Dynamic temperature selection for parallel-tempering in Markov chain Monte Carlo simulations (1501.05823v3)

Abstract: Modern problems in astronomical Bayesian inference require efficient methods for sampling from complex, high-dimensional, often multi-modal probability distributions. Most popular methods, such as Markov chain Monte Carlo sampling, perform poorly on strongly multi-modal probability distributions, rarely jumping between modes or settling on just one mode without finding others. Parallel tempering addresses this problem by sampling simultaneously with separate Markov chains from tempered versions of the target distribution with reduced contrast levels. Gaps between modes can be traversed at higher temperatures, while individual modes can be efficiently explored at lower temperatures. In this paper, we investigate how one might choose the ladder of temperatures to achieve more efficient sampling, as measured by the autocorrelation time of the sampler. In particular, we present a simple, easily-implemented algorithm for dynamically adapting the temperature configuration of a sampler while sampling. This algorithm dynamically adjusts the temperature spacing to achieve a uniform rate of exchanges between chains at neighbouring temperatures. We compare the algorithm to conventional geometric temperature configurations on a number of test distributions and on an astrophysical inference problem, reporting efficiency gains by a factor of 1.2-2.5 over a well-chosen geometric temperature configuration and by a factor of 1.5-5 over a poorly chosen configuration. On all of these problems a sampler using the dynamical adaptations to achieve uniform acceptance ratios between neighbouring chains outperforms one that does not.

• The paper introduces a dynamic temperature algorithm that adjusts the temperature ladder to achieve a uniform exchange rate between chains.
• The paper demonstrates efficiency gains with autocorrelation improvements ranging from 1.2 to 5 times compared to static geometric configurations.
• The paper applies the method to gravitational wave parameter estimation, significantly boosting PTMCMC performance in astrophysical inference.

Dynamic Temperature Selection for Parallel-Tempering in MCMC Simulations

The paper evaluates the challenges and proposes a method for optimizing temperature selection in Parallel-Tempering Markov chain Monte Carlo (PTMCMC) simulations. PTMCMC is frequently utilized in astronomical Bayesian inference, particularly when sampling from high-dimensional, multi-modal probability distributions. The inherent challenge with PTMCMC algorithms is their dependence on a predefined ladder of temperatures, which influences the efficiency of sampling from complex posterior distributions.

Key Contributions

1. Dynamic Temperature Algorithm: The authors introduce a simple and effective algorithm for the dynamic selection of temperatures in PTMCMC. This algorithm aims to achieve a uniform rate of exchanges between chains at neighboring temperatures, which is hypothesized to improve sampling efficiency.
2. Efficiency Gains: Using a variety of test distributions, both simple and complex, the paper quantifies the efficiency gains of the proposed method in terms of autocorrelation time (ACT). The results show that the dynamic temperature selection algorithm yields efficiency improvements by factors ranging from 1.2 to 2.5 over a well-chosen geometric temperature configuration, and by 1.5 to 5 over a poorly chosen configuration.
3. Application to Gravitational Wave Analysis: The algorithm's applicability is further demonstrated in a practical, computationally demanding astrophysical inference problem: parameter estimation in gravitational wave (GW) data analysis related to compact binary coalescences (CBC). The implementation showed a notable reduction in ACT, enhancing the PTMCMC sampler's performance compared to using static geometric temperature ladders.

Implications and Future Directions

• Enhanced Sampling in Complex Distributions: The dynamic temperature selection approach effectively addresses one of the significant drawbacks of conventional PTMCMC methods—choosing a suitable temperature ladder. By dynamically adjusting temperatures to maintain uniform acceptance rates between adjacent chains, the method optimizes the communication between chains, facilitating more efficient sampling in complex posterior landscapes.
• Theoretical Insights and Practical Applications: This research provides both theoretical insights into the behavior of PTMCMC samplers and practical guidance on improving their efficiency. The adaptability of the algorithm makes it potentially advantageous for a broader range of applications beyond the specific test cases studied.
• Further Research and Development: Despite the improvements shown, the algorithm's performance in varied settings and its integration with existing parallel tempering frameworks provide avenues for future research. Additionally, exploring its potential to enhance other Monte Carlo-based sampling techniques could be a valuable extension of this work.

In conclusion, the paper contributes a novel perspective on tackling the temperature selection problem in PTMCMC, resulting in significant advancements in sampling efficiency for complex astronomical inference problems. The method promises practical benefits for researchers working on high-dimensional Bayesian inference tasks and lays a foundation for continued exploration and optimization in the field of statistical computing and applied astronomy.