Langevin and Hamiltonian Based Sequential MCMC for Efficient Bayesian Filtering in High-Dimensional Spaces

Abstract

Nonlinear non-Gaussian state-space models arise in numerous applications in statistics and signal processing. In this context, one of the most successful and popular approximation techniques is the Sequential Monte Carlo (SMC) algorithm, also known as particle filtering. Nevertheless, this method tends to be inefficient when applied to high dimensional problems.In this paper, we focus on another class of sequential inference methods, namely the Sequential Markov Chain Monte Carlo (SMCMC) techniques, which represent a promising alternative to SMC methods. After providing a unifying framework for the class of SMCMC approaches, we propose novel efficient strategies based on the principle of Langevin diffusion andHamiltonian dynamics in order to cope with the increasing number of high-dimensional applications. Simulation results show that the proposed algorithms achieve significantly better performance compared to existing algorithms.

Invertible Particle-Flow-Based Sequential MCMC With Extension to Gaussian Mixture Noise Models

“…Sequential Markov chain Monte Carlo (SMCMC) methods were proposed as a general MCMC approach for approximating the joint posterior distribution π k (x 0:k ) = p(x 0:k |z 1:k ) recursively. A unifying framework of the various SMCMC methods was provided in [33]. At time step k, π k (x 0:k ) can be computed pointwise up to a constant in a recursive manner:…”

“…The Metropolis-Hastings (MH) algorithm used within SMCMC to generate one sample is summarized in Algorithm 1. 1) Composite MH kernel in SMCMC: Different choices of the MCMC kernel for high dimensional SMCMC are discussed in [33]. In most SMCMC algorithms, an independent MH kernel is adopted [33], i.e., q k (x 0:k−1 |x i−1 k,0:k ) = q k (x 0:k ), meaning that the proposal is independent of the state of the Markov chain at the previous iteration.…”

“…1) Composite MH kernel in SMCMC: Different choices of the MCMC kernel for high dimensional SMCMC are discussed in [33]. In most SMCMC algorithms, an independent MH kernel is adopted [33], i.e., q k (x 0:k−1 |x i−1 k,0:k ) = q k (x 0:k ), meaning that the proposal is independent of the state of the Markov chain at the previous iteration. The ideal choice is the optimal independent MH kernel, but it is usually impossible to sample from [33].…”

“…In most SMCMC algorithms, an independent MH kernel is adopted [33], i.e., q k (x 0:k−1 |x i−1 k,0:k ) = q k (x 0:k ), meaning that the proposal is independent of the state of the Markov chain at the previous iteration. The ideal choice is the optimal independent MH kernel, but it is usually impossible to sample from [33]. It is difficult to identify an effective approximation to the optimal independent MH kernel.…”

“…The choice of independent MH kernel using the prior as the proposal can lead to very low acceptance rates if the state dimension is very high or the measurements are highly informative. [32], [33] propose the use of a composite MH kernel which is constituted of a joint proposal q k,1 (to update x 0:k ) followed by two individual state variable refinements using proposals q k,2 (to update x 0:k−1 ) and q k,3 (to update x k ), based on the Metropolis within Gibbs approach, within a single MCMC iteration. The composite kernel is summarized in Algorithm 2.…”