Smoothing with Couplings of Conditional Particle Filters
Pierre E. Jacob, Fredrik Lindsten, Thomas B. Schön
备注：
This document is a self-contained and direct description of the smoothing method introduced in Coupling of Particle Filters (arXiv:1606.01156). Code is available at github.com/pierrejacob/CoupledCPF. Compared to the previous version, a bug was fixed in the code, and the numerical results were updated
In state space models, smoothing refers to the task of estimating a latent stochastic process given noisy measurements related to the process. We propose an unbiased estimator of smoothing expectations. The lack-of-bias property has methodological benefits: independent estimators can be generated in parallel, and confidence intervals can be constructed from the central limit theorem to quantify the approximation error. To design unbiased estimators, we combine a generic debiasing technique for Markov chains with a Markov chain Monte Carlo algorithm for smoothing. The resulting procedure is widely applicable and we show in numerical experiments that the removal of the bias comes at a manageable increase in variance. We establish the validity of the proposed estimators under mild assumptions. Numerical experiments are provided on toy models, including a setting of highly-informative observations, and a realistic Lotka-Volterra model with an intractable transition density.
In this note we introduce an estimate for the marginal likelihood associated to hidden Markov models (HMMs) using sequential Monte Carlo (SMC) approximations of the generalized two-filter smoothing decomposition (Briers, 2010). This estimate is shown to be unbiased and a central limit theorem (CLT) is established. This latter CLT also allows one to prove a CLT associated to estimates of expectations w.r.t. a marginal of the joint smoothing distribution; these form some of the first theoretical results associated to the SMC approximation of the generalized two-filter smoothing decomposition. Th…
We consider approximate maximum likelihood parameter estimation in nonlinear state-space models. We discuss both direct optimization of the likelihood and expectation--maximization (EM). For EM, we also give closed-form expressions for the maximization step in a class of models that are linear in parameters and have additive noise. To obtain approximations to the filtering and smoothing distributions needed in the likelihood-maximization methods, we focus on using Gaussian filtering and smoothing algorithms that employ sigma-points to approximate the required integrals. We discuss different si…
Recently, Fasano, Rebaudo, Durante and Petrone (2019) provided closed-form expressions for the filtering, predictive and smoothing distributions of multivariate dynamic probit models, leveraging on unified skew-normal distribution properties. This allows to develop algorithms to draw independent and identically distributed samples from such distributions, as well as sequential Monte Carlo procedures for the filtering and predictive distributions, allowing to overcome computational bottlenecks that may arise for large sample sizes. In this paper, we briefly review the above-mentioned closed-for…
"Particle methods" are sequential Monte Carlo algorithms, typically involving importance sampling, that are used to estimate and sample from joint and marginal densities from a collection of a, presumably increasing, number of random variables. In particular, a particle filter aims to estimate the current state $X_{n}$ of a stochastic system that is not directly observable by estimating a posterior distribution $π(x_{n}|y_{1},y_{2}, \ldots, y_{n})$ where the $\{Y_{n}\}$ are observations related to the $\{X_{n}\}$ through some measurement model $π(y_{n}|x_{n})$. A particle smoother aims to esti…
Particle learning (PL) provides state filtering, sequential parameter learning and smoothing in a general class of state space models. Our approach extends existing particle methods by incorporating the estimation of static parameters via a fully-adapted filter that utilizes conditional sufficient statistics for parameters and/or states as particles. State smoothing in the presence of parameter uncertainty is also solved as a by-product of PL. In a number of examples, we show that PL outperforms existing particle filtering alternatives and proves to be a competitor to MCMC.
Gerber and Chopin (2015) recently introduced Sequential quasi-Monte Carlo (SQMC) algorithms as an efficient way to perform filtering in state-space models. The basic idea is to replace random variables with low-discrepancy point sets, so as to obtain faster convergence than with standard particle filtering. Gerber and Chopin (2015) describe briefly several ways to extend SQMC to smoothing, but do not provide supporting theory for this extension. We discuss more thoroughly how smoothing may be performed within SQMC, and derive convergence results for the so-obtained smoothing algorithms. We con…
Smoothing in state-space models amounts to computing the conditional distribution of the latent state trajectory, given observations, or expectations of functionals of the state trajectory with respect to this distributions. For models that are not linear Gaussian or possess finite state space, smoothing distributions are in general infeasible to compute as they involve intergrals over a space of dimensionality at least equal to the number of observations. Recent years have seen an increased interest in Monte Carlo-based methods for smoothing, often involving particle filters. One such method …
This paper focuses on Sequential Monte Carlo approximations of smoothing distributions in conditionally linear and Gaussian state spaces. To reduce Monte Carlo variance of smoothers, it is typical in these models to use Rao-Blackwellization: particle approximation is used to sample sequences of hidden regimes while the Gaussian states are explicitly integrated conditional on the sequence of regimes and observations, using variants of the Kalman filter / smoother. The first successful attempt to use Rao-Blackwellization for smoothing extends the Bryson-Frazier smoother for Gaussian linear state…
We introduce a methodology for online estimation of smoothing expectations for a class of additive functionals, in the context of a rich family of diffusion processes (that may include jumps) -- observed at discrete-time instances. We overcome the unavailability of the transition density of the underlying SDE by working on the augmented pathspace. The new method can be applied, for instance, to carry out online parameter inference for the designated class of models. Algorithms defined on the infinite-dimensional pathspace have been developed in the last years mainly in the context of MCMC tech…
In this article we consider recursive approximations of the smoothing distribution associated to partially observed stochastic differential equations (SDEs), which are observed discretely in time. Such models appear in a wide variety of applications including econometrics, finance and engineering. This problem is notoriously challenging, as the smoother is not available analytically and hence require numerical approximation. This usually consists by applying a time-discretization to the SDE, for instance the Euler method, and then applying a numerical (e.g. Monte Carlo) method to approximate t…
Particle smoothing methods are used for inference of stochastic processes based on noisy observations. Typically, the estimation of the marginal posterior distribution given all observations is cumbersome and computational intensive. In this paper, we propose a simple algorithm based on path integral control theory to estimate the smoothing distribution of continuous-time diffusion processes with partial observations. In particular, we use an adaptive importance sampling method to improve the effective sampling size of the posterior over processes given the observations and the reliability of …
We provide a new strategy built on the divide-and-conquer approach by Lindsten et al. (2017) to investigate the smoothing problem in a hidden Markov model. We employ this approach to decompose a hidden Markov model into sub-models with intermediate target distributions based on an auxiliary tree structure and produce independent samples from the sub-models at the leaf nodes towards the original model of interest at the root. We review the target distribution in the sub-models suggested by Lindsten et al. and propose two new classes of target distributions, which are the estimates of the (joint…
Sequential Monte Carlo (SMC) methods, such as the particle filter, are by now one of the standard computational techniques for addressing the filtering problem in general state-space models. However, many applications require post-processing of data offline. In such scenarios the smoothing problem--in which all the available data is used to compute state estimates--is of central interest. We consider the smoothing problem for a class of conditionally linear Gaussian models. We present a forward-backward-type Rao-Blackwellized particle smoother (RBPS) that is able to exploit the tractable subst…
Non-Gaussian state-space models arise in several applications, and within this framework the binary time series setting provides a relevant example. However, unlike for Gaussian state-space models - where filtering, predictive and smoothing distributions are available in closed form - binary state-space models require approximations or sequential Monte Carlo strategies for inference and prediction. This is due to the apparent absence of conjugacy between the Gaussian states and the likelihood induced by the observation equation for the binary data. In this article we prove that the filtering, …
The conditional particle filter (CPF) is a promising algorithm for general hidden Markov model smoothing. Empirical evidence suggests that the variant of CPF with backward sampling (CBPF) performs well even with long time series. Previous theoretical results have not been able to demonstrate the improvement brought by backward sampling, whereas we provide rates showing that CBPF can remain effective with a fixed number of particles independent of the time horizon. Our result is based on analysis of a new coupling of two CBPFs, the coupled conditional backward sampling particle filter (CCBPF). …
Columbia University（哥伦比亚大学）；Cornell University（康奈尔大学）
A body of recent work has focused on constructing a variational family of filtered distributions using Sequential Monte Carlo (SMC). Inspired by this work, we introduce Particle Smoothing Variational Objectives (SVO), a novel backward simulation technique and smoothed approximate posterior defined through a subsampling process. SVO augments support of the proposal and boosts particle diversity. Recent literature argues that increasing the number of samples K to obtain tighter variational bounds may hurt the proposal learning, due to a signal-to-noise ratio (SNR) of gradient estimators decreasi…
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