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arXiv · 2405.03096

Exact Sampling of Spanning Trees via Fast-forwarded Random Walks

Edric Tam·David B. Dunson·Leo L. Duan
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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.

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Edric Tam, David B. Dunson, Leo L. Duan. 2024-05-06. Exact Sampling of Spanning Trees via Fast-forwarded Random Walks. https://arxiv.org/abs/2405.03096

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