Weighted Riesz Particles

Markov chain Monte Carlo (MCMC) methods are simulated by local exploration of complex statistical distributions, and while bypassing the cumbersome requirement of a specific analytical expression for the target, this stochastic exploration of an uncertain parameter space comes at the expense of a large number of samples, and this computational complexity increases with parameter dimensionality. Although at the exploration level, some methods are proposed to accelerate the convergence of the algorithm, such as tempering, Hamiltonian Monte Carlo, Rao-redwellization, and scalable methods for better performance, it cannot avoid the stochastic nature of this exploration. We consider the target distribution as a mapping where the infinite-dimensional Eulerian space of the parameters consists of a number of deterministic submanifolds and propose a generalized energy metric, termed weighted Riesz energy, where a number of points is generated through pairwise interactions, to discretize rectifiable submanifolds. We study the properties of the point, called Riesz particle, and embed it into sequential MCMC, and we find that there will be higher acceptance rates with fewer evaluations, we validate it through experimental comparative analysis from a linear Gaussian state-space model with synthetic data and a non-linear stochastic volatility model with real-world data.

Variance Reduction of Resampling for Sequential Monte Carlo

A resampling scheme provides a way to switch low-weight particles for sequential Monte Carlo with higher-weight particles representing the objective distribution. The less the variance of the weight distribution is, the more concentrated the effective particles are, and the quicker and more accurate it is to approximate the hidden Markov model, especially for the nonlinear case. We propose a repetitive deterministic domain with median ergodicity for resampling and have achieved the lowest variances compared to the other resampling methods. As the size of the deterministic domain $M\ll N$ (the size of population), given a feasible size of particles, our algorithm is faster than the state of the art, which is verified by theoretical deduction and experiments of a hidden Markov model in both the linear and non-linear cases.

Chebyshev Particles

Markov chain Monte Carlo (MCMC) provides a feasible method for inferring Hidden Markov models, however, it is often computationally prohibitive, especially constrained by the curse of dimensionality, as the Monte Carlo sampler traverses randomly taking small steps within uncertain regions in the parameter space. We are the first to consider the posterior distribution of the objective as a mapping of samples in an infinite-dimensional Euclidean space where deterministic submanifolds are embedded and propose a new criterion by maximizing the weighted Riesz polarization quantity, to discretize rectifiable submanifolds via pairwise interaction. We study the characteristics of Chebyshev particles and embed them into sequential MCMC, a novel sampler with a high acceptance ratio that proposes only a few evaluations. We have achieved high performance from the experiments for parameter inference in a linear Gaussian state-space model with synthetic data and a non-linear stochastic volatility model with real-world data.