An Adaptive Sequential Monte Carlo Sampler

Sequential Monte Carlo (SMC) methods are not only a popular tool in the analysis of state space models, but offer an alternative to MCMC in situations where Bayesian inference must proceed via simulation. This paper introduces a new SMC method that uses adaptive MCMC kernels for particle dynamics. The proposed algorithm features an online stochastic optimization procedure to select the best MCMC kernel and simultaneously learn optimal tuning parameters. Theoretical results are presented that justify the approach and give guidance on how it should be implemented. Empirical results, based on analysing data from mixture models, show that the new adaptive SMC algorithm (ASMC) can both choose the best MCMC kernel, and learn an appropriate scaling for it. ASMC with a choice between kernels outperformed the adaptive MCMC algorithm of Haario et al. (1998) in 5 out of the 6 cases considered.

💡 Research Summary

The paper introduces an Adaptive Sequential Monte Carlo (ASMC) algorithm that integrates adaptive Markov chain Monte Carlo (MCMC) kernels into the particle dynamics of a standard SMC framework. Traditional SMC methods rely on a fixed proposal kernel to move particles between resampling steps, which can be inefficient when the target posterior distribution is multimodal, high‑dimensional, or exhibits strong correlations. Conversely, adaptive MCMC algorithms such as the Haario‑Roberts‑Suter scheme adjust proposal scales on the fly but keep the underlying kernel type unchanged. The authors bridge this gap by allowing the SMC sampler to select, at each iteration, the most suitable MCMC kernel from a predefined pool and to learn its optimal tuning parameters through an online stochastic optimization routine.

The algorithm proceeds as follows. At time (t) a set of particles ({x_t^{(i)}}{i=1}^N) approximates the current target distribution (\pi_t). For each particle a collection of candidate kernels ({K_j}{j=1}^J) (e.g., random‑walk Metropolis, independent Metropolis, Hamiltonian Monte Carlo) is considered, each equipped with a set of tunable parameters (\theta_{j}) (step‑size, mass matrix, etc.). A loss function (L_{j}(x,\theta_j)) is evaluated for each kernel–parameter pair; the loss can be based on the empirical acceptance probability, the estimated Kullback‑Leibler divergence between the proposal and the target, or a combination thereof. The losses are transformed into selection probabilities via a soft‑max or exponential weighting scheme, yielding a stochastic policy (\pi_j^{(i)}) that determines which kernel each particle will use. After a kernel is sampled, the particle is propagated using the selected MCMC move, and the importance weight is updated in the usual SMC manner.

Crucially, the parameters (\theta_j) are updated online using a Robbins‑Monro stochastic gradient step: \