## 1. Introduction

Monte Carlo methods have become one of the standard tools of the statistician's apparatus and among other things have allowed the Bayesian paradigm to be routinely applied to ever more sophisticated models. However, expectations are constantly rising and such methods are now expected to deal with high dimensionality and complex patterns of dependence in statistical models. In this paper we propose a novel addition to the Monte Carlo toolbox named particle Markov chain Monte Carlo (PMCMC) methods. They rely on a non-trivial and non-standard combination of MCMC and sequential Monte Carlo (SMC) methods which takes advantage of the strength of its two components. Several algorithms combining MCMC and SMC approaches have already been proposed in the literature. In particular, MCMC kernels have been used to build proposal distributions for SMC algorithms (Gilks and Berzuini, 2001). Our approach is entirely different as we use SMC algorithms to design efficient high dimensional proposal distributions for MCMC algorithms. As we shall see, our framework is particularly suitable for inference in state space models (SSMs) but extends far beyond this application of choice and allows us to push further the boundaries of the class of problems that can be routinely addressed by using MCMC methods.

To be more specific, the successful design of most practical Monte Carlo algorithms to sample from a target distribution, say *π*, in scenarios involving both high dimension and complex patterns of dependence relies on the appropriate choice of proposal distributions. As a rule of thumb, to lead to efficient algorithms, such distributions should both be easy to sample from and capture some of the important characteristics of *π*, such as its scale or dependence structure. Whereas the design of such efficient proposal distributions is often feasible in small dimensions, this proves to be much more difficult in larger scenarios. The classical solution that is exploited by both MCMC and SMC methods, albeit in differing ways, consists of breaking up the original sampling problem into smaller and simpler sampling problems by focusing on some of the subcomponents of *π*. This results in an easier design of proposal distributions. This relative ease of implementation comes at a price, however, as such local strategies inevitably ignore some of the global features of the target distribution *π*, resulting in potentially poor performance. The art of designing Monte Carlo algorithms mainly resides in the adoption of an adequate trade-off between simplicity of implementation and the often difficult incorporation of important characteristics of the target distribution. Our novel approach exploits differing strengths of MCMC and SMC algorithms, which allow us to design efficient and flexible MCMC algorithms for important classes of statistical models, while typically requiring limited design effort on the user's part. This is illustrated later in the paper (Section 3) where, even using standard off-the-shelf components, our methodology allows us straightforwardly to develop efficient MCMC algorithms for important models for which no satisfactory solution is currently available.

The rest of the paper is organized as follows. Section 2 is entirely dedicated to inference in SSMs. This class of models is ubiquitous in applied science and lends itself particularly well to the exposition of our methodology. We show that PMCMC algorithms can be thought of as natural approximations to standard and ‘idealized’ MCMC algorithms which cannot be implemented in practice. This section is entirely descriptive both for pedagogical purposes and to demonstrate the conceptual and implementational simplicity of the resulting algorithms. In Section 3, we demonstrate the efficiency of our methodology on a non-linear SSM and a Lévy-driven stochastic volatility model. We first show that PMCMC sampling allows us to perform Bayesian inference simply in non-linear non-Gaussian scenarios where standard MCMC methods can fail. Second, we demonstrate that it is an effective method in situations where using the prior distribution of the underlying latent process as the proposal distribution is the only known practical possibility. In Section 4 we provide a simple and complete formal justification for the validity and properties of PMCMC algorithms. Key to our results is the realization that such seemingly approximate algorithms sample from an artificial distribution which admits our initial target distribution of interest as one of its components. The framework that is considered is somewhat more abstract and general than that for SSMs but has the advantage of applicability far beyond this class of models. In Section 5 we discuss connections to previous work and potential extensions.