Importance Sampling Algorithm Research Articles

Sampling and filtering with Markov chains

A new continuous-time Markov chain rate change formula is proven. This theorem is used to derive existence and uniqueness of novel filtering equations akin to the Duncan–Mortensen–Zakai equation and the Fujisaki–Kallianpur–Kunita equation but for Markov signals with general continuous-time Markov chain observations. The equations in this second theorem have the unique feature of being driven by both the observations and the process counting the observation transitions. A direct method of solving these filtering equations is also derived. Most results apply as special cases to the continuous-time Hidden Markov Models (CTHMM), which have become important in applications like disease progression tracking, The corresponding CTHMM results are stated as corollaries. Finally, application of our general theorems to Markov chain importance sampling, rejection sampling and branching particle filtering algorithms is also explained and these are illustrated by way of disease tracking simulations.

Efficient augmented radial basis function for reliability analysis with dynamic importance sampling

Conducting reliability analysis on complex systems, particularly those characterized by minimal failure probabilities and computationally intensive models, presents significant challenges. This article introduces a reliability analysis method combining the adaptive augmented radial basis function and the dynamic importance sampling (ARBF-DIS) algorithm. This method improves the generalization capacity of the augmented radial basis function at both local and global scales through the application of an anisotropic technique and orthogonal conditions. By using efficient cross-validation, the methodology facilitates the training of the metamodel, enabling it to provide predictive variance essential for constructing an adaptive sampling learning function. A key aspect of this approach is its capability to address the issue of low failure probabilities through dynamic importance sampling, which involves calculating an optimal importance sampling function derived from the current metamodel. The effectiveness and precision of the ARBF-DIS method are demonstrated through two analytical examples and a solid rocket motor reliability problem.

A distributed particle filter with sampling-based consensus density fusion for speaker tracking in distributed microphone networks

This paper proposes a novel distributed particle filter with sampling-based consensus density fusion for speaker tracking in distributed microphone networks (DMNs). Firstly, to combat the reverberation and ambient noise, multi-hypothesis likelihood model is applied to describe the likelihood distribution in the state-space model of speaker tracking problem. Then, the sequential importance resampling (SIR) is performed on each node to obtain the local posteriors of the state vector of the speaker. Since the non-Gaussian noise and the reverberation may make local posteriors presenting non-Gaussian and multi-modal characters, in this paper, the local posterior densities are approximated by Gaussian mixtures whose parameters are further transmitted to neighboring nodes to complete the distributed fusion based on the consensus method. We investigate a novel adaptive multiple importance sampling (AMIS) algorithm combining parallel tempering Monte Carlo (PTMC) samplers to achieve the sampling-based consensus density fusion of the non-Gaussian local posteriors and estimate the global posterior locally on each node. At last, the optimal global estimation of the speaker state vector could be calculated by using the estimated global posterior on each node in DMNs. Simulation and real-world recording experiment results show that the proposed speaker tracking method exhibits satisfactory tracking performance under reverberant and noisy environments. Moreover, it is easy and feasible to extend the proposed tracking algorithm to solve the multi-speaker tracking problem.

Meta model-based and cross entropy-based importance sampling algorithms for efficiently solving system failure probability function

The multi-mode system failure probability function (SFPF) can quantify how the distribution parameters of the random input vector affect the system safety and decouple the system reliability-based design optimization model. However, for a problem with a time-consuming implicit performance function and rare failure domain, efficiently solving the SFPF remains significantly challenging. Therefore, in this study, two efficient algorithms are proposed, namely, the meta model-based importance sampling and cross entropy-based importance sampling. The contributions of this study are twofold. The first is constructing a single-loop optimal importance sampling density (SL-OISD) method to decouple the double-loop framework for analyzing the SFPF. The second is establishing two methods to efficiently approximate the SL-OISD and complete the SFPF estimation. The first method is based on the meta model of the system performance function, which is abbreviated as SL-Meta-IS. The second method is based on minimizing the cross entropy between the Gaussian mixture density model and SL-OISD, which is abbreviated as SL-CE-IS. To reduce the number of evaluating the system performance function when approximating the SL-OISD, sampling the SL-OISD, and identifying the state of the samples for completing the SFPF estimation, an adaptive Kriging model of the system performance function is introduced into SL-Meta-IS and SL-CE-IS. Owing to decoupling the double-loop framework into a single-loop framework, replacing the time-consuming system performance function with the economic Kriging model, and employing importance sampling variance reduction techniques to address issues related to the rare failure domain, the proposed SL-Meta-IS and SL-CE-IS methods greatly enhance the efficiency of SFPF estimations. The numerical and practical examples demonstrate that the two proposed methods are superior to the existing algorithms; moreover, the efficiency of SL-CE-IS is higher than that of SL-Meta-IS.

Inference for Bayesian Nonparametric Models with Binary Response Data via Permutation Counting

Since the beginning of Bayesian nonparametrics in the early 1970s, there has been a wide interest in constructing models for binary response data. Such data arise naturally in problems dealing with bioassay, current status data and sensitivity testing, and are equivalent to left and right censored observations if the inputs are one-dimensional. For models based on the Dirichlet process, inference is possible via Markov chain Monte Carlo (MCMC) simulations. However, there exist multiple processes based on different principles, for which such MCMC-based methods fail. Examples include logistic Gaussian processes and quantile pyramids. These require MCMC for posterior inference given exact observations, and thus become intractable when the data comprise both left and right censored observations. Here we present a new importance sampling algorithm for nonparametric models given exchangeable binary response data. It can be applied to any model from which samples can be generated, or even only approximately generated. The main idea behind the algorithm is to exploit the symmetries introduced by exchangeability. Calculating the importance weights turns out to be equivalent to evaluating the permanent of a certain class of (0,1)-matrix, which we prove can be done in polynomial time by deriving an explicit algorithm.

Probing the large deviations for the beta random walk in random medium.

We consider a discrete-time random walk on a one-dimensional lattice with space- and time-dependent random jump probabilities, known as the beta random walk. We are interested in the probability that, for a given realization of the jump probabilities (a sample), a walker starting at the origin at time t=0 is at position beyond ξsqrt[T/2] at time T. This probability fluctuates from sample to sample and we study the large-deviation rate function, which characterizes the tails of its distribution at large time T≫1. It is argued that, up to a simple rescaling, this rate function is identical to the one recently obtained exactly by two of the authors for the continuum version of the model. That continuum model also appears in the macroscopic fluctuation theory of a class of lattice gases, e.g., in the so-called KMP model of heat transfer. An extensive numerical simulation of the beta random walk, based on an importance sampling algorithm, is found in good agreement with the detailed analytical predictions. A first-order transition in the tilted measure, predicted to occur in the continuum model, is also observed in the numerics.

A multi-fidelity approach for reliability-based risk assessment of single-vehicle crashes

Road vehicles are highly susceptible to single-vehicle crashes (SVCs) under complex road geometry and inclement weather, which can significantly threaten traffic safety and mobility of the whole traffic system. Most existing studies involve various simplifications and approximations to assess the associated SVC risks promptly, and therefore the assessment accuracy is often compromised. A novel multi-fidelity approach is developed for the reliability-based risk assessment of SVCs to balance the simulation accuracy and efficiency. Specifically, a high-fidelity transient dynamic vehicle model is introduced for a robust estimation of the vehicle dynamics under various driving environments, assisted by a low-fidelity simplified physics-based vehicle model to improve the computational efficiency. Based on the simulations of the two models, a new multi-fidelity improved cross entropy-based importance sampling (MFICE) algorithm is proposed for integrating multi-fidelity information and facilitating accurate and efficient reliability analysis. Five demonstrative cases are studied to evaluate the performance of the proposed approach, including the comparison with existing representative approaches. The results show that the proposed innovative multi-fidelity approach can provide a reliability evaluation of SVCs both accurately and efficiently, with obviously superior performance over typical state-of-the-art counterparts. Therefore, the proposed approach bears great potential on developing proactive and near real-time intelligent traffic operation and management strategies against SVCs in both normal and hazardous conditions.

Interval Reliability Evaluation of Hybrid AC/DC Grids With Integrated Renewable Energy

Reliability evaluation of a hybrid AC/DC grid (HADG) with renewable energy is critical because of prevailing uncertainties and complex operations of voltage source converter-based high voltage direct current (DC) transmission. We propose an interval reliability evaluation model to accelerate the speed of HADG reliability evaluation with uncertainties. First, the general nonconvex optimal load shedding (OLS) model is linearized. Second, the linearized OLS model is converted into an interval model, which is decomposed into two bi-level sub-models to quantify the impact of renewable energy uncertainties at HADG. Third, an Average and Scattered Sampling-Adaptive Importance Sampling (AS-AIS) algorithm is proposed to accelerate the convergence of the reliability evaluation. Simulation results for several test systems show the effectiveness and computational tractability of the proposed interval reliability model.

Influential Observations in Bayesian Regression Tree Models

Bayesian Classification and Regression Trees (BCART) and Bayesian Additive Regression Trees (BART) are popular Bayesian regression models widely applicable in modern regression problems. Their popularity is intimately tied to the ability to flexibly model complex responses depending on high-dimensional inputs while simultaneously being able to quantify uncertainties. This ability to quantify uncertainties is key, as it allows researchers to perform appropriate inferential analyses in settings that have generally been too difficult to handle using the Bayesian approach. However, surprisingly little work has been done to evaluate the sensitivity of these modern regression models to violations of modeling assumptions. In particular, we will consider influential observations, which one reasonably would imagine to be common—or at least a concern—in the big-data setting. In this article, we consider both the problem of detecting influential observations and adjusting predictions to not be unduly affected by such potentially problematic data. We consider three detection diagnostics for Bayesian tree models, one an analogue of Cook’s distance and the others taking the form of a divergence measure and a conditional predictive density metric, and then propose an importance sampling algorithm to re-weight previously sampled posterior draws so as to remove the effects of influential data in a computationally efficient manner. Finally, our methods are demonstrated on real-world data where blind application of the models can lead to poor predictions and inference. Supplementary materials for this article are available online.

Forecasting Pathogen Dynamics with Bayesian Model-Averaging: Application to Xylella fastidiosa.

Forecasting invasive-pathogen dynamics is paramount to anticipate eradication and containment strategies. Such predictions can be obtained using a model grounded on partial differential equations (PDE; often exploited to model invasions) and fitted to surveillance data. This framework allows the construction of phenomenological but concise models relying on mechanistic hypotheses and real observations. However, it may lead to models with overly rigid behavior and possible data-model mismatches. Hence, to avoid drawing a forecast grounded on a single PDE-based model that would be prone to errors, we propose to apply Bayesian model averaging (BMA), which allows us to account for both parameter and model uncertainties. Thus, we propose a set of different competing PDE-based models for representing the pathogen dynamics, we use an adaptive multiple importance sampling algorithm (AMIS) to estimate parameters of each competing model from surveillance data in a mechanistic-statistical framework, we evaluate the posterior probabilities of models by comparing different approaches proposed in the literature, and we apply BMA to draw posterior distributions of parameters and a posterior forecast of the pathogen dynamics. This approach is applied to predict the extent of Xylella fastidiosa in South Corsica, France, a phytopathogenic bacterium detected in situ in Europe less than 10 years ago (Italy 2013, France 2015). Separating data into training and validation sets, we show that the BMA forecast outperforms competing forecast approaches.

Feasibility of Monte-Carlo maximum likelihood for fitting spatial log-Gaussian Cox processes

Log-Gaussian Cox processes (LGCPs) are a popular and flexible tool for modelling point pattern data. While maximum likelihood estimation of the parameters of such a model is attractive in principle, the likelihood function is not available in closed form. Various Monte Carlo approximations have been proposed, but these have seen very limited use in the literature and are often dismissed as impractical. This article provides a comprehensive study of the computational properties of Monte Carlo maximum likelihood estimation (MCMLE) for LGCPs. We compare various importance sampling algorithms for MCMLE, and also consider their performance against other methods of inference (such as minimum contrast) in numerical studies. We find that the best MCMLE algorithm is a practical proposition for parameter estimation given modern computing power, but the performance of this methodology is rather sensitive to the choice of reference parameters defining the importance sampling distribution.

Copula‐based importance sampling method for efficient slope reliability analysis

AbstractEfficient slope reliability analysis that accounts for the interdependencies of random variables is challenging. In this study, an efficient reliability method is proposed based on importance sampling (IS) and copulas. First, the IS method based on the design point is introduced to construct the marginal IS density. Copulas are subsequently adopted to capture the interdependencies and establish both the initial joint distributions and the joint IS densities of random variable pairs. The IS algorithm is modified to consider the dependencies through not only the estimation of the probability distribution but also random sampling. Finally, a framework of the copula‐based IS method is constructed. The applicability, accuracy, efficiency, and robustness of the proposed method were validated by a classical slope case. The results showed that with the proposed method, slope reliability could be efficiently assessed with a relatively high accuracy. The dependencies of the variables significantly affected the IS and final slope reliability, whereas the IS density had a significantly lower effect. The proposed method combines the advantages of IS and copulas well. Interdependencies of the variables can be properly characterized during sampling. Moreover, the proposed method was shown to be robust regardless of the system failure probability level. It can be further applied to improve the random finite element method in the future.

Importance sampling for McKean-Vlasov SDEs

This paper deals with Monte-Carlo (MC) methods for evaluating expectations of functionals of solutions to McKean-Vlasov Stochastic Differential Equations (MV-SDE) including those with super-linearly growing drifts. Underpinned by an interacting particle system approximation, we propose two importance sampling (IS) algorithms to reduce the variance of an associated MC estimator.The “complete measure change” algorithm sees the IS measure change applied to the target expectation to be calculated and to the MV-SDE coefficients simultaneously. The “decoupling algorithm” consists of estimating the law of the MV-SDE’s solution via standard simulation under the initial measure, then fixing the law component of the MV-SDE via that simulation, and finally simulating the new equation under the IS measure.Methodologically, large deviations and Pontryagin principle are employed to determine and solve the variance minimisation problem that yields the required measure change. The optimisation problem associated to the complete measure change is more complex than that for the decoupling algorithm, nonetheless, symmetry arguments allow for non-trivial complexity reduction. As an example, both algorithms are tested using the Kuramoto model from statistical physics. For the functionals tested, we see a reduction of up to 3 orders of magnitude on the variance of both IS schemes in comparison to the standard Monte Carlo approximation. In terms of computational cost, the complete measure change is akin to standard Monte Carlo whilst the decoupled approach increases the cost by a factor of around 2 if one uses the same number of particles for both steps. The statistical error of the method dominates the propagation of chaos error by 1 order of magnitude.

Sequential importance sampling for estimating expectations over the space of perfect matchings

This paper makes three contributions to estimating the number of perfect matching in bipartite graphs. First, we prove that the popular sequential importance sampling algorithm works in polynomial time for dense bipartite graphs. More carefully, our algorithm gives a (1±ϵ)-approximation for the number of perfect matchings of a λ-dense bipartite graph, using O(n1−2λλϵ−2) samples. With size n on each side and for 1 2>λ>0, a λ-dense bipartite graph has all degrees greater than (λ+1 2)n. Second, practical applications of the algorithm requires many calls to matching algorithms. A novel preprocessing step is provided which makes significant improvements. Third, three applications are provided. The first is for counting Latin squares, the second is a practical way of computing the greedy algorithm for a card guessing game with feedback and the third is for stochastic block models. In all three examples, sequential importance sampling allows treating practical problems of reasonably large sizes.

State-dependent importance sampling for estimating expectations of functionals of sums of independent random variables

Estimating the expectations of functionals applied to sums of random variables (RVs) is a well-known problem encountered in many challenging applications. Generally, closed-form expressions of these quantities are out of reach. A naive Monte Carlo simulation is an alternative approach. However, this method requires numerous samples for rare event problems. Therefore, it is paramount to use variance reduction techniques to develop fast and efficient estimation methods. In this work, we use importance sampling (IS), known for its efficiency in requiring fewer computations to achieve the same accuracy requirements. We propose a state-dependent IS scheme based on a stochastic optimal control formulation, where the control is dependent on state and time. We aim to calculate rare event quantities that could be written as an expectation of a functional of the sums of independent RVs. The proposed algorithm is generic and can be applied without restrictions on the univariate distributions of RVs or the functional applied to the sum. We apply this approach to the log-normal distribution to compute the left tail and cumulative distribution of the ratio of independent RVs. For each case, we numerically demonstrate that the proposed state-dependent IS algorithm compares favorably to most well-known estimators dealing with similar problems.

Reliability Estimation in Inverse Pareto Distribution Using Progressively First Failure Censored Data

The progressively first failure censored (PFFC) data have become very popular in the past decade due to its usefulness in life testing experiments and in reliability theory. The PFFC data record the number of failures and increase the efficiency of the estimators. The inverse Pareto distribution (IPD) is useful when empirical data suggest a decreasing or upside-down bathtub-shaped failure rate functions. In this article, we consider the classical and Bayesian estimation of the model parameter and the reliability characteristics of the IPD using the PFFC data. The maximum likelihood estimators, asymptotic confidence, and bootstrap confidence intervals are considered in the classical estimation. Under the Bayesian paradigm, Bayes estimators based on non-informative and the gamma informative priors under the squared error loss function using Tierney-Kadane approximation, importance sampling, and Metropolis-Hasting (M-H) algorithm are assessed. In addition, the highest probability density (HPD) intervals based on the M-H algorithm are also constructed. To investigate the efficacy of each of the estimation procedures, numerical computations are performed based on a simulation study. Finally, a real data set is re-analyzed to show the applicability of the IPD model under a censoring scheme.

Fuzzy Stress‐Strength Model and Mean Remaining Strength for Lindley Distribution: Estimation and Application in Cancer of Benign Endocrine

This paper is interested in the Bayesian and non‐Bayesian estimation of the stress‐strength model and the mean remaining strength when there is fuzziness for stress and strength random variables having Lindley’s distribution with different parameters. A fuzzy is defined as a function of the difference between stress and strength variables. In the context of Bayesian estimation, two approximate algorithms are used importance sampling algorithm and the Monte Carlo Markov chain algorithm. For non‐Bayesian estimation, maximum likelihood estimation and maximum product of spacing method are used. The Monte Carlo simulation study is performed to compare between different estimators for our proposed models using statistical criteria. Finally, to show the ability of our proposed models in real life, real medical application is introduced.

Numerical solution of the Fokker–Planck equation using physics-based mixture models

The Fokker–Planck equation governs the uncertainty propagation of dynamical systems driven by stochastic processes. The solution of the Fokker–Planck equation is a time-varying Probability Density Function (PDF) that is usually of high dimension with unbounded support. There are also properties that must be conserved in time for the joint and any marginal solution PDF (i.e., a time-varying PDF is a non-negative function that must integrate to unity at any given time.) Satisfying these properties poses a significant challenge to the numerical solution of the Fokker–Planck equation. Another challenge is to capture the tail behavior of the solution PDF with unbounded support, required to predict the probability of rare events like a system failure. Satisfying the conditions of a proper PDF with unbounded support limits the application of traditional grid-based numerical methods, like finite difference and finite element methods. This paper develops a novel numerical method based on physics-based mixture models for the transient and steady-state solutions of the Fokker–Planck equation. In the proposed numerical method, the trial function space consists of the convex combinations of some parametric PDFs (i.e., mixture models) that trivially satisfy the necessary conditions of the solution. Estimating the unknown parameters of the mixture model is via Bayesian inference while considering the constraints on model parameters. Bayesian inference facilitates integrating data on the responses of dynamical systems (e.g., from simulations) with the governing Fokker–Planck equation to estimate the unknown parameters. Since the solution PDF is not observable, combining it with observable data on system responses is far from trivial based on current numerical methods. The formulation of Bayesian inference also introduces a weighting function to reduce the estimation error in specific regions of interest like the tail of the solution PDF. To reduce the computational demand, the paper develops an importance sampling algorithm that generates a small set of collocation points at which the residual of the Fokker–Planck equation is evaluated. The performance of the proposed numerical method is demonstrated with several benchmark problems. The obtained results are compared with analytical solutions when available and otherwise with simulations.

Fitting double hierarchical models with the integrated nested Laplace approximation

Double hierarchical generalized linear models (DHGLM) are a family of models that are flexible enough as to model hierarchically the mean and scale parameters. In a Bayesian framework, fitting highly parameterized hierarchical models is challenging when this problem is addressed using typical Markov chain Monte Carlo (MCMC) methods due to the potential high correlation between different parameters and effects in the model. The integrated nested Laplace approximation (INLA) could be considered instead to avoid dealing with these problems. However, DHGLM do not fit within the latent Gaussian Markov random field (GMRF) models that INLA can fit. In this paper, we show how to fit DHGLM with INLA by combining INLA and importance sampling (IS) algorithms. In particular, we will illustrate how to split DHGLM into submodels that can be fitted with INLA so that the remainder of the parameters are fit using adaptive multiple IS (AMIS) with the aid of the graphical representation of the hierarchical model. This is illustrated using a simulation study on three different types of models and two real data examples.

Binding mechanism of oseltamivir and influenza neuraminidase suggests perspectives for the design of new anti-influenza drugs.

Oseltamivir is a widely used influenza virus neuraminidase (NA) inhibitor that prevents the release of new virus particles from host cells. However, oseltamivir-resistant strains have emerged, but effective drugs against them have not yet been developed. Elucidating the binding mechanisms between NA and oseltamivir may provide valuable information for the design of new drugs against NA mutants resistant to oseltamivir. Here, we conducted large-scale (353.4 μs) free-binding molecular dynamics simulations, together with a Markov State Model and an importance-sampling algorithm, to reveal the binding process of oseltamivir and NA. Ten metastable states and five major binding pathways were identified that validated and complemented previously discovered binding pathways, including the hypothesis that oseltamivir can be transferred from the secondary sialic acid binding site to the catalytic site. The discovery of multiple new metastable states, especially the stable bound state containing a water-mediated hydrogen bond between Arg118 and oseltamivir, may provide new insights into the improvement of NA inhibitors. We anticipated the findings presented here will facilitate the development of drugs capable of combating NA mutations.