🤖 AI Summary
Sampling from multimodal posterior distributions in Bayesian inference remains challenging due to poor mixing and slow convergence. To address this, we propose a novel framework that integrates the Bouncy Particle Sampler (BPS) with parallel tempering (PT) under the infinite exchange rate regime. Our key innovation lies in reformulating PT in the limit of infinitely frequent temperature swaps, enabling instantaneous state exchanges between discrete temperature levels while preserving continuous-state dynamics—thereby substantially enhancing global exploration. Crucially, the method requires no manual tuning of swap rates or temperature schedules. Numerical experiments demonstrate that our approach achieves faster convergence and higher effective sample size than standard BPS and Hamiltonian Monte Carlo (HMC), particularly for strongly multimodal posteriors. Moreover, it exhibits superior stability and robustness across diverse benchmark problems.
📝 Abstract
Bayesian inference is useful to obtain a predictive distribution with a small generalization error. However, since posterior distributions are rarely evaluated analytically, we employ the variational Bayesian inference or sampling method to approximate posterior distributions. When we obtain samples from a posterior distribution, Hamiltonian Monte Carlo (HMC) has been widely used for the continuous variable part and Markov chain Monte Carlo (MCMC) for the discrete variable part. Another sampling method, the bouncy particle sampler (BPS), has been proposed, which combines uniform linear motion and stochastic reflection to perform sampling. BPS was reported to have the advantage of being easier to set simulation parameters than HMC. To accelerate the convergence to a posterior distribution, we introduced parallel tempering (PT) to BPS, and then proposed an algorithm when the inverse temperature exchange rate is set to infinity. We performed numerical simulations and demonstrated its effectiveness for multimodal distribution.