Ensemble-Based Annealed Importance Sampling

Overview of Ensemble-Based Annealed Importance Sampling

Sampling from complex, multimodal distributions is a challenging task in computational science and statistics. In practice, regions with high probability are often separated by regions where probability densities are low, leading to difficulties in efficiently transitioning between these high-probability modes. Common Monte Carlo methods such as Markov Chain Monte Carlo (MCMC) often struggle to adequately sample from these distributions due to metastability. This paper introduces Ensemble-Based Annealed Importance Sampling (AIS) to address this challenge by leveraging population-based Monte Carlo methods.

Annealed Importance Sampling (AIS)

AIS is a continuation method that gradually transforms an easy-to-sample initial distribution into a target distribution through a series of intermediate distributions. The method interpolates between the initial and target distributions and uses Metropolis-Hastings (MH) algorithms to sample from these intermediate distributions. Standard AIS generates independent samples along this path, necessitating reweighting of the samples to align with the target distribution.

However, the efficiency of AIS depends on effectively handling two key problems:

• Controlling the variance of importance weights.
• Enhancing the mixing of the transition kernels (T(_l)) for the intermediate distributions.

Ensemble-Based Methods

Ensemble-based methods maintain multiple samplers instead of a single one, enabling interactions between different samplers to encourage global exploration of the state space and transition between high-probability modes. This paper proposes two novel algorithms that integrate AIS with ensemble-based methods utilizing:

• Local exploitation by standard MCMC methods (e.g., Langevin dynamics or Glauber dynamics).
• Global exploration using algorithms from the Evolutionary Monte Carlo framework, such as the snooker and genetic algorithms.
• Birth-Death dynamics to manage the weights of particles more effectively, thereby avoiding the need for reweighting.

Theoretical Contributions

Key theoretical advancements of the ensemble-based AIS include:

• A new sampling algorithm combining AIS with ensemble-based methods to encourage exploration of undiscovered modes while maintaining efficient local exploitation.
• Derivation of a partial differential equation (PDE) governing the ensemble's empirical density in the continuous time and mean-field limit.

Specifically, the derived PDE helps understand the behavior and convergence properties of the proposed algorithms, offering insights into their theoretical performance.

Numerical Experiments

The paper presents a series of numerical experiments to demonstrate the efficacy of the proposed ensemble-based AIS algorithms. These experiments span various continuous and discrete distributions, showing that the new algorithms outperform both standard AIS and ensemble-based AIS without explicit exploration mechanisms.

Implications and Future Directions

The practical implications of ensemble-based AIS are significant for fields requiring efficient sampling of multimodal distributions, such as computational physics, Bayesian inference, and machine learning. The usage of population-based methods enhances the exploration capabilities, leading to more accurate and representative samples.

Future avenues of research may involve:

• Extending the approach to other continuation methods like Simulated Tempering and Tempered Transitions.
• Investigating the use of ensemble-based AIS for distributions with approximate symmetries.
• Studying the detailed theoretical properties of the derived mean-field PDE.

In conclusion, Ensemble-Based Annealed Importance Sampling stands as a solid theoretical and practical improvement over existing sampling methods, effectively addressing the challenges of sampling from multimodal distributions. The integration of ensemble-based techniques offers a promising direction for further research and development in efficient sampling algorithms.