[Paper Review] Initializing adaptive importance sampling with Markov chains

Frederik Beaujean, A. Caldwell|arXiv (Cornell University)|Apr 29, 2013

Bayesian Methods and Mixture Models22 references16 citations

TL;DR

This paper proposes a robust initialization method for adaptive importance sampling using Markov chains and hierarchical clustering to generate accurate, multimodal proposal densities. The approach enables rapid convergence and precise estimation of normalization and marginals in high-dimensional, challenging posterior distributions, outperforming nested sampling in multimodal 20D examples.

ABSTRACT

Adaptive importance sampling is a powerful tool to sample from complicated target densities, but its success depends sensitively on the initial proposal density. An algorithm is presented to automatically perform the initialization using Markov chains and hierarchical clustering. The performance is checked on challenging multimodal examples in up to 20 dimensions and compared to results from nested sampling. Our approach yields a proposal that leads to rapid convergence and accurate estimation of overall normalization and marginal distributions.

Motivation & Objective

• Address the critical challenge of poor initialization in adaptive importance sampling (PMC), which often leads to failure due to poor proposal density matching.
• Develop a fully automated, robust initialization method that reliably identifies multiple modes and degeneracies in complex, high-dimensional target densities.
• Enable efficient, parallelizable sampling for Bayesian inference in high-dimensional problems (up to 20D) with multimodal posteriors, common in particle physics and cosmology.
• Improve the accuracy of marginal distributions and normalization (evidence) estimation compared to existing methods like nested sampling.
• Pave the way toward a 'black-box' sampling algorithm by minimizing manual tuning and enhancing reliability across diverse target distributions.

Proposed method

• Use multiple short Markov chains initialized from random or uniform starting points to explore the target density and generate representative samples.
• Apply hierarchical clustering to group samples from the Markov chains into distinct clusters, each corresponding to a potential mode of the target distribution.
• Use cluster centers and covariance matrices as the initial components of a mixture proposal density in population Monte Carlo (PMC).
• Construct the initial proposal as a Gaussian or Student’s t mixture, with components derived from cluster statistics to match the target’s multimodal structure.
• Integrate the initialized PMC with standard EM-based updates to iteratively refine the proposal and minimize KL divergence between proposal and target.
• Leverage massive parallelization by distributing importance weight evaluations across multiple cores or GPUs, with each sample drawn from the mixture proposal.

Experimental results

Research questions

• RQ1 How can a reliable, automated initialization be achieved for adaptive importance sampling in high-dimensional, multimodal posterior distributions?
• RQ2 Can Markov chains combined with hierarchical clustering effectively identify multiple modes and degeneracies in complex target densities?
• RQ3 Does the proposed initialization method lead to faster convergence and more accurate estimation of normalization and marginals compared to nested sampling?
• RQ4 To what extent can the method be scaled to high-dimensional problems (e.g., d=20) without manual tuning?
• RQ5 Can the initialization method reduce the number of PMC iterations required for convergence while maintaining accuracy in challenging Bayesian inference tasks?

Key findings

• The proposed initialization method successfully identifies multiple isolated modes and degeneracies in multimodal target densities up to 20 dimensions.
• The method achieved more accurate evidence (normalization) estimates than a publicly available nested sampling implementation on the same test problems.
• Marginal distributions estimated using the initialized PMC closely matched the true target distributions, indicating high fidelity in sampling.
• The algorithm converged rapidly, requiring only a few PMC iterations due to the high-quality initial proposal, even in high-dimensional settings.
• The method is highly parallelizable, with importance weight evaluations and proposal updates amenable to distribution across thousands of cores or GPUs.
• The approach significantly reduces reliance on manual tuning and 'educated guesses' for initialization, moving toward a more automated, robust sampling pipeline.