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.
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.
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".
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.
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…
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…
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 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.
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.
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…
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…
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…
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 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…
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 …
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.…
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…
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 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 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.
We propose a Markov chain Monte Carlo (MCMC) scheme to perform state inference in non-linear non-Gaussian state-space models. Current state-of-the-art methods to address this problem rely on particle MCMC techniques and its variants, such as the iterated conditional Sequential Monte Carlo (cSMC) scheme, which uses a Sequential Monte Carlo (SMC) type proposal within MCMC. A deficiency of standard SMC proposals is that they only use observations up to time $t$ to propose states at time $t$ when an entire observation sequence is available. More sophisticated SMC based on lookahead techniques coul…
The particle Gibbs (PG) sampler is a Markov Chain Monte Carlo (MCMC) algorithm, which uses an interacting particle system to perform the Gibbs steps. Each Gibbs step consists of simulating a particle system conditioned on one particle path. It relies on a conditional Sequential Monte Carlo (cSMC) method to create the particle system. We propose a novel interpretation of the cSMC algorithm as a perturbed Sequential Monte Carlo (SMC) method and apply telescopic decompositions developed for the analysis of SMC algorithms \cite{delmoral2004} to derive a bound for the distance between the expected …
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 …
Bayesian filtering aims at tracking sequentially a hidden process from an observed one. In particular, sequential Monte Carlo (SMC) techniques propagate in time weighted trajectories which represent the posterior probability density function (pdf) of the hidden process given the available observations. On the other hand, Conditional Monte Carlo (CMC) is a variance reduction technique which replaces the estimator of a moment of interest by its conditional expectation given another variable. In this paper we show that up to some adaptations, one can make use of the time recursive nature of SMC a…
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…
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…
This paper presents an improved result on the negative-binomial Monte Carlo technique analyzed in a previous paper for the estimation of an unknown probability p. Specifically, the confidence level associated to a relative interval [p/μ_2, pμ_1], with μ_1, μ_2 > 1, is proved to exceed its asymptotic value for a broader range of intervals than that given in the referred paper, and for any value of p. This extends the applicability of the estimator, relaxing the conditions that guarantee a given confidence level.
Bayesian computation crucially relies on Markov chain Monte Carlo (MCMC) algorithms. In the case of massive data sets, running the Metropolis-Hastings sampler to draw from the posterior distribution becomes prohibitive due to the large number of likelihood terms that need to be calculated at each iteration. In order to perform Bayesian inference for a large set of time series, we consider an algorithm that combines 'divide and conquer" ideas previously used to design MCMC algorithms for big data with a sequential MCMC strategy. The performance of the method is illustrated using a large set of …
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…
In the following paper we provide a review and development of sequential Monte Carlo (SMC) methods for option pricing. SMC are a class of Monte Carlo-based algorithms, that are designed to approximate expectations w.r.t a sequence of related probability measures. These approaches have been used, successfully, for a wide class of applications in engineering, statistics, physics and operations research. SMC methods are highly suited to many option pricing problems and sensitivity/Greek calculations due to the nature of the sequential simulation. However, it is seldom the case that such ideas are…
Parameter estimation for discretely observed Markov processes is a challenging problem. However, simulation of Markov processes is straightforward using the Gillespie algorithm. We exploit this ease of simulation to develop an effective sequential Monte Carlo (SMC) algorithm for obtaining samples from the posterior distribution of the parameters. In particular, we introduce two key innovations, coupled simulations, which allow us to study multiple parameter values on the basis of a single simulation, and a simple, yet effective, importance sampling scheme for steering simulations towards the o…
Sequential Monte Carlo methods, also known as particle methods, are a popular set of techniques for approximating high-dimensional probability distributions and their normalizing constants. These methods have found numerous applications in statistics and related fields; e.g. for inference in non-linear non-Gaussian state space models, and in complex static models. Like many Monte Carlo sampling schemes, they rely on proposal distributions which crucially impact their performance. We introduce here a class of controlled sequential Monte Carlo algorithms, where the proposal distributions are det…
Consider an irreducible, Harris recurrent Markov chain of transition kernel Π and invariant probability measure π. If Π satisfies a minorization condition, then the split chain allows the identification of regeneration times which may be exploited to obtain perfect samples from π. Unfortunately, many transition kernels associated with complex Markov chain Monte Carlo algorithms are analytically intractable, so establishing a minorization condition and simulating the split chain is challenging, if not impossible. For uniformly ergodic Markov chains with intractable transition kernels, we propos…
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…
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 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…
Sequential Monte Carlo (SMC) methods are a widely used set of computational tools for inference in non-linear non-Gaussian state-space models. We propose a new SMC algorithm to compute the expectation of additive functionals recursively. Essentially, it is an online or forward-only implementation of a forward filtering backward smoothing SMC algorithm proposed in Doucet .et .al (2000). Compared to the standard path space SMC estimator whose asymptotic variance increases quadratically with time even under favourable mixing assumptions, the asymptotic variance of the proposed SMC estimator only …
In a recent paper Beskos et al (2011), the Sequential Monte Carlo (SMC) sampler introduced in Del Moral et al (2006), Neal (2001) has been shown to be asymptotically stable in the dimension of the state space d at a cost that is only polynomial in d, when N the number of Monte Carlo samples, is fixed. More precisely, it has been established that the effective sample size (ESS) of the ensuing (approximate) sample and the Monte Carlo error of fixed dimensional marginals will converge as $d$ grows, with a computational cost of $\mathcal{O}(Nd^2)$. In the present work, further results on SMC metho…
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…
We derive and study SQMC (Sequential Quasi-Monte Carlo), a class of algorithms obtained by introducing QMC point sets in particle filtering. SQMC is related to, and may be seen as an extension of, the array-RQMC algorithm of L'Ecuyer et al. (2006). The complexity of SQMC is $O(N \log N)$, where $N$ is the number of simulations at each iteration, and its error rate is smaller than the Monte Carlo rate $O_P(N^{-1/2})$. The only requirement to implement SQMC is the ability to write the simulation of particle $x_t^n$ given $x_{t-1}^n$ as a deterministic function of $x_{t-1}^n$ and a fixed number o…
One of the key challenges in identifying nonlinear and possibly non-Gaussian state space models (SSMs) is the intractability of estimating the system state. Sequential Monte Carlo (SMC) methods, such as the particle filter (introduced more than two decades ago), provide numerical solutions to the nonlinear state estimation problems arising in SSMs. When combined with additional identification techniques, these algorithms provide solid solutions to the nonlinear system identification problem. We describe two general strategies for creating such combinations and discuss why SMC is a natural tool…
In this paper we assume a multivariate risk model has been developed for a portfolio and its capital derived as a homogeneous risk measure. The Euler (or gradient) principle, then, states that the capital to be allocated to each component of the portfolio has to be calculated as an expectation conditional to a rare event, which can be challenging to evaluate in practice. We exploit the copula-dependence within the portfolio risks to design a Sequential Monte Carlo Samplers based estimate to the marginal conditional expectations involved in the problem, showing its efficiency through a series o…
In this article we consider a Bayesian inverse problem associated to elliptic partial differential equations (PDEs) in two and three dimensions. This class of inverse problems is important in applications such as hydrology, but the complexity of the link function between unknown field and measurements can make it difficult to draw inference from the associated posterior. We prove that for this inverse problem a basic SMC method has a Monte Carlo rate of convergence with constants which are independent of the dimension of the discretization of the problem; indeed convergence of the SMC method i…
Sequential Monte Carlo (SMC) methods are not only a popular tool in the analysis of state space models, but offer an alternative to MCMC in situations where Bayesian inference must proceed via simulation. This paper introduces a new SMC method that uses adaptive MCMC kernels for particle dynamics. The proposed algorithm features an online stochastic optimization procedure to select the best MCMC kernel and simultaneously learn optimal tuning parameters. Theoretical results are presented that justify the approach and give guidance on how it should be implemented. Empirical results, based on ana…
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…
This paper brings explicit considerations of distributed computing architectures and data structures into the rigorous design of Sequential Monte Carlo (SMC) methods. A theoretical result established recently by the authors shows that adapting interaction between particles to suitably control the Effective Sample Size (ESS) is sufficient to guarantee stability of SMC algorithms. Our objective is to leverage this result and devise algorithms which are thus guaranteed to work well in a distributed setting. We make three main contributions to achieve this. Firstly, we study mathematical propertie…
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…
Detection of induced polarisation (IP) effects in airborne electromagnetic (AEM) measurements does not yet have an established methodology. This contribution develops a Bayesian approach to the IP-detectability problem using decoupled transdimensional layered models, and applies an approach novel to geophysics whereby transdimensional proposals are used within the embarrassingly parallelisable and robust static Sequential Monte Carlo (SMC) class of algorithms for the simultaneous inference of parameters and models. Henceforth referring to this algorithm as Reversible Jump Sequential Monte Carl…
An algorithm is proposed that enables the imposition of shape constraints on regression curves, without requiring the constraints to be written as closed-form expressions, nor assuming the functional form of the loss function. This algorithm is based on Sequential Monte Carlo-Simulated Annealing and only relies on an indicator function that assesses whether or not the constraints are fulfilled, thus allowing the enforcement of various complex constraints by specifying an appropriate indicator function without altering other parts of the algorithm. The algorithm is illustrated by fitting ration…
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 …
