Sequential Monte Carlo (SMC) is a powerful technique originally developed for particle filtering and Bayesian inference. As a generic optimizer for statistical and nonstatistical objectives, its role is far less known. Density-tempered SMC is a highly efficient sampling technique ideally suited for challenging global optimization problems and is implementable with a somewhat arbitrary initialization sampler instead of relying on a prior distribution. SMC optimization is anchored at the fact that all optimization tasks (continuous, discontinuous, combinatorial, or noisy objective function) can be turned into sampling under a density or probability function short of a norming constant. The point with the highest functional value is the SMC estimate for the maximum. Through examples, we systematically present various density-tempered SMC algorithms and their superior performance vs. other techniques like Markov Chain Monte Carlo. Data cloning and k-fold duplication are two easily implementable accuracy accelerators, and their complementarity is discussed. The Extreme Value Theorem on the maximum order statistic can also help assess the quality of the SMC optimum. Our coverage includes the algorithmic essence of the density-tempered SMC with various enhancements and solutions for (1) a bimodal nonstatistical function without and with constraints, (2) a multidimensional step function, (3) offline and online optimizations, (4) combinatorial variable selection, and (5) noninvertibility of the Hessian. This article is categorized under: Statistical and Graphical Methods of Data Analysis > Monte Carlo Methods Algorithms and Computational Methods > Stochastic Optimization Algorithms and Computational Methods > Integer Programming

In this paper we consider sequential joint state and static parameter estimation given discrete time observations associated to a partially observed stochastic partial differential equation (SPDE). It is assumed that one can only estimate the hidden state using a discretization of the model. In this context, it is known that the multi-index Monte Carlo (MIMC) method of [11] can be used to improve over direct Monte Carlo from the most precise discretizaton. However, in the context of interest, it cannot be directly applied, but rather must be used within another advanced method such as sequential Monte Carlo (SMC). We show how one can use the MIMC method by renormalizing the MI identity and approximating the resulting identity using the SMC$^2$ method of [5]. We prove that our approach can reduce the cost to obtain a given mean square error (MSE), relative to just using SMC$^2$ on the most precise discretization. We demonstrate this with some numerical examples.

This paper considers a new approach to using Markov chain Monte Carlo (MCMC) in contexts where one may adopt multilevel (ML) Monte Carlo. The underlying problem is to approximate expectations w.r.t. an underlying probability measure that is associated to a continuum problem, such as a continuous-time stochastic process. It is then assumed that the associated probability measure can only be used (e.g. sampled) under a discretized approximation. In such scenarios, it is known that to achieve a target error, the computational effort can be reduced when using MLMC relative to exact sampling from the most accurate discretized probability. The ideas rely upon introducing hierarchies of the discretizations where less accurate approximations cost less to compute, and using an appropriate collapsing sum expression for the target expectation. If a suitable coupling of the probability measures in the hierarchy is achieved, then a reduction in cost is possible. This article focused on the case where exact sampling from such coupling is not possible. We show that one can construct suitably coupled MCMC kernels when given only access to MCMC kernels which are invariant with respect to each discretized probability measure. We prove, under assumptions, that this coupled MCMC approach in a ML context can reduce the cost to achieve a given error, relative to exact sampling. Our approach is illustrated on a numerical example.

We consider a class of finite-time horizon nonlinear stochastic optimal control problem. Although the optimal control admits a path integral representation for this class of control problems, efficient computation of the associated path integrals remains a challenging task. We propose a new Monte Carlo approach that significantly improves upon existing methodology. We tackle the issue of exponential growth in variance with the time horizon by casting optimal control estimation as a smoothing problem for a state-space model, and applying smoothing algorithms based on particle Markov chain Monte Carlo. To further reduce the cost, we then develop a multilevel Monte Carlo method which allows us to obtain an estimator of the optimal control with mean squared error with a cost of . In contrast, a cost of is required for the existing methodology to achieve the same mean squared error. Our approach is illustrated on two numerical examples.

Multivariate stochastic volatility models are a popular and well-known class of models in the analysis of financial time series because of their abilities to capture the important stylized facts of financial returns data. We consider the problems of filtering distribution estimation and also marginal likelihood calculation for multivariate stochastic volatility models with cross-leverage effects in the high dimensional case, that is when the number of financial time series that we analyze simultaneously (denoted by d) is large. The standard particle filter has been widely used in the literature to solve these intractable inference problems. It has excellent performance in low to moderate dimensions, but collapses in the high dimensional case. In this article, two new and advanced particle filters proposed in [4], named the space-time particle filter and the marginal space-time particle filter, are explored for these estimation problems. The better performance in both the accuracy and stability for the two advanced particle filters are shown using simulation and empirical studies in comparison with the standard particle filter. In addition, Bayesian static model parameter estimation problem is considered with the advances in particle Markov chain Monte Carlo methods. The particle marginal Metropolis-Hastings algorithm is applied together with the likelihood estimates from the space-time particle filter to infer the static model parameter successfully when that using the likelihood estimates from the standard particle filter fails.

We consider the problem of the computation of smoothed additive functional, which are some integrals with respect to the joint smoothing distribution. It is a key issue in inference for general state-space models as these quantities appear naturally for maximum likelihood parameter inference. The computation of smoothed additive functionals is very challenging as exact computations are not possible for non-linear non-Gaussian state-space models. It becomes even more difficult when the hidden state lies in a high dimensional space because traditional numerical methods suffer from the curse of dimensionality. We propose a new algorithm to efficiently calculate the smoothed additive functionals in an online manner for a specific family of high-dimensional state-space models in discrete time, which is named the Space-Time Forward Smoothing (STFS) algorithm. The cost of this algorithm is at least O(N2d2T), which is polynomial in d. T and N denote the number of time steps and the number of particles respectively, while d is the dimension of the hidden state space. Its superior performance over other existing methods is illustrated by various simulation studies. Moreover, STFS algorithm is successfully applied to perform Maximum Likelihood estimation for static model parameters both in an online and an offline manner.