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. Copyright 2006 Royal Statistical Society.

Approximate Bayesian computation (ABC) is a popular approach to address inference problems where the likelihood function is intractable, or expensive to calculate. To improve over Markov chain Monte Carlo (MCMC) implementations of ABC, the use of sequential Monte Carlo (SMC) methods has recently been suggested. Effective SMC algorithms that are currently available for ABC have a computational complexity that is quadratic in the number of Monte Carlo samples [4,17,19,21] and require the careful choice of simulation parameters. In this article an adaptive SMC algorithm is proposed which admits a computational complexity that is linear in the number of samples and determines on-the-fly the simulation parameters. We demonstrate our algorithm on a toy example and a population genetics example.

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In this paper we present a review of populationbased simulation for static inference problems. Such methods can be described as generating a collection of random variables {X n } n=1,...,N in parallel in order to simulate from some target density π (or potentially sequence of target densities). Population-based simulation is important as many challenging sampling problems in applied statistics cannot be dealt with successfully by conventional Markov chain Monte Carlo (MCMC) methods. We summarize population-based MCMC (Geyer, Computing Science and Statistics: The 23rd Symposium on the Interface, pp. 156-163, 1991; Liang and Wong, J. Am. Stat. Assoc. 96, 653-666, 2001) and sequential Monte Carlo samplers (SMC) (Del Moral, Doucet and Jasra, J. Roy. Stat. Soc. Ser. B 68, 411-436, 2006a), providing a comparison of the approaches. We give numerical examples from Bayesian mixture modelling (Richardson and Green, J. Roy. Stat. Soc. Ser. B 59, 731-792, 1997).

We investigate the stability of a Sequential Monte Carlo (SMC) method applied to the problem of sampling from a target distribution on R d for large d. It is well known [9,14,56] that using a single importance sampling step one produces an approximation for the target that deteriorates as the dimension d increases, unless the number of Monte Carlo samples N increases at an exponential rate in d. We show that this degeneracy can be avoided by introducing a sequence of artificial targets, starting from a 'simple' density and moving to the one of interest, using an SMC method to sample from the sequence (see e.g. [20,27,38,48]). Using this class of SMC methods with a fixed number of samples, one can produce an approximation for which the effective sample size (ESS) converges to a random variable εN as d → ∞ with 1 < εN < N . The convergence is achieved with a computational cost proportional to N d 2 . If εN ≪ N , we can raise its value by introducing a number of resampling steps, say m (where m is independent of d). In this case, ESS converges to a random variable εN,m as d → ∞ and limm→∞ εN,m = N . Also, we show that the Monte Carlo error for estimating a fixed dimensional marginal expectation is of order 1 √ N uniformly in d. The results imply that, in high dimensions, SMC algorithms can efficiently control the variability of the importance sampling weights and estimate fixed dimensional marginals at a cost which is less than exponential in d and indicate that, in high dimensions, resampling leads to a reduction in the Monte Carlo error and increase in the ESS.

Abstract. In this paper the filtering of partially observed diffusions, with discrete-time observations, is considered. It is assumed that only biased approximations of the diffusion can be obtained, for choice of an accuracy parameter indexed by l. A multilevel estimator is proposed, consisting of a telescopic sum of increment estimators associated to the successive levels. The work associated to O(ε 2 ) mean-square error between the multilevel estimator and average with respect to the filtering distribution is shown to scale optimally, for example as O(ε −2 ) for optimal rates of convergence of the underlying diffusion approximation. The method is illustrated on some toy examples as well as estimation of interest rate based on real S&P 500 stock price data.

In this article we consider the approximation of expectations w.r.t. probability distributions associated to the solution of partial differential equations (PDEs); this scenario appears routinely in Bayesian inverse problems. In practice, one often has to solve the associated PDE numerically, using, for instance finite element methods and leading to a discretisation bias, with the step-size level h L . In addition, the expectation cannot be computed analytically and one often resorts to Monte Carlo methods. In the context of this problem, it is known that the introduction of the multilevel Monte Carlo (MLMC) method can reduce the amount of computational effort to estimate expectations, for a given level of error. This is achieved via a telescoping identity associated to a Monte Carlo approximation of a sequence of probability distributions with discretisation levels ∞ > h 0 > h 1 · · · > h L . In many practical problems of interest, one cannot achieve an i.i.d. sampling of the associated sequence of probability distributions. A sequential Monte Carlo (SMC) version of the MLMC method is introduced to deal with this problem. It is shown that under appropriate assumptions, the attractive property of a reduction of the amount of computational effort to estimate expectations, for a given level of error, can be maintained within the SMC context. The approach is numerically illustrated on a Bayesian inverse problem.

We consider the inverse problem of estimating the initial condition of a partial differential equation, which is only observed through noisy measurements at discrete time intervals. In particular, we focus on the case where Eulerian measurements are obtained from the time and space evolving vector field, whose evolution obeys the two-dimensional Navier-Stokes equations defined on a torus. This context is particularly relevant to the area of numerical weather forecasting and data assimilation. We will adopt a Bayesian formulation resulting from a particular regularization that ensures the problem is well posed. In the context of Monte Carlo based inference, it is a challenging task to obtain samples from the resulting high dimensional posterior on the initial condition. In real data assimilation applications it is common for computational methods to invoke the use of heuristics and Gaussian approximations. As a result, the resulting inferences are biased and not well-justified in the presence of non-linear dynamics and observations. On the other hand, Monte Carlo methods can be used to assimilate data in a principled manner, but are often perceived as inefficient in this context due to the high-dimensionality of the problem. In this work we will propose a generic Sequential Monte Carlo (SMC) sampling approach for high dimensional inverse problems that overcomes these difficulties. The method builds upon "state of the art" Markov chain Monte Carlo (MCMC) techniques, which are currently considered as benchmarks for evaluating data assimilation algorithms used in practice. SMC samplers can improve in terms of efficiency as they possess greater flexibility and one can include steps like sequential tempering, adaptation and parallelization with relatively low amount of extra computations. We will illustrate this using numerical examples, where our proposed SMC approach can achieve the same accuracy as MCMC but in a much more efficient manner.