Twisting the Alive Particle Filter

TL;DR

This paper introduces the alive twisted particle filter and its integration into a particle marginal Metropolis-Hastings scheme, improving variance reduction and convergence speed for sampling in complex hidden Markov models with intractable densities.

Contribution

It develops a novel alive twisted particle filter and combines it with PMMH, enhancing variance reduction and efficiency in challenging HMM scenarios.

Findings

Lower variance estimates of normalising constants.

Faster convergence of the alive twisted PMMH.

Improved performance in stochastic volatility models.

Abstract

This work focuses on sampling from hidden Markov models (Cappe et al, 2005) whose observations have intractable density functions. We develop a new sequential Monte Carlo (Doucet et al, 2000 and Gordon et al, 1993) algorithm and a new particle marginal Metropolis-Hastings (Andrieu et al, 2010) algorithm for these purposes. We build from Jasra, et al (2013) and Whiteley, et al (2013) to construct the sequential Monte Carlo (SMC) algorithm (which we call the alive twisted particle filter). Like the alive particle filter of Jasra, et al (2013), our new SMC algorithm adopts an approximate Bayesian computation (Tavare et al, 1997) estimate of the HMM. Our alive twisted particle filter also uses a twisted proposal as in Whiteley, et al (2013) to obtain a low-variance estimate of the HMM normalising constant. We demonstrate via numerical examples that, in some scenarios, this estimate has a much lower variance than that of the estimate obtained via the alive particle filter. The low variance of this normalising constant estimate encourages the implementation of our SMC algorithm within a particle marginal Metropolis-Hastings (PMMH) scheme, and we call the resulting methodology ``alive twisted PMMH''. We numerically demonstrate on a stochastic volatility model how our alive twisted PMMH can converge faster than the standard alive PMMH of Jasra, et al (2013).