When implementing Markov Chain Monte Carlo (MCMC) algorithms, perturbation caused by numerical errors is sometimes inevitable. This paper studies how the perturbation of MCMC affects the convergence speed and approximation accuracy. Our results show that when the original Markov chain converges to stationarity fast enough and the perturbed transition kernel is a good approximation to the original transition kernel, the corresponding perturbed sampler has fast convergence speed and high approximation accuracy as well. Our convergence analysis is conducted under either the Wasserstein metric or the $\chi^2$ metric, both are widely used in the literature. The results can be extended to obtain non-asymptotic error bounds for MCMC estimators. We demonstrate how to apply our convergence and approximation results to the analysis of specific sampling algorithms, including Random walk Metropolis, Metropolis adjusted Langevin algorithm with perturbed target densities, and parallel tempering Monte Carlo with perturbed densities. Finally, we present some simple numerical examples to verify our theoretical claims.

In this paper we consider the filtering of partially observed multidimensional diffusion processes that are observed regularly at discrete times. This is a challenging problem which requires the use of advanced numerical schemes based upon time-discretization of the diffusion process and then the application of particle filters. Perhaps the state-of-the-art method for moderate-dimensional problems is the multilevel particle filter of Jasra et al. (SIAM J. Numer. Anal. 55 (2017), 3068–3096). This is a method that combines multilevel Monte Carlo and particle filters. The approach in that article is based intrinsically upon an Euler discretization method. We develop a new particle filter based upon the antithetic truncated Milstein scheme of Giles and Szpruch (Ann. Appl. Prob. 24 (2014), 1585–1620). We show empirically for a class of diffusion problems that, for $\epsilon>0$ given, the cost to produce a mean squared error (MSE) of $\mathcal{O}(\epsilon^2)$ in the estimation of the filter is $\mathcal{O}(\epsilon^{-2}\log(\epsilon)^2)$. In the case of multidimensional diffusions with non-constant diffusion coefficient, the method of Jasra et al. (2017) requires a cost of $\mathcal{O}(\epsilon^{-2.5})$ to achieve the same MSE.

In this article we consider the estimation of the log-normalization constant associated to a class of continuous-time filtering models. In particular, we consider ensemble Kalman–Bucy filter estimates based upon several nonlinear Kalman–Bucy diffusions. Using new conditional bias results for the mean of the aforementioned methods, we analyze the empirical log-scale normalization constants in terms of their $\mathbb{L}_n$ -errors and $\mathbb{L}_n$ -conditional bias. Depending on the type of nonlinear Kalman–Bucy diffusion, we show that these are bounded above by terms such as $\mathsf{C}(n)\left[t^{1/2}/N^{1/2} + t/N\right]$ or $\mathsf{C}(n)/N^{1/2}$ ( $\mathbb{L}_n$ -errors) and $\mathsf{C}(n)\left[t+t^{1/2}\right]/N$ or $\mathsf{C}(n)/N$ ( $\mathbb{L}_n$ -conditional bias), where t is the time horizon, N is the ensemble size, and $\mathsf{C}(n)$ is a constant that depends only on n, not on N or t. Finally, we use these results for online static parameter estimation for the above filtering models and implement the methodology for both linear and nonlinear models.

In this article we consider a Monte-Carlo-based method to filter partially observed diffusions observed at regular and discrete times. Given access only to Euler discretizations of the diffusion process, we present a new procedure which can return online estimates of the filtering distribution with no time-discretization bias and finite variance. Our approach is based upon a novel double application of the randomization methods of Rhee and Glynn (Operat. Res.63, 2015) along with the multilevel particle filter (MLPF) approach of Jasra et al. (SIAM J. Numer. Anal.55, 2017). A numerical comparison of our new approach with the MLPF, on a single processor, shows that similar errors are possible for a mild increase in computational cost. However, the new method scales strongly to arbitrarily many processors.

In this article we prove new central limit theorems (CLTs) for several coupled particle filters (CPFs). CPFs are used for the sequential estimation of the difference of expectations with respect to filters which are in some sense close. Examples include the estimation of the filtering distribution associated to different parameters (finite difference estimation) and filters associated to partially observed discretized diffusion processes (PODDP) and the implementation of the multilevel Monte Carlo (MLMC) identity. We develop new theory for CPFs, and based upon several results, we propose a new CPF which approximates the maximal coupling (MCPF) of a pair of predictor distributions. In the context of ML estimation associated to PODDP with time-discretization $\Delta_l=2^{-l}$ , $l\in\{0,1,\dots\}$ , we show that the MCPF and the approach of Jasra, Ballesio, et al. (2018) have, under certain assumptions, an asymptotic variance that is bounded above by an expression that is of (almost) the order of $\Delta_l$ ( $\mathcal{O}(\Delta_l)$ ), uniformly in time. The $\mathcal{O}(\Delta_l)$ bound preserves the so-called forward rate of the diffusion in some scenarios, which is not the case for the CPF in Jasra et al. (2017).

In this paper we present results on the concentration properties of the smoothing and filtering distributions of some partially observed chaotic dynamical systems. We show that, rather surprisingly, for the geometric model of the Lorenz equations, as well as some other chaotic dynamical systems, the smoothing and filtering distributions do not concentrate around the true position of the signal, as the number of observations tends to ∞. Instead, under various assumptions on the observation noise, we show that the expected value of the diameter of the support of the smoothing and filtering distributions remains lower bounded by a constant multiplied by the standard deviation of the noise, independently of the number of observations. Conversely, under rather general conditions, the diameter of the support of the smoothing and filtering distributions are upper bounded by a constant multiplied by the standard deviation of the noise. To some extent, applications to the three-dimensional Lorenz 63 model and to the Lorenz 96 model of arbitrarily large dimension are considered.

We consider the numerical approximation of the filtering problem in high dimensions, that is, when the hidden state lies in ℝd with large d. For low-dimensional problems, one of the most popular numerical procedures for consistent inference is the class of approximations termed particle filters or sequential Monte Carlo methods. However, in high dimensions, standard particle filters (e.g. the bootstrap particle filter) can have a cost that is exponential in d for the algorithm to be stable in an appropriate sense. We develop a new particle filter, called the space‒time particle filter, for a specific family of state-space models in discrete time. This new class of particle filters provides consistent Monte Carlo estimates for any fixed d, as do standard particle filters. Moreover, when there is a spatial mixing element in the dimension of the state vector, the space‒time particle filter will scale much better with d than the standard filter for a class of filtering problems. We illustrate this analytically for a model of a simple independent and identically distributed structure and a model of an L-Markovian structure (L≥ 1, L independent of d) in the d-dimensional space direction, when we show that the algorithm exhibits certain stability properties as d increases at a cost 𝒪(nNd2), where n is the time parameter and N is the number of Monte Carlo samples, which are fixed and independent of d. Our theoretical results are also supported by numerical simulations on practical models of complex structures. The results suggest that it is indeed possible to tackle some high-dimensional filtering problems using the space‒time particle filter that standard particle filters cannot handle.

We study the asymptotic behavior of a new particle filter approach for the estimation of hidden Markov models. In particular, we develop an algorithm where the latent-state sequence is segmented into multiple shorter portions, with an estimation technique based upon a separate particle filter in each portion. The partitioning facilitates the use of parallel processing, which reduces the wall-clock computational time. Based upon this approach, we introduce new estimators of the latent states and likelihood which have similar or better variance properties compared to estimators derived from standard particle filters. We show that the likelihood function estimator is unbiased, and show asymptotic normality of the underlying estimators.

We consider the time behaviour associated to the sequential Monte Carlo estimate of the backward interpretation of Feynman-Kac formulae. This is particularly of interest in the context of performing smoothing for hidden Markov models. We prove a central limit theorem under weaker assumptions than adopted in the literature. We then show that the associated asymptotic variance expression for additive functionals grows at most linearly in time under hypotheses that are weaker than those currently existing in the literature. The assumptions are verified for some hidden Markov models.

In this paper we develop a collection of results associated to the analysis of the sequential Monte Carlo (SMC) samplers algorithm, in the context of high-dimensional independent and identically distributed target probabilities. The SMC samplers algorithm can be designed to sample from a single probability distribution, using Monte Carlo to approximate expectations with respect to this law. Given a target density in d dimensions our results are concerned with d → ∞, while the number of Monte Carlo samples, N, remains fixed. We deduce an explicit bound on the Monte-Carlo error for estimates derived using the SMC sampler and the exact asymptotic relative -error of the estimate of the normalising constant associated to the target. We also establish marginal propagation of chaos properties of the algorithm. These results are deduced when the cost of the algorithm is O(Nd2).