- The paper introduces a new Gibbs self-tuning framework that adaptively adjusts HMC parameters through conditional sampling and Metropolis corrections.
- The paper demonstrates that integrating adaptive parameter updates within a Gibbs sampler preserves detailed balance and enhances sampling efficiency.
- The paper unifies various HMC variants under one framework and empirically shows its competitive performance on both synthetic and real-world problems.
Localized Tuning of Hamiltonian Monte Carlo through a Gibbs Self-Tuning Framework
Introduction to Gibbs Self-Tuning HMC (GIST)
Hamiltonian Monte Carlo (HMC) is well-known for its efficacy in drawing samples from high-dimensional probability distributions. The tuning of HMC parameters such as path length, step size, and mass matrix is crucial for optimal performance, yet poses a significant challenge in practice. This paper introduces the Gibbs Self-Tuning Hamiltonian Monte Carlo (GIST-HMC) framework designed to adaptively tune these parameters within a Gibbs sampling paradigm.
Description of the GIST Framework
The GIST framework extends the state space of the traditional HMC to include algorithm tuning parameters. Each iteration of GIST samples not only the position and momentum but also tuning parameters from their conditional distributions tailored to the current state of the system. This enlarged state representation facilitates the adaptation of key HMC parameters on-the-fly, fostering an efficient exploration of the target distribution.
Conditional Sampling and Metropolis Adjustment
In GIST, the adaptive sampling of tuning parameters like path length is achieved via conditional distributions, and these choices are subsequently corrected through Metropolis-Hastings adjustments. This ensures the correct stationary distribution of the Markov chain is maintained. The application is straightforward: after initial parameter setup and momentum refreshment, the tuning parameter is sampled from its conditional distribution, followed by a Metropolis-adjusted proposal step ensuring detailed balance.
Theoretical Foundation and Correctness Proof
A formal proof of the theoretical correctness of the GIST sampler is presented, revealing that each Gibbs sampling step within GIST preserves the desired invariant distribution. The key is maintaining the detailed balance and the symmetrical properties of the Markov transition mechanism, leveraging Gibbs sampling for conditional updates of tuning parameters.
Special Cases and Variants
The framework encapsulates several existing HMC variants, such as Randomized HMC, the No-U-Turn Sampler (NUTS), and the Apogee-to-Apogee Path Sampler, as special cases under various settings of the adaptive sampling mechanism. This unification under the GIST framework allows for new insights into the operational similarities and differences among these methods.
Experimental Evaluation
The new GIST approach is empirically compared to well-established techniques like NUTS. The GIST framework demonstrates competitiveness with these techniques, providing robust performance across various synthetic and real-world problems. This section carefully analyzes the efficiency and accuracy of sampling, reinforced by detailed numerical results and comparative analyses.
Implications and Future Directions
This paper's introduction of a flexible and theoretically sound framework for self-tuning HMC samplers provides a significant step towards more adaptive, efficient MCMC algorithms. Looking forward, the GIST framework opens avenues for further research into more sophisticated conditional distributions for tuning parameters and their impact on sampling efficiency. There is also potential for extending this framework beyond HMC to other forms of MCMC, potentially enhancing a broader range of applications in Bayesian computation.
In conclusion, the GIST framework not only strengthens the theoretical foundations of adaptive MCMC methods but also offers practical tools for enhancing the performance of HMC samplers, promising improvements in both the speed and accuracy of Bayesian inference across diverse applications.