Sequential Exchange Monte Carlo: Sampling Method for Multimodal Distribution without Parameter Tuning

TL;DR

The paper tackles the challenge of sampling multimodal Bayesian posteriors without manual tuning. It introduces Sequential Exchange Monte Carlo (SEMC), which fuses REMC and SMCS ideas to automatically determine temperatures and step sizes while enabling parallel updates. Empirical results across artificial multimodal distributions, spectral deconvolution, and exhaustive search show SEMC achieves comparable or superior sampling accuracy and Bayesian free energy estimates relative to tuned REMC and waste-free SMC, while eliminating the need for problem-specific parameter tuning. The method promises to broaden the practical adoption of Bayesian inference for complex models by non-experts and supports scalable, automated sequential experimentation.

Abstract

The Replica Exchange Monte Carlo (REMC) method, a Markov Chain Monte Carlo (MCMC) algorithm for sampling multimodal distributions, is typically employed in Bayesian inference for complex models. Using the REMC method, multiple probability distributions with different temperatures are defined to enhance sampling efficiency and allow for the high-precision computation of Bayesian free energy. However, the REMC method requires the tuning of many parameters, including the number of distributions, temperature, and step size, which makes it difficult for nonexperts to effectively use. Thus, we propose the Sequential Exchange Monte Carlo (SEMC) method, which automates the tuning of parameters by sequentially determining the temperature and step size. Numerical experiments showed that SEMC is as efficient as parameter-tuned REMC and parameter-tuned Sequential Monte Carlo Samplers (SMCS), which is also effective for the Bayesian inference of complex models.

Figures (11)

• Figure 1: Sketch illustrating the difference between the SEMC and waste-free SMC algorithms. Left: waste-free SMC algorithm when $S = 2, n = 3$. Right: SEMC algorithm with $S = 1$. The SEMC algorithm incorporates exchange steps, which mitigate the likelihood of bias caused by resampling when $S$ is small.
• Figure 2: Two-dimensional multimodal distribution for sampling (left) and histogram of $\theta_1$ (right).
• Figure 3: Observed data for spectral deconvolution (left) and histogram of posterior distribution of $\mu_1, \mu_2, \mu_3$ (right) for $K = 3$.
• Figure 4: Observed data for spectral deconvolution (left) and histogram of posterior distribution of $\mu_1, \mu_2, ..., \mu_{10}$ (right) for $K = 10$.
• Figure 5: Inverse temperature, $\{\beta_l\}_{l=2}^L$, and exchange rate between the $p_{\beta_l}, p_{\beta_{l-1}}$ of the SEMC and REMC methods with 10, 30, and 100 temperatures. (I) Sampling from a multimodal distribution, (II-(a)) Spectral deconvolution ($K=3$), (II-(b)) Spectral deconvolution ($K=10$), (III) Exhaustive search. (a) SEMC method, (b) REMC method with 10 temperatures, (c) REMC method with 30 temperatures, (d) REMC method with 100 temperatures. The vertical and horizontal axes represent the exchange rate and inverse temperature, respectively.