High-Dimensional Filtering Using Nested Sequential Monte Carlo

Abstract

Sequential Monte Carlo (SMC) methods comprise one of the most successful approaches to approximate Bayesian filtering. However, SMC without good proposal distributions struggle in high dimensions. We propose nested sequential Monte Carlo (NSMC), a methodology that generalises the SMC framework by requiring only approximate, properly weighted, samples from the SMC proposal distribution, while still resulting in a correct SMC algorithm. This way we can exactly approximate the locally optimal proposal, and extend the class of models for which we can perform efficient inference using SMC. We show improved accuracy over other state-of-the-art methods on several spatio-temporal state space models.Inference in complex and high-dimensional statistical models is a very challenging problem that is ubiquitous in applications such as climate informatics [Monteleoni et al., 2013], bioinformatics [Cohen, 2004] and machine learning [Wainwright and Jordan, 2008], to mention a few.We are interested in sequential Bayesian inference in settings where we have a sequence of posterior distributions that we need to compute. To be specific, we are focusing on settings where the model (or state variable) is high-dimensional, but where there are local dependencies. One example of the type of models we consider are the so-called spatiotemporal models [Wikle, 2015, Cressie and Wikle, 2011, Rue and Held, 2005.Sequential Monte Carlo (SMC) methods comprise one of the most successful methodologies for sequential Bayesian inference. However, SMC struggles in high dimensions and these methods are rarely used for dimensions, say, higher than ten . The purpose of the NSMC methodology is to push this limit well beyond the single digits.The basic strategy is to mimic the behavior of a so-called fully adapted (or locally optimal) SMC algorithm. Full adaptation can drastically improve the efficiency of SMC in high dimensions [Snyder et al., 2015]. Unfortunately, it can rarely be implemented in practice since the fully adapted proposal distributions are typically intractable. NSMC addresses this difficulty by requiring only approximate, properly weighted, samples from the proposal distribution. This enables us to use a second layer of SMC to simulate approximately from the proposal. The proper weighting condition ensures the validity of NSMC, thus providing a generalisation of the family of SMC methods. This paper extends preliminary work [Naesseth et al., 2015a] with the ability to handle more expressive models, more informative 2 central limit theorems and convergence proofs, as well as new experiments.

Sampling methods for solving Bayesian model updating problems: A tutorial

“…And furthermore, the SMC sampler is less efficient and effective in sampling from a high-dimensional posterior [188,189] as shown in Section 4.3 which is attributed to the inefficiency and inapplicability of the Importance sampling procedure to samples in high dimensions [187]. To overcome this issue, one can turn to advanced SMC sampling strategies such as the through the use of an adaptive MCMC mutation kernel proposed in [193], or the nested SMC sampling approach [158,159].…”

“…online learning) at every iteration j [153,154]. Over the years, numerous advanced SMC sampling strategies have been proposed such as the block sampling strategies [155], adaptive resampling strategies [156], adaptive SMC sampler [157], and nested SMC strategies [158,159]. However, in this paper, we shall only discuss the basic SMC sampler algorithm proposed by [153] which will be adopted to sample from static posteriors as per the case of the numerical examples presented in this paper.…”

A survey of Monte Carlo methods for parameter estimation

“…The idea of a summary sample/weight has been implicitly used in different SMC schemes proposed in literature, for instance, for the communication among parallel particle filters [305][306][307], and in the particle island methods [297,308,309]. GIS also appears indirectly in particle filtering for model selection [304,310,311], and in the so-called Nested Sequential Monte Carlo techniques [302,312,313]. For further observations and applications of GIS see [301,303].…”

A Rao-Blackwellized Particle Filter With Variational Inference for State Estimation With Measurement Model Uncertainties

“…Several approaches can be found in the literature for jointly estimating state and static parameter in the frame of the RBPF. The common way is to replace analytical methods with different types of SMC samplers for implementing the parameter estimation, such as the particle Markov chain Monte Carlo [26], the twisted particle filter [27] and the nested particle filter [28], [29]. However, these approaches cannot be easily applied to cases where the parameters appearing in the measurement equation are time-varying.…”