In the last decade, sequential Monte-Carlo methods (SMC) emerged as a key tool in computational statistics. These algorithms approximate a sequence of distributions by a sequence of weighted empirical measures associated to a weighted population of particles. These particles and weights are generated recursively according to elementary transformations: mutation and selection. Examples of applications include the sequential Monte-Carlo techniques to solve optimal non-linear filtering problems in state-space models, molecular simulation, genetic optimization, etc.
Despite many theoretical advances the asymptotic property of these approximations remains of course a question of central interest. In this paper, we analyze sequential Monte Carlo methods from an asymptotic perspective, that is, we establish law of large numbers and invariance principle as the number of particles gets large. We introduce the concepts of "weighted sample" consistency and asymptotic normality, and derive conditions under which the mutation and the selection procedure used in the sequential Monte-Carlo build-up preserve these properties. To illustrate our findings, we analyze SMC algorithms to approximate the filtering distribution in state-space models. We show how our techniques allow to relax restrictive technical conditions used in previously reported works and provide grounds to analyze more sophisticated sequential sampling strategies.
The main question concerns another recent advance in sequential Monte Carlo, the use of a mixture transition kernel that automatically adapts to the target distribution (Douc et al. 2006). Is there a class of static inference problems for which the backward-kernel approach is better suited, or is it too early to predict which method may have better performance in a particular situation?
We propose a sequential Markov chain Monte Carlo (SMCMC) algorithm to sample from a sequence of probability distributions, corresponding to posterior distributions at different times in on-line applications. SMCMC proceeds as in usual MCMC but with the stationary distribution updated appropriately each time new data arrive. SMCMC has advantages over sequential Monte Carlo (SMC) in avoiding particle degeneracy issues. We provide theoretical guarantees for the marginal convergence of SMCMC under various settings, including parametric and nonparametric models. The proposed approach is compared to…
We present a sequential Monte Carlo sampler variant of the partial rejection control algorithm, and show that this variant can be considered as a sequential Monte Carlo sampler with a modified mutation kernel. We prove that the new sampler can reduce the variance of the incremental importance weights when compared with standard sequential Monte Carlo samplers. We provide a study of theoretical properties of the new algorithm, and make connections with some existing algorithms. Finally, the sampler is adapted for application under the challenging "likelihood free," approximate Bayesian computat…
Sequential Monte Carlo (SMC) is a methodology for sampling approximately from a sequence of probability distributions of increasing dimension and estimating their normalizing constants. We propose here an alternative methodology named Sequentially Interacting Markov Chain Monte Carlo (SIMCMC). SIMCMC methods work by generating interacting non-Markovian sequences which behave asymptotically like independent Metropolis-Hastings (MH) Markov chains with the desired limiting distributions. Contrary to SMC, SIMCMC allows us to iteratively improve our estimates in an MCMC-like fashion. We establish c…
This paper introduces an open-ended sequential algorithm for computing the p-value of a test using Monte Carlo simulation. It guarantees that the resampling risk, the probability of a different decision than the one based on the theoretical p-value, is uniformly bounded by an arbitrarily small constant. Previously suggested sequential or non-sequential algorithms, using a bounded sample size, do not have this property. Although the algorithm is open-ended, the expected number of steps is finite, except when the p-value is on the threshold between rejecting and not rejecting. The algorithm is s…
Sequential Monte Carlo (SMC) methods represent a classical set of techniques to simulate a sequence of probability measures through a simple selection/mutation mechanism. However, the associated selection functions and mutation kernels usually depend on tuning parameters that are of first importance for the efficiency of the algorithm. A standard way to address this problem is to apply Adaptive Sequential Monte Carlo (ASMC) methods, which consist in exploiting the information given by the history of the sample to tune the parameters. This article is concerned with variance estimation in such A…
A Monte Carlo algorithm is said to be adaptive if it automatically calibrates its current proposal distribution using past simulations. The choice of the parametric family that defines the set of proposal distributions is critical for good performance. In this paper, we present such a parametric family for adaptive sampling on high-dimensional binary spaces. A practical motivation for this problem is variable selection in a linear regression context. We want to sample from a Bayesian posterior distribution on the model space using an appropriate version of Sequential Monte Carlo. Raw versions …
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 functio…
We provide a framework which admits a number of ``marginal'' sequential Monte Carlo (SMC) algorithms as particular cases -- including the marginal particle filter [Klaas et al., 2005, in: Proceedings of Uncertainty in Artificial Intelligence, pp. 308--315], , the independent particle filter [Lin et al., 2005, Journal of the American Statistical Association 100, pp. 1412--1421] and linear-cost Approximate Bayesian Computation SMC [Sisson et al., 2007, Proceedings of the National Academy of Sciences (USA) 104, pp. 1760--1765.]. We provide conditions under which such algorithms obey laws of large…
We present bounds for the finite sample error of sequential Monte Carlo samplers on static spaces. Our approach explicitly relates the performance of the algorithm to properties of the chosen sequence of distributions and mixing properties of the associated Markov kernels. This allows us to give the first finite sample comparison to other Monte Carlo schemes. We obtain bounds for the complexity of sequential Monte Carlo approximations for a variety of target distributions including finite spaces, product measures, and log-concave distributions including Bayesian logistic regression. The bounds…
We present simple conditions under which the limiting genealogical process associated with a class of interacting particle systems with non-neutral selection mechanisms, as the number of particles grows, is a time-rescaled Kingman coalescent. Sequential Monte Carlo algorithms are popular methods for approximating integrals in problems such as non-linear filtering and smoothing which employ this type of particle system. Their performance depends strongly on the properties of the induced genealogical process. We verify the conditions of our main result for standard sequential Monte Carlo algorit…
The paper proposes a new Monte-Carlo simulator combining the advantages of Sequential Monte Carlo simulators and Hamiltonian Monte Carlo simulators. The result is a method that is robust to multimodality and complex shapes to use for inference in presence of difficult likelihoods or target functions. Several examples are provided.
This paper concerns the use of sequential Monte Carlo methods (SMC) for smoothing in general state space models. A well-known problem when applying the standard SMC technique in the smoothing mode is that the resampling mechanism introduces degeneracy of the approximation in the path space. However, when performing maximum likelihood estimation via the EM algorithm, all functionals involved are of additive form for a large subclass of models. To cope with the problem in this case, a modification of the standard method (based on a technique proposed by Kitagawa and Sato) is suggested. Our algor…
In this paper we consider a fractional stochastic volatility model, that is a model in which the volatility may exhibit a long-range dependent or a rough/antipersistent behavior. We propose a dynamic sequential Monte Carlo methodology that is applicable to both long memory and antipersistent processes in order to estimate the volatility as well as the unknown parameters of the model. We establish a central limit theorem for the state and parameter filters and we study asymptotic properties (consistency and asymptotic normality) for the filter. We illustrate our results with a simulation study …
We investigate the properties of a sequential Monte Carlo method where the particle weight that appears in the algorithm is estimated by a positive, unbiased estimator. We present broadly-applicable convergence results, including a central limit theorem, and we discuss their relevance for applications in statistical physics. Using these results, we show that the resampling step reduces the impact of the randomness of the weights on the asymptotic variance of the estimator. In addition, we explore the limits of convergence of the sequential Monte Carlo method, with a focus on almost sure conver…
A discussion on the possibility of reducing the variance of quasi-Monte Carlo estimators in applications. Further details are provided in the accompanying paper "Variance Reduction for Quasi-Monte Carlo".
Statisticians often use Monte Carlo methods to approximate probability distributions, primarily with Markov chain Monte Carlo and importance sampling. Sequential Monte Carlo samplers are a class of algorithms that combine both techniques to approximate distributions of interest and their normalizing constants. These samplers originate from particle filtering for state space models and have become general and scalable sampling techniques. This article describes sequential Monte Carlo samplers and their possible implementations, arguing that they remain under-used in statistics, despite their ab…
Sequential Monte Carlo (SMC) samplers form an attractive alternative to MCMC for Bayesian computation. However, their performance depends strongly on the Markov kernels used to rejuvenate particles. We discuss how to calibrate automatically (using the current particles) Hamiltonian Monte Carlo kernels within SMC. To do so, we build upon the adaptive SMC approach of Fearnhead and Taylor (2013), and we also suggest alternative methods. We illustrate the advantages of using HMC kernels within an SMC sampler via an extensive numerical study.
This paper addresses finite sample stability properties of sequential Monte Carlo methods for approximating sequences of probability distributions. The results presented herein are applicable in the scenario where the start and end distributions in the sequence are fixed and the number of intermediate steps is a parameter of the algorithm. Under assumptions which hold on non-compact spaces, it is shown that the effect of the initial distribution decays exponentially fast in the number of intermediate steps and the corresponding stochastic error is stable in \mathbb{L}_{p} norm.
This paper proposes a family of weighted batch means variance estimators, which are computationally efficient and can be conveniently applied in practice. The focus is on Markov chain Monte Carlo simulations and estimation of the asymptotic covariance matrix in the Markov chain central limit theorem, where conditions ensuring strong consistency are provided. Finite sample performance is evaluated through auto-regressive, Bayesian spatial-temporal, and Bayesian logistic regression examples, where the new estimators show significant computational gains with a minor sacrifice in variance compared…
We describe an "embarrassingly parallel" method for Bayesian phylogenetic inference, annealed Sequential Monte Carlo, based on recent advances in the Sequential Monte Carlo literature such as adaptive determination of annealing parameters. The algorithm provides an approximate posterior distribution over trees and evolutionary parameters as well as an unbiased estimator for the marginal likelihood. This unbiasedness property can be used for the purpose of testing the correctness of posterior simulation software. We evaluate the performance of phylogenetic annealed Sequential Monte Carlo by rev…
Sequential Monte Carlo samplers represent a compelling approach to posterior inference in Bayesian models, due to being parallelisable and providing an unbiased estimate of the posterior normalising constant. In this work, we significantly accelerate sequential Monte Carlo samplers by adopting the L-BFGS Hessian approximation which represents the state-of-the-art in full-batch optimisation techniques. The L-BFGS Hessian approximation has only linear complexity in the parameter dimension and requires no additional posterior or gradient evaluations. The resulting sequential Monte Carlo algorithm…
SMC (Sequential Monte Carlo) is a class of Monte Carlo algorithms for filtering and related sequential problems. Gerber and Chopin (2015) introduced SQMC (Sequential quasi-Monte Carlo), a QMC version of SMC. This paper has two objectives: (a) to introduce Sequential Monte Carlo to the QMC community, whose members are usually less familiar with state-space models and particle filtering; (b) to extend SQMC to the filtering of continuous-time state-space models, where the latent process is a diffusion. A recurring point in the paper will be the notion of dimension reduction, that is how to implem…
Methods of approximate Bayesian computation (ABC) are increasingly used for analysis of complex models. A major challenge for ABC is over-coming the often inherent problem of high rejection rates in the accept/reject methods based on prior:predictive sampling. A number of recent developments aim to address this with extensions based on sequential Monte Carlo (SMC) strategies. We build on this here, introducing an ABC SMC method that uses data-based adaptive weights. This easily implemented and computationally trivial extension of ABC SMC can very substantially improve acceptance rates, as is d…
Sequential Monte Carlo algorithms (also known as particle filters) are popular methods to approximate filtering (and related) distributions of state-space models. However, they converge at the slow $1/\sqrt{N}$ rate, which may be an issue in real-time data-intensive scenarios. We give a brief outline of SQMC (Sequential Quasi-Monte Carlo), a variant of SMC based on low-discrepancy point sets proposed by Gerber and Chopin (2015), which converges at a faster rate, and we illustrate the greater performance of SQMC on autonomous positioning problems.
Many real-world problems require one to estimate parameters of interest, in a Bayesian framework, from data that are collected sequentially in time. Conventional methods for sampling from posterior distributions, such as {Markov Chain Monte Carlo} can not efficiently address such problems as they do not take advantage of the data's sequential structure. To this end, sequential methods which seek to update the posterior distribution whenever a new collection of data become available are often used to solve these types of problems. Two popular choices of sequential method are the Ensemble Kalman…
This is a collection of three written discussions of the paper "sequential quasi-Monte Carlo sampling" by M. Gerber and N. Chopin, following the presentation given before the Royal Statistical Society in London on December 10th, 2014.
Sequential Monte Carlo is a family of algorithms for sampling from a sequence of distributions. Some of these algorithms, such as particle filters, are widely used in the physics and signal processing researches. More recent developments have established their application in more general inference problems such as Bayesian modeling. These algorithms have attracted considerable attentions in recent years as they admit natural and scalable parallelizations. However, these algorithms are perceived to be difficult to implement. In addition, parallel programming is often unfamiliar to many research…
We prove finite sample complexities for sequential Monte Carlo (SMC) algorithms which require only local mixing times of the associated Markov kernels. Our bounds are particularly useful when the target distribution is multimodal and global mixing of the Markov kernel is slow; in such cases our approach establishes the benefits of SMC over the corresponding Markov chain Monte Carlo (MCMC) estimator. The lack of global mixing is addressed by sequentially controlling the bias introduced by SMC resampling procedures. We apply these results to obtain complexity bounds for approximating expectation…
Pricing options is an important problem in financial engineering. In many scenarios of practical interest, financial option prices associated to an underlying asset reduces to computing an expectation w.r.t.~a diffusion process. In general, these expectations cannot be calculated analytically, and one way to approximate these quantities is via the Monte Carlo method; Monte Carlo methods have been used to price options since at least the 1970's. It has been seen in Del Moral, P. \& Shevchenko, P.V. (2014) `Valuation of barrier options using Sequential Monte Carlo' and Jasra, A. \& Del Moral, P.…
In this article we consider importance sampling (IS) and sequential Monte Carlo (SMC) methods in the context of 1-dimensional random walks with absorbing barriers. In particular, we develop a very precise variance analysis for several IS and SMC procedures. We take advantage of some explicit spectral formulae available for these models to derive sharp and explicit estimates; this provides stability properties of the associated normalized Feynman-Kac semigroups. Our analysis allows one to compare the variance of SMC and IS techniques for these models. The work in this article, is one of the few…
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 …
In this paper we propose a flexible and efficient framework for handling multi-armed bandits, combining sequential Monte Carlo algorithms with hierarchical Bayesian modeling techniques. The framework naturally encompasses restless bandits, contextual bandits, and other bandit variants under a single inferential model. Despite the model's generality, we propose efficient Monte Carlo algorithms to make inference scalable, based on recent developments in sequential Monte Carlo methods. Through two simulation studies, the framework is shown to outperform other empirical methods, while also natural…
We derive a new Sequential-Monte-Carlo-based algorithm to estimate the capacity of two-dimensional channel models. The focus is on computing the noiseless capacity of the 2-D one-infinity run-length limited constrained channel, but the underlying idea is generally applicable. The proposed algorithm is profiled against a state-of-the-art method, yielding more than an order of magnitude improvement in estimation accuracy for a given computation time.
We present a sequential Monte Carlo sampler algorithm for the Bayesian analysis of generalised linear mixed models (GLMMs). These models support a variety of interesting regression-type analyses, but performing inference is often extremely difficult, even when using the Bayesian approach combined with Markov chain Monte Carlo (MCMC). The Sequential Monte Carlo sampler (SMC) is a new and general method for producing samples from posterior distributions. In this article we demonstrate use of the SMC method for performing inference for GLMMs. We demonstrate the effectiveness of the method on both…
Bayesian phylogenetic inference is often conducted via local or sequential search over topologies and branch lengths using algorithms such as random-walk Markov chain Monte Carlo (MCMC) or Combinatorial Sequential Monte Carlo (CSMC). However, when MCMC is used for evolutionary parameter learning, convergence requires long runs with inefficient exploration of the state space. We introduce Variational Combinatorial Sequential Monte Carlo (VCSMC), a powerful framework that establishes variational sequential search to learn distributions over intricate combinatorial structures. We then develop nes…
We build on auto-encoding sequential Monte Carlo (AESMC): a method for model and proposal learning based on maximizing the lower bound to the log marginal likelihood in a broad family of structured probabilistic models. Our approach relies on the efficiency of sequential Monte Carlo (SMC) for performing inference in structured probabilistic models and the flexibility of deep neural networks to model complex conditional probability distributions. We develop additional theoretical insights and introduce a new training procedure which improves both model and proposal learning. We demonstrate that…
We propose a new framework for how to use sequential Monte Carlo (SMC) algorithms for inference in probabilistic graphical models (PGM). Via a sequential decomposition of the PGM we find a sequence of auxiliary distributions defined on a monotonically increasing sequence of probability spaces. By targeting these auxiliary distributions using SMC we are able to approximate the full joint distribution defined by the PGM. One of the key merits of the SMC sampler is that it provides an unbiased estimate of the partition function of the model. We also show how it can be used within a particle Marko…
Sequential Monte Carlo (SMC) methods are a class of Monte Carlo methods that are used to obtain random samples of a high dimensional random variable in a sequential fashion. Many problems encountered in applications often involve different types of constraints. These constraints can make the problem much more challenging. In this paper, we formulate a general framework of using SMC for constrained sampling problems based on forward and backward pilot resampling strategies. We review some existing methods under the framework and develop several new algorithms. It is noted that all information o…
In this comment we consider whether QMC methods can be further embedded within SMC schemes in settings in which the transition density of the latent process is intractable and pseudo-marginal methods are deployed.
We explore a general framework in Markov chain Monte Carlo (MCMC) sampling where sequential proposals are tried as a candidate for the next state of the Markov chain. This sequential-proposal framework can be applied to various existing MCMC methods, including Metropolis-Hastings algorithms using random proposals and methods that use deterministic proposals such as Hamiltonian Monte Carlo (HMC) or the bouncy particle sampler. Sequential-proposal MCMC methods construct the same Markov chains as those constructed by the delayed rejection method under certain circumstances. In the context of HMC,…
We propose sequential Monte Carlo based algorithms for maximum likelihood estimation of the static parameters in hidden Markov models with an intractable likelihood using ideas from approximate Bayesian computation. The static parameter estimation algorithms are gradient based and cover both offline and online estimation. We demonstrate their performance by estimating the parameters of three intractable models, namely the alpha-stable distribution, g-and-k distribution, and the stochastic volatility model with alpha-stable returns, using both real and synthetic data.
We introduce a new sequential Monte Carlo algorithm we call the particle cascade. The particle cascade is an asynchronous, anytime alternative to traditional particle filtering algorithms. It uses no barrier synchronizations which leads to improved particle throughput and memory efficiency. It is an anytime algorithm in the sense that it can be run forever to emit an unbounded number of particles while keeping within a fixed memory budget. We prove that the particle cascade is an unbiased marginal likelihood estimator which means that it can be straightforwardly plugged into existing pseudomar…
This paper introduces methodology for performing Bayesian inference sequentially on a sequence of posteriors on spaces of different dimensions. We show how this may be achieved through the use of sequential Monte Carlo (SMC) samplers (Del Moral et al., 2006, 2007), making use of the full flexibility of this framework in order that the method is computationally efficient. In particular, we introduce the innovation of using deterministic transformations to move particles effectively between target distributions with different dimensions. This approach, combined with adaptive methods, yields an e…
We propose kernel sequential Monte Carlo (KSMC), a framework for sampling from static target densities. KSMC is a family of sequential Monte Carlo algorithms that are based on building emulator models of the current particle system in a reproducing kernel Hilbert space. We here focus on modelling nonlinear covariance structure and gradients of the target. The emulator's geometry is adaptively updated and subsequently used to inform local proposals. Unlike in adaptive Markov chain Monte Carlo, continuous adaptation does not compromise convergence of the sampler. KSMC combines the strengths of s…
Multifidelity approximate Bayesian computation (MF-ABC) is a likelihood-free technique for parameter inference that exploits model approximations to significantly increase the speed of ABC algorithms (Prescott and Baker, 2020). Previous work has considered MF-ABC only in the context of rejection sampling, which does not explore parameter space particularly efficiently. In this work, we integrate the multifidelity approach with the ABC sequential Monte Carlo (ABC-SMC) algorithm into a new MF-ABC-SMC algorithm. We show that the improvements generated by each of ABC-SMC and MF-ABC to the efficien…
We propose two new Bayesian smoothing methods for general state-space models with unknown parameters. The first approach is based on the particle learning and smoothing algorithm, but with an adjustment in the backward resampling weights. The second is a new method combining sequential parameter learning and smoothing algorithms for general state-space models. This method is straightforward but effective, and we find it is the best existing Sequential Monte Carlo algorithm to solve the joint Bayesian smoothing problem. We first illustrate the methods on three benchmark models using simulated d…
Closed-form stochastic filtering equations can be derived in a general setting where probability distributions are replaced by some specific outer measures. In this article, we study how the principles of the sequential Monte Carlo method can be adapted for the purpose of practical implementation of these equations. In particular, we explore how sampling can be used to provide support points for the approximation of these outer measures. This step enables practical algorithms to be derived in the spirit of particle filters. The performance of the obtained algorithms is demonstrated in simulati…
We use a multivariate formulation of sequential Monte Carlo filter that utilizes mechanistic models for Ebola virus propagation and available incidence data to simultaneously estimate the disease progression states and the model parameters. This method has the advantage of performing the inference online as the new data becomes available and estimates the evolution of basic reproductive ratio $R_0(t)$ of the Ebola outbreak through time. Our analysis identifies a peak in the basic reproductive ratio close to the time when Ebola cases were reported in Europe and the USA.
Many recent advances in large scale probabilistic inference rely on variational methods. The success of variational approaches depends on (i) formulating a flexible parametric family of distributions, and (ii) optimizing the parameters to find the member of this family that most closely approximates the exact posterior. In this paper we present a new approximating family of distributions, the variational sequential Monte Carlo (VSMC) family, and show how to optimize it in variational inference. VSMC melds variational inference (VI) and sequential Monte Carlo (SMC), providing practitioners with…
