Iterative Importance Sampling Research Articles

Iterative importance sampling with Markov chain Monte Carlo sampling in robust Bayesian analysis

Bayesian inference under a set of priors, called robust Bayesian analysis, allows for estimation of parameters within a model and quantification of epistemic uncertainty in quantities of interest by bounded (or imprecise) probability. Iterative importance sampling can be used to estimate bounds on the quantity of interest by optimizing over the set of priors. A method for iterative importance sampling when the robust Bayesian inference relies on Markov chain Monte Carlo (MCMC) sampling is proposed. To accommodate the MCMC sampling in iterative importance sampling, a new expression for the effective sample size of the importance sampling is derived, which accounts for the correlation in the MCMC samples. To illustrate the proposed method for robust Bayesian analysis, iterative importance sampling with MCMC sampling is applied to estimate the lower bound of the overall effect in a previously published meta-analysis with a random effects model. The performance of the method compared to a grid search method and under different degrees of prior-data conflict is also explored.

Quasi‐Online Groundwater Model Optimization Under Constraints of Geological Consistency Based on Iterative Importance Sampling

AbstractThe increasing use of wireless sensor networks and remote sensing permits real‐time access to environmental observations. Data assimilation frameworks tap into such data streams to autonomously update and gradually improve numerical models. In hydrogeology, such methods are relevant in areas of long‐term interest in water quality and quantity, for example, in drinking water production. Unfortunately, accurate hydrogeological predictions often demand a degree of geological realism, which is difficult to reconcile with the operational limitations of many data assimilation frameworks. Alluvial aquifers, for example, are sometimes characterized by paleo‐channels of unknown extent and properties, which may act as preferential flow paths. Gradually optimizing such fields in real‐time or quasi‐real‐time settings is a formidable task. Besides subsurface properties, ill‐specified model forcings are a further source of predictive bias, which an optimizer could learn to compensate. In this study, we explore the use of a quasi‐online optimizer based on the iterative batch importance sampling framework for a groundwater model of a field site near Valdobbiadene, Italy. This site is characterized by the presence of paleo‐channels and heavily exploited for drinking water production and irrigation. We use Markov chain Monte Carlo steps to explore new parameterizations while maintaining consistency between states and parameters as well as conformance to a multipoint statistics training image. We also optimize a preprocessor designed to compensate for potential bias in the model forcing. We achieve promising and geologically consistent quasi‐real‐time optimization, albeit at the loss of parameter uncertainty.

A surrogate‐based iterative importance sampling method for structural reliability analysis

AbstractIn this paper, a surrogate‐based iterative importance sampling (IS) method is proposed to efficiently evaluate failure probability (FP) of engineering structures. The surrogate is constructed by the training points centered at the most probable failure point (MPP), which is estimated by an approximate algorithm without any additional evaluations of the real limit state function (LSF), and the IS method using the estimated MPP is further obtained; and an iterative procedure combined with the surrogate and the IS method is finally proposed to improve the accuracy of the estimated result for the FP. Three numerical examples are employed to demonstrate the accuracy of the proposed method, and further, an inside flap of an aircraft is employed to demonstrate the practical application of the proposed method in engineering problems. The results show that the reasonable estimated result of the FP within the range of 10−3‐10−5 can be evaluated by the proposed method with only hundreds of the evaluations of the real LSF.

Imprecise Monte Carlo simulation and iterative importance sampling for the estimation of lower previsions

We develop a theoretical framework for studying numerical estimation of lower previsions, generally applicable to two-level Monte Carlo methods, importance sampling methods, and a wide range of other sampling methods one might devise. We link consistency of these estimators to Glivenko–Cantelli classes, and for the sub-Gaussian case we show how the correlation structure of this process can be used to bound the bias and prove consistency. We also propose a new upper estimator, which can be used along with the standard lower estimator, in order to provide a simple confidence interval. As a case study of this framework, we then discuss how importance sampling can be exploited to provide accurate numerical estimates of lower previsions. We propose an iterative importance sampling method to drastically improve the performance of imprecise importance sampling. We demonstrate our results on the imprecise Dirichlet model.

Trajectory reweighting for non-equilibrium steady states

Modern methods for sampling rugged landscapes in state space mainly rely on knowledge of the relative probabilities of microstates, which is given by the Boltzmann factor for equilibrium systems. In principle, trajectory reweighting provides an elegant way to extend these algorithms to non-equilibrium systems, by numerically calculating the relative weights that can be directly substituted for the Boltzmann factor. We show that trajectory reweighting has many commonalities with Rosenbluth sampling for chain macromolecules, including practical problems which stem from the fact that both are iterated importance sampling schemes: for long trajectories the distribution of trajectory weights becomes very broad and trajectories carrying high weights are infrequently sampled, yet long trajectories are unavoidable in rugged landscapes. For probing the probability landscapes of genetic switches and similar systems, these issues preclude the straightforward use of trajectory reweighting. The analogy to Rosenbluth sampling suggests though that path-ensemble methods such as PERM (pruned-enriched Rosenbluth method) could provide a way forward.

On the asymptotic efficiency of approximate Bayesian computation estimators

Many statistical applications involve models for which it is difficult to evaluate the likelihood, but from which it is relatively easy to sample. Approximate Bayesian computation is a likelihood-free method for implementing Bayesian inference in such cases. We present results on the asymptotic variance of estimators obtained using approximate Bayesian computation in a large-data limit. Our key assumption is that the data are summarized by a fixed-dimensional summary statistic that obeys a central limit theorem. We prove asymptotic normality of the mean of the approximate Bayesian computation posterior. This result also shows that, in terms of asymptotic variance, we should use a summary statistic that is the same dimension as the parameter vector, p; and that any summary statistic of higher dimension can be reduced, through a linear transformation, to dimension p in a way that can only reduce the asymptotic variance of the posterior mean. We look at how the Monte Carlo error of an importance sampling algorithm that samples from the approximate Bayesian computation posterior affects the accuracy of estimators. We give conditions on the importance sampling proposal distribution such that the variance of the estimator will be the same order as that of the maximum likelihood estimator based on the summary statistics used. This suggests an iterative importance sampling algorithm, which we evaluate empirically on a stochastic volatility model.

An Improved Localization Scheme Based on PMCL Method for Large-Scale Mobile Wireless Aquaculture Sensor Networks

Localization is crucial to many applications in wireless sensor networks (WSNs) because measurement data or information exchanges happened in WSNs without location information are meaningless. Most localization schemes for mobile WSNs are based on Sequential Monte Carlo (SMC) algorithm. These SMC-based methods often suffer from too many iterations, sample impoverishment and less sample diversity, which leads to low sampling and filtering efficiency, and consequently low localization accuracy and high localization costs. In this paper, we propose an improved range-free localization scheme for mobile WSNs based on improved Population Monte Carlo localization (PMCL) method, accompanying with Hidden Terminal Couple scheme. A population of probability density functions is proposed to approximate the distribution of unknown locations based on a set of observations through an iterative importance sampling procedure. Behaviors are enhanced by adopting three improved methods to increase accuracy, enhance delay and save cost. Firstly, resampling, with importance weights, is introduced in PMCL method to avoid sample degeneracy. Secondly, twofold constraints, constraining the number of random samples in initialized step and constraining valid observations in resampling step, are proposed to decrease the number of iterations. Thirdly, mixture perspective is introduced to maintain the diversity of samples in resampling weighted process. Then, localization error, delay and consumption, especial delay, are predicted based on the statistic point of view, which takes mobile model of RWP into account. Moreover, performance comparisons of PMCL with other SMC-based schemes are also proposed. Simulation results show that delay of PMCL has some superiorities to that of other schemes, and accuracy and energy consumption is improved in some cases of less anchor rate and lower mobile velocity.

Analysis of a nonlinear importance sampling scheme for Bayesian parameter estimation in state-space models

The Bayesian estimation of the unknown parameters of state-space (dynamical) systems has received considerable attention over the past decade, with a handful of powerful algorithms being introduced. In this paper we tackle the theoretical analysis of the recently proposed nonlinear population Monte Carlo (NPMC). This is an iterative importance sampling scheme whose key features, compared to conventional importance samplers, are (i) the approximate computation of the importance weights (IWs) assigned to the Monte Carlo samples and (ii) the nonlinear transformation of these IWs in order to prevent the degeneracy problem that flaws the performance of conventional importance samplers. The contribution of the present paper is a rigorous proof of convergence of the nonlinear IS (NIS) scheme as the number of Monte Carlo samples, M, increases. Our analysis reveals that the NIS approximation errors converge to 0 almost surely and with the optimal Monte Carlo rate of M−12. Moreover, we prove that this is achieved even when the mean estimation error of the IWs remains constant, a property that has been termed exact approximation in the Markov chain Monte Carlo literature. We illustrate these theoretical results by means of a computer simulation example involving the estimation of the parameters of a state-space model typically used for target tracking.

An Adaptive Metamodel-Based Subset Importance Sampling approach for the assessment of the functional failure probability of a thermal-hydraulic passive system

An Adaptive Metamodel-Based Subset Importance Sampling (AM-SIS) approach, previously developed by the authors, is here employed to assess the (small) functional failure probability of a thermal-hydraulic (T-H) nuclear passive safety system. The approach relies on an iterative Importance Sampling (IS) scheme that efficiently couples the powerful characteristics of Subset Simulation (SS) and fast-running Artificial Neural Networks (ANNs). In particular, SS and ANNs are intelligently employed to build and progressively refine a fully nonparametric estimator of the ideal, zero-variance Importance Sampling Density (ISD), in order to: (i) increase the robustness of the failure probability estimates and (ii) decrease the number of expensive T-H simulations needed (together with the related computational burden). The performance of the approach is thoroughly compared to that of other efficient Monte Carlo (MC) techniques on a case study involving an advanced nuclear reactor cooled by helium.

A population Monte Carlo scheme with transformed weights and its application to stochastic kinetic models

This paper addresses the Monte Carlo approximation of posterior probability distributions. In particular, we consider the population Monte Carlo (PMC) technique, which is based on an iterative importance sampling (IS) approach. An important drawback of this methodology is the degeneracy of the importance weights (IWs) when the dimension of either the observations or the variables of interest is high. To alleviate this difficulty, we propose a new method that performs a nonlinear transformation of the IWs. This operation reduces the weight variation, hence it avoids degeneracy and increases the efficiency of the IS scheme, specially when drawing from proposal functions which are poorly adapted to the true posterior. For the sake of illustration, we have applied the proposed algorithm to the estimation of the parameters of a Gaussian mixture model. This is a simple problem that enables us to discuss the main features of the proposed technique. As a practical application, we have also considered the challenging problem of estimating the rate parameters of a stochastic kinetic model (SKM). SKMs are multivariate systems that model molecular interactions in biological and chemical problems. We introduce a particularization of the proposed algorithm to SKMs and present numerical results.

Adaptive Multiple Importance Sampling

Abstract. The Adaptive Multiple Importance Sampling algorithm is aimed at an optimal recycling of past simulations in an iterated importance sampling (IS) scheme. The difference with earlier adaptive IS implementations like Population Monte Carlo is that the importance weights of all simulated values, past as well as present, are recomputed at each iteration, following the technique of the deterministic multiple mixture estimator of Owen & Zhou (J. Amer. Statist. Assoc., 95, 2000, 135). Although the convergence properties of the algorithm cannot be investigated, we demonstrate through a challenging banana shape target distribution and a population genetics example that the improvement brought by this technique is substantial.

Iterated filtering

Inference for partially observed Markov process models has been a longstanding methodological challenge with many scientific and engineering applications. Iterated filtering algorithms maximize the likelihood function for partially observed Markov process models by solving a recursive sequence of filtering problems. We present new theoretical results pertaining to the convergence of iterated filtering algorithms implemented via sequential Monte Carlo filters. This theory complements the growing body of empirical evidence that iterated filtering algorithms provide an effective inference strategy for scientific models of nonlinear dynamic systems. The first step in our theory involves studying a new recursive approach for maximizing the likelihood function of a latent variable model, when this likelihood is evaluated via importance sampling. This leads to the consideration of an iterated importance sampling algorithm which serves as a simple special case of iterated filtering, and may have applicability in its own right.

Efficient identification of ocean thermodynamics in a physical/biogeochemical ocean model with an iterative Importance Sampling method

Efficient identification of parameters in numerical models remains a computationally demanding problem. Here we present an iterative Importance Sampling approach and demonstrate its application to estimating parameters that control the heat uptake efficiency of a physical/biogeochemical ocean model coupled to a simple atmosphere. The algorithm has similarities to a previously-developed ensemble Kalman filtering (EnKF) method applied to similar problems, but is more flexible and powerful in the case of nonlinear models and non-Gaussian uncertainties. The method is somewhat more computationally demanding than the EnKF but may be preferred in cases where the approximations that the EnKF relies upon are unsound. Our results suggest that the three-dimensional structure of ocean tracer fields may act as a useful constraint on ocean mixing and consequently the heat uptake of the climate system under anthropogenic forcing.

Population Monte Carlo Algorithm in High Dimensions

The population Monte Carlo algorithm is an iterative importance sampling scheme for solving static problems. We examine the population Monte Carlo algorithm in a simplified setting, a single step of the general algorithm, and study a fundamental problem that occurs in applying importance sampling to high-dimensional problem. The precision of the computed estimate from the simplified setting is measured by the asymptotic variance of estimate under conditions on the importance function. We demonstrate the exponential growth of the asymptotic variance with the dimension and show that the optimal covariance matrix for the importance function can be estimated in special cases.

Integration of Ranked Lists via Cross Entropy Monte Carlo with Applications to mRNA and microRNA Studies

One of the major challenges facing researchers studying complex biological systems is integration of data from -omics platforms. Omic-scale data include DNA variations, transcriptom profiles, and RAomics. Selection of an appropriate approach for a data-integration task is problem dependent, primarily dictated by the information contained in the data. In situations where modeling of multiple raw datasets jointly might be extremely challenging due to their vast differences, rankings from each dataset would provide a commonality based on which results could be integrated. Aggregation of microRNA targets predicted from different computational algorithms is such a problem. Integration of results from multiple mRNA studies based on different platforms is another example that will be discussed. Formulating the problem of integrating ranked lists as minimizing an objective criterion, we explore the usage of a cross entropy Monte Carlo method for solving such a combinatorial problem. Instead of placing a discrete uniform distribution on all the potential solutions, an iterative importance sampling technique is utilized "to slowly tighten the net" to place most distributional mass on the optimal solution and its neighbors. Extensive simulation studies were performed to assess the performance of the method. With satisfactory simulation results, the method was applied to the microRNA and mRNA problems to illustrate its utility.

Iterated importance sampling in missing data problems

Missing variable models are typical benchmarks for new computational techniques in that the ill-posed nature of missing variable models offer a challenging testing ground for these techniques. This was the case for the EM algorithm and the Gibbs sampler, and this is also true for importance sampling schemes. A population Monte Carlo scheme taking advantage of the latent structure of the problem is proposed. The potential of this approach and its specifics in missing data problems are illustrated in settings of increasing difficulty, in comparison with existing approaches. The improvement brought by a general Rao–Blackwellisation technique is also discussed.

Use of multicanonical Monte Carlo simulations to obtain accurate bit error rates in optical communications systems.

We apply the multicanonical Monte Carlo (MMC) method to compute the probability distribution of the received voltage in a chirped return-to-zero system. When computing the probabilities of very rare events, the MMC technique greatly enhances the efficiency of Monte Carlo simulations by biasing the noise realizations. Our results agree with the covariance matrix method over 20 orders of magnitude. The MMC method can be regarded as iterative importance sampling that automatically converges toward the optimal bias so that it requires less a priori knowledge of the simulated system than importance sampling requires. A second advantage is that the merging of different regions of a probability distribution function to obtain the entire function is not necessary in many cases.

An Algorithm for Least-Squares Estimation of Nonlinear Parameters

An Algorithm for Least-Squares Estimation of Nonlinear Parameters