Exact Sampling of Spanning Trees via Fast-forwarded Random Walks

TL;DR

This paper introduces an exact sampling algorithm for spanning trees that overcomes the limitations of traditional random walk methods by fast-forwarding to new nodes, improving efficiency in Bayesian model applications.

Contribution

The paper presents a novel fast-forwarded cover algorithm that provides exact samples of spanning trees, addressing bottleneck issues in random walk-based samplers.

Findings

The new algorithm efficiently breaks bottlenecks in random walks.

It yields exact samples unlike approximation methods.

Applied successfully to Bayesian dendrogram modeling.

Abstract

Tree graphs are routinely used in statistics. When estimating a Bayesian model with a tree component, sampling the posterior remains a core difficulty. Existing Markov chain Monte Carlo methods tend to rely on local moves, often leading to poor mixing. A promising approach is to instead directly sample spanning trees on an auxiliary graph. Current spanning tree samplers, such as the celebrated Aldous--Broder algorithm, predominantly rely on simulating random walks that are required to visit all the nodes of the graph. Such algorithms are prone to getting stuck in certain sub-graphs. We formalize this phenomenon using the bottlenecks in the random walk's transition probability matrix. We then propose a novel fast-forwarded cover algorithm that can break free from bottlenecks. The core idea is a marginalization argument that leads to a closed-form expression which allows for fast-forwarding to the event of visiting a new node. Unlike many existing approximation algorithms, our algorithm yields exact samples. We demonstrate the enhanced efficiency of the fast-forwarded cover algorithm, and illustrate its application in fitting a Bayesian dendrogram model on a Massachusetts crimes and communities dataset.