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Practical use of Hierarchical Bayes models require the availability of efficient methods for posterior inference. Sequential Monte Carlo methods have appeared as an extremely robust way to simulate complicated Bayesian posteriors. The simplest version successively weights and resample draws called particles from a sequence of target distributions. The method has a main weakness keeping it from being used in complex hierarchical models and large data sets: sample impoverishment. This issue is usually alleviated through the use of a "refreshing" MCMC step. However, to preserve effectiveness, the computational costs are quadratically increasing with the total number of observations. We propose a new SMC-within-SMC method. In a first step, each individual-level parameter is estimated separately using standard SMC and a non-hierarchical auxiliary prior. In a second step, we use weighting methods to replace the auxiliary prior with the hierarchical one without the need to recompute any likelihood. In addition to allowing for the separate processing of individual data, this approach drastically reduces the computational costs. A MATLAB package is provided.

In behavioral experiments, carefully choosing the stimuli is critical for success. Recently, several "adaptive" Bayesian methods gained popularity by proposing to optimally select the stimulus in each trial based on the results of the preceding trials. However, current methods are computationally expensive and might require a long waiting period between each question. Moreover, they are often tailored to a particular model and a particular objective, such as parameter estimation, prediction or model selection. It is left to the researcher to extend these approaches to other models by providing a suitable Bayesian inference method. We propose to apply the Sequential Monte Carlo (SMC) framework to solve both the inference problem and the optimal experimental design problem. This new method, called Sequential Optimal Inference (SOI) provides gains in computational efficiency and allows for the use of a broad class of complex models and objectives. We demonstrate its validity with simulation studies. An implementation of the method in MATLAB and Python is provided.

The paper proposes a new Monte-Carlo simulator combining the advantages of Sequential Monte Carlo simulators and Hamiltonian Monte Carlo simulators. The result is a method that is robust to multimodality and complex shapes to use for inference in presence of difficult likelihoods or target functions. Several examples are provided.