Proceedings Article•
Unbiased Smoothing using Particle Independent Metropolis-Hastings
11 Apr 2019-Vol. 89, pp 2378-2387
TL;DR: A simple way of coupling two MCMC chains built using Particle Independent Metropolis–Hastings (PIMH) to produce unbiased smoothing estimators is proposed, which is easier to implement than recently proposed unbiased smoothers.
Abstract: We consider the approximation of expectations with respect to the distribution of a latent Markov process given noisy measurements. This is known as the smoothing problem and is often approached with particle and Markov chain Monte Carlo (MCMC) methods. These methods provide consistent but biased estimators when run for a finite time. We propose a simple way of coupling two MCMC chains built using Particle Independent Metropolis-Hastings (PIMH) to produce unbiased smoothing estimators. Unbiased estimators are appealing in the context of parallel computing, and facilitate the construction of confidence intervals. The proposed scheme only requires access to off-the-shelf Particle Filters (PF) and is thus easier to implement than recently proposed unbiased smoothers. The approach is demonstrated on a Levy-driven stochastic volatility model and a stochastic kinetic model.
Citations
More filters
TL;DR: In this paper, the convergence analysis of a class of sequential Monte Carlo (SMC) methods where the times at which resampling occurs are computed online using criteria such as the effective sample size is studied.
Abstract: Sequential Monte Carlo (SMC) methods are a class of techniques to sample approximately from any sequence of probability distributions using a combination of importance sampling and resampling steps. This paper is concerned with the convergence analysis of a class of SMC methods where the times at which resampling occurs are computed online using criteria such as the effective sample size. This is a popular approach amongst practitioners but there are very few convergence results available for these methods. By combining semigroup techniques with an original coupling argument, we obtain functional central limit theorems and uniform exponential concentration estimates for these algorithms.
153 citations
TL;DR: The theoretical validity of the estimators proposed and their efficiency relative to the underlying MCMC algorithms are established and the performance and limitations of the method are illustrated.
Abstract: Markov chain Monte Carlo (MCMC) methods provide consistent approximations of integrals as the number of iterations goes to ∞. MCMC estimators are generally biased after any fixed number of iterations. We propose to remove this bias by using couplings of Markov chains together with a telescopic sum argument of Glynn and Rhee. The resulting unbiased estimators can be computed independently in parallel. We discuss practical couplings for popular MCMC algorithms. We establish the theoretical validity of the estimators proposed and study their efficiency relative to the underlying MCMC algorithms. Finally, we illustrate the performance and limitations of the method on toy examples, on an Ising model around its critical temperature, on a high dimensional variable‐selection problem, and on an approximation of the cut distribution arising in Bayesian inference for models made of multiple modules.
107 citations
TL;DR: This paper presents a coupling construction between two particle Gibbs updates from different starting points and shows that the coupling probability may be made arbitrarily close to one by increasing the number of particles, and extends particle Gibbs to work with lower variance resampling schemes.
Abstract: The particle Gibbs sampler is a Markov chain Monte Carlo (MCMC) algorithm to sample from the full posterior distribution of a state-space model. It does so by executing Gibbs sampling steps on an extended target distribution defined on the space of the auxiliary variables generated by an interacting particle system. This paper makes the following contributions to the theoretical study of this algorithm. Firstly, we present a coupling construction between two particle Gibbs updates from different starting points and we show that the coupling probability may be made arbitrarily close to one by increasing the number of particles. We obtain as a direct corollary that the particle Gibbs kernel is uniformly ergodic. Secondly, we show how the inclusion of an additional Gibbs sampling step that reselects the ancestors of the particle Gibbs' extended target distribution, which is a popular approach in practice to improve mixing, does indeed yield a theoretically more efficient algorithm as measured by the asymptotic variance. Thirdly, we extend particle Gibbs to work with lower variance resampling schemes. A detailed numerical study is provided to demonstrate the efficiency of particle Gibbs and the proposed variants.
70 citations
Proceedings Article•
01 Jan 2019TL;DR: In this paper, the authors introduce L-lag couplings to generate computable, nonasymptotic upper bound estimates for the total variation or the Wasserstein distance of general Markov chains.
Abstract: Markov chain Monte Carlo (MCMC) methods generate samples that are asymptotically distributed from a target distribution of interest as the number of iterations goes to infinity. Various theoretical results provide upper bounds on the distance between the target and marginal distribution after a fixed number of iterations. These upper bounds are on a case by case basis and typically involve intractable quantities, which limits their use for practitioners. We introduce L-lag couplings to generate computable, non-asymptotic upper bound estimates for the total variation or the Wasserstein distance of general Markov chains. We apply L-lag couplings to the tasks of (i) determining MCMC burn-in, (ii) comparing different MCMC algorithms with the same target, and (iii) comparing exact and approximate MCMC. Lastly, we (iv) assess the bias of sequential Monte Carlo and self-normalized importance samplers.
30 citations
Posted Content•
TL;DR: This article presents this class of methods and a number of recent advances, with the goal of helping statisticians assess the applicability and usefulness of these methods for their purposes.
Abstract: Sequential Monte Carlo samplers provide consistent approximations of sequences of probability distributions and of their normalizing constants, via particles obtained with a combination of importance weights and Markov transitions. This article presents this class of methods and a number of recent advances, with the goal of helping statisticians assess the applicability and usefulness of these methods for their purposes. Our presentation emphasizes the role of bridging distributions for computational and statistical purposes. Numerical experiments are provided on simple settings such as multivariate Normals, logistic regression and a basic susceptible-infected-recovered model, illustrating the impact of the dimension, the ability to perform inference sequentially and the estimation of normalizing constants.
26 citations
Cites background from "Unbiased Smoothing using Particle I..."
...Such assembly of algorithms leads to estimators with new properties, for example lack of bias [Middleton et al., 2019]....
[...]
References
More filters
TL;DR: In this article, a simulation algorithm for the stochastic formulation of chemical kinetics is proposed, which uses a rigorously derived Monte Carlo procedure to numerically simulate the time evolution of a given chemical system.
Abstract: There are two formalisms for mathematically describing the time behavior of a spatially homogeneous chemical system: The deterministic approach regards the time evolution as a continuous, wholly predictable process which is governed by a set of coupled, ordinary differential equations (the “reaction-rate equations”); the stochastic approach regards the time evolution as a kind of random-walk process which is governed by a single differential-difference equation (the “master equation”). Fairly simple kinetic theory arguments show that the stochastic formulation of chemical kinetics has a firmer physical basis than the deterministic formulation, but unfortunately the stochastic master equation is often mathematically intractable. There is, however, a way to make exact numerical calculations within the framework of the stochastic formulation without having to deal with the master equation directly. It is a relatively simple digital computer algorithm which uses a rigorously derived Monte Carlo procedure to numerically simulate the time evolution of the given chemical system. Like the master equation, this “stochastic simulation algorithm” correctly accounts for the inherent fluctuations and correlations that are necessarily ignored in the deterministic formulation. In addition, unlike most procedures for numerically solving the deterministic reaction-rate equations, this algorithm never approximates infinitesimal time increments df by finite time steps At. The feasibility and utility of the simulation algorithm are demonstrated by applying it to several well-known model chemical systems, including the Lotka model, the Brusselator, and the Oregonator.
10,275 citations
"Unbiased Smoothing using Particle I..." refers methods in this paper
...To simulate synthetic data and to run the bootstrap PF, we sample the latent process Xt using Gillespie’s direct method [16], whereby the time to the next event is exponential with rate ∑R r=1 fr(X, c) and reaction r occurs with probability fr(X, c)/ ∑R r=1 fr(X, c)....
[...]
TL;DR: In this paper, the moments and the asymptotic distribution of the realized volatility error were derived under the assumption of a rather general stochastic volatility model, and the difference between realized volatility and the discretized integrated volatility (which is called actual volatility) were estimated.
Abstract: Summary. The availability of intraday data on the prices of speculative assets means that we can use quadratic variation-like measures of activity in financial markets, called realized volatility, to study the stochastic properties of returns. Here, under the assumption of a rather general stochastic volatility model, we derive the moments and the asymptotic distribution of the realized volatility error—the difference between realized volatility and the discretized integrated volatility (which we call actual volatility). These properties can be used to allow us to estimate the parameters of stochastic volatility models without recourse to the use of simulation-intensive methods.
2,207 citations
"Unbiased Smoothing using Particle I..." refers background in this paper
...In the terminology of [4], the integrated volatility is the integral of the spot volatility and the actual volatility is an increment of the integrated volatility over some unit time....
[...]
TL;DR: The authors construct continuous time stochastic volatility models for financial assets where the volatility processes are superpositions of positive Ornstein-Uhlenbeck (OU) processes, and study these models in relation to financial data and theory.
Abstract: Non-Gaussian processes of Ornstein–Uhlenbeck (OU) type offer the possibility of capturing important distributional deviations from Gaussianity and for flexible modelling of dependence structures. This paper develops this potential, drawing on and extending powerful results from probability theory for applications in statistical analysis. Their power is illustrated by a sustained application of OU processes within the context of finance and econometrics. We construct continuous time stochastic volatility models for financial assets where the volatility processes are superpositions of positive OU processes, and we study these models in relation to financial data and theory.
1,991 citations
"Unbiased Smoothing using Particle I..." refers background in this paper
...2 Lévy-driven stochastic volatility Introduced in [3], Lévy-driven stochastic volatility models provide a flexible model for the log-returns of a financial asset....
[...]
Book Chapter•
01 Jan 2008TL;DR: A complete, up-to-date survey of particle filtering methods as of 2008, including basic and advanced particle methods for filtering as well as smoothing.
Abstract: Optimal estimation problems for non-linear non-Gaussian state-space models do not typically admit analytic solutions. Since their introduction in 1993, particle filtering methods have become a very popular class of algorithms to solve these estimation problems numerically in an online manner, i.e. recursively as observations become available, and are now routinely used in fields as diverse as computer vision, econometrics, robotics and navigation. The objective of this tutorial is to provide a complete, up-to-date survey of this field as of 2008. Basic and advanced particle methods for filtering as well as smoothing are presented.
1,860 citations
"Unbiased Smoothing using Particle I..." refers background or methods in this paper
...h(x1:T ) = x1, we have H̄k:m = Hk:m with high probability if N is moderate because of the particle degeneracy problem [13, 23, 25]....
[...]
...All standard resampling schemes -multinomial, residual and systematic- are unbiased [13]....
[...]
...2 Particle methods Particle methods are often used to approximate smoothing expectations [13, 25]....
[...]
...It is also established in [26] that one can even used adaptive resampling procedures [13, 11]....
[...]
TL;DR: In this paper, the authors propose a methodology to sample sequentially from a sequence of probability distributions that are defined on a common space, each distribution being known up to a normalizing constant.
Abstract: Summary. We propose a methodology to sample sequentially from a sequence of probability distributions that are defined on a common space, each distribution being known up to a normalizing constant. These probability distributions are approximated by a cloud of weighted random samples which are propagated over time by using sequential Monte Carlo methods. This methodology allows us to derive simple algorithms to make parallel Markov chain Monte Carlo algorithms interact to perform global optimization and sequential Bayesian estimation and to compute ratios of normalizing constants. We illustrate these algorithms for various integration tasks arising in the context of Bayesian inference.
1,684 citations