited to parallel computation.

Abstract

We construct a zig-zag process targeting a posterior distribution defined on a hybrid state space consisting of both discrete and continuous variables. The construction does not require any assumptions on the structure among discrete variables. We demonstrate our method on two examples in genetics based on the Kingman coalescent, showing that the zig-zag process can lead to efficiency gains of up to several orders of magnitude over classical Metropolis-Hastings algorithms, and that it is well suited to parallel computation. Our construction resembles existing techniques for Hamiltonian Monte Carlo on a hybrid state space, which suffers from implementationally and analytically complex boundary crossings when applied to the coalescent. We demonstrate that the continuous-time zig-zag process avoids these complications.

Zig-zag sampling for discrete structures and non-reversible phylogenetic MCMC

TL;DR

A zig-zag process targeting a posterior distribution defined on a hybrid state space consisting of both discrete and continuous variables that can lead to efficiency gains over classical Metropolis–Hastings algorithms, and that it is well suited to parallel computation.

Abstract

We construct a zig-zag process targeting a posterior distribution defined on a hybrid state space consisting of both discrete and continuous variables. The construction does not require any assumptions on the structure among discrete variables. We demonstrate our method on two examples in genetics based on the Kingman coalescent, showing that the zig-zag process can lead to efficiency gains of up to several orders of magnitude over classical Metropolis-Hastings algorithms, and that it is well suited to parallel computation. Our construction resembles existing techniques for Hamiltonian Monte Carlo on a hybrid state space, which suffers from implementationally and analytically complex boundary crossings when applied to the coalescent. We demonstrate that the continuous-time zig-zag process avoids these complications.

Paper Structure

This paper contains 6 sections, 2 theorems, 32 equations, 10 figures, 3 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Zig-zag on hybrid spaces
  3. The coalescent and a geometric embedding
  4. The infinite sites model
  5. The finite sites model
  6. Discussion

Key Result

Theorem 1

Suppose $\mathbb{F}$ is finite, $\tilde{ \pi }( m, \cdot, \mathbf{v} ) \in C^1( \Omega_m^o )$ for each $\mathbf{v} \in \{ -1, 1 \}^d$ and $m \in \mathbb{F}$, $\tilde{ \pi } > 0$ on $\Omega^o \times \{ -1, 1 \}^d$, that $Q( m, \mathbf{x}, \mathbf{v}; \cdot )$ has compact support for each $( m, \mathb Suppose the initial distribution of $( m, \mathbf{x} )$ has a density on $\Omega^*$ and that skew_d

Figures (10)

  • Figure 1: Operators $p_2^{\uparrow}$, $p_2^{\downarrow}$, and $s_2$. The horizontal arrangement of leaves is arbitrary throughout this paper; only vertical distance is meaningful.
  • Figure 2: (Left) $\tau$-space with $n = 3$ embedded into $\mathbb{R}^3$. Each square is a copy of $[ 0, \infty )^2$ associated with the given topology. The coordinates $( t_1, t_2 )$ are the respective time of the first merger, and the time between the first and second merger. The dot is the origin, and the line on which all three orthants intersect is a type 3 boundary consisting of trees in which all three leaves merge simultaneously at time $t_1$. The dashed lines are boundaries at $\infty$. (Right) A segment of $\tau$-space depicting a type 2 boundary, in which each square represents $[ 0, \infty )^{ n - 1 }$. Only the two orthants adjacent to the boundary are shown.
  • Figure 3: A realization of the infinite sites model with $n = 4$, two mutations, three types, and $D_n = ( \{ 0.7 \}, \{ 0.7 \}, \{\}, \{ 0.2 \} )$. The holding times $\mathbf{t}_3$ are shown on the left.
  • Figure 4: Trace plots under the infinite sites model and the data set of wardetal:1991.
  • Figure 5: Trace plots for the infinite sites model and the data set with $n = 550$, $\theta = 5.5$, 30 distinct types, and 38 mutations.
  • ...and 5 more figures

Theorems & Definitions (7)

  • Theorem 1
  • proof
  • Remark 1
  • Proposition 1
  • proof
  • proof : Proof of Theorem \ref{['stationary_theorem']}
  • proof : Proof of Proposition \ref{['prop_1']}