The Polya Tree Sampler: Towards Efficient and Automatic Independent Metropolis-Hastings Proposals

Area of Science:

• Computational statistics
• Bayesian inference
• Machine learning

Background:

• Exploring un-normalized multivariate densities is crucial for Bayesian inference.
• Existing methods like Metropolis-within-Gibbs can be computationally intensive.
• Approximating complex probability distributions requires efficient algorithms.

Purpose of the Study:

• To present a simple, efficient, and computationally cheap sampling method for multivariate densities.
• To introduce the Polya tree sampler for exploring un-normalized densities.
• To compare the Polya tree sampler with existing algorithms.

Main Methods:

• The Polya tree sampler constructs an independent proposal based on an approximation of the target density.
• An initial "warming-up" phase iteratively refines support points to minimize distribution distance.
• The "sampling" phase uses samples from an approximating mixture of finite Polya trees with Metropolis-Hastings acceptance.

Main Results:

• The Polya tree sampler demonstrates efficiency and computational affordability.
• Illustrations show its applicability in exploring multivariate densities.
• Comparisons indicate competitive performance against Metropolis-within-Gibbs and delayed rejection adaptive Metropolis.

Conclusions:

• The Polya tree sampler is a viable and efficient method for exploring un-normalized multivariate densities.
• This approach offers a computationally inexpensive alternative for complex density approximation.
• The method shows promise for applications in Bayesian statistics and machine learning.