Importance Sampling Estimator Research Articles

Data-based importance sampling estimates for extreme events

An accurate, efficient, and conceptually simple method is developed to estimate distributions of maxima of solutions of stochastic equations, i.e., ordinary or partial differential equations with random entries. The method is data-based. It constructs importance sampling (IS) or biasing measures from samples of surrogates of full model solutions of stochastic equations and uses these measures and mixtures of surrogate and full model samples to estimate probabilities of extreme events. Numerical examples are presented to illustrate the implementations of the proposed method and demonstrate numerically its performance.

Reliability and reliability sensitivity analysis of structure by combining adaptive linked importance sampling and Kriging reliability method

The application of reliability analysis and reliability sensitivity analysis methods to complicated structures faces two main challenges: small failure probability (typical less than 10−5) and time-demanding mechanical models. This paper proposes an improved active learning surrogate model method, which combines the advantages of the classical Active Kriging – Monte Carlo Simulation (AK-MCS) procedure and the Adaptive Linked Importance Sampling (ALIS) procedure. The proposed procedure can, on the one hand, adaptively produce a series of intermediate sampling density approaching the quasi-optimal Importance Sampling (IS) density, on the other hand, adaptively generate a set of intermediate surrogate models approaching the true failure surface of the rare failure event. Then, the small failure probability and the corresponding reliability sensitivity indices are efficiently estimated by their IS estimators based on the quasi-optimal IS density and the surrogate models. Compared with the classical AK-MCS and Active Kriging – Importance Sampling (AK-IS) procedure, the proposed method neither need to build very large sample pool even when the failure probability is extremely small, nor need to estimate the Most Probable Points (MPPs), thus it is computationally more efficient and more applicable especially for problems with multiple MPPs. The effectiveness and engineering applicability of the proposed method are demonstrated by one numerical test example and two engineering applications.

Coupling the reduced-order model and the generative model for an importance sampling estimator

In this work, we develop an importance sampling estimator by coupling the reduced-order model and the generative model in a problem setting of uncertainty quantification. The target is to estimate the probability that the quantity of interest (QoI) in a complex system is beyond a given threshold. To avoid the prohibitive cost of sampling a large scale system, the reduced-order model is usually considered for a trade-off between efficiency and accuracy. However, the Monte Carlo estimator given by the reduced-order model is biased due to the error from dimension reduction. To correct the bias, we still need to sample the fine model. An effective technique to reduce the variance reduction is importance sampling, where we employ the generative model to estimate the distribution of the data from the reduced-order model and use it for the change of measure in the importance sampling estimator. To compensate the approximation errors of the reduced-order model, more data that induce a slightly smaller QoI than the threshold need to be included into the training set. Although the amount of these data can be controlled by a posterior error estimate, redundant data, which may outnumber the effective data, will be kept due to the epistemic uncertainty. To deal with this issue, we introduce a weighted empirical distribution to process the data from the reduced-order model. The generative model is then trained by minimizing the cross entropy between it and the weighted empirical distribution. We also introduce a penalty term into the objective function to deal with the overfitting for more robustness. Numerical results are presented to demonstrate the effectiveness of the proposed methodology.

Information Reuse for Importance Sampling in Reliability-Based Design Optimization

This paper introduces a new approach for importance-sampling-based reliability-based design optimization (RBDO) that reuses information from past optimization iterations to reduce computational effort. RBDO is a two-loop process—an uncertainty quantification loop embedded within an optimization loop—that can be computationally prohibitive due to the numerous evaluations of expensive high-fidelity models to estimate the probability of failure in each optimization iteration. In this work, we use the existing information from past optimization iterations to create efficient biasing densities for importance sampling estimates of probability of failure. The method involves two levels of information reuse: (1) reusing the current batch of samples to construct an a posteriori biasing density with optimal parameters, and (2) reusing the a posteriori biasing densities of the designs visited in past optimization iterations to construct the biasing density for the current design. We demonstrate for the RBDO of a benchmark speed reducer problem and a combustion engine problem that the proposed method leads to computational savings in the range of 51% to 76%, compared to building biasing densities with no reuse in each iteration.

Class-Specific Interferometric Phase Estimation Using Patch-Based Importance Sampling

Interferometric phase (InPhase) estimation, that is, the denoising of modulo- $2\pi $ phase images from sinusoidal $2\pi $ -periodic and noisy observations, is a challenging inverse problem with wide applications in many coherent imaging techniques. This paper introduces a novel approach to InPhase restoration based on an external data set and importance sampling. In the proposed method, a class-specific data set of clean patches is clustered using a mixture of circular symmetric Gaussian (csMoG) distributions. For each noisy patch, a “home-cluster”, i.e., the closest cluster in the external data set, is identified. An InPhase estimator, termed as Shift-invariant Importance Sampling (SIS) estimator, is developed using the principles of importance sampling. The SIS estimator uses samples from the home-cluster to perform the denoising operation. Both the clustering mechanism and the estimation technique are developed for complex-valued signals by taking into account patch shift invariance, which is an important property for an efficient InPhase denoiser. The effectiveness of the proposed algorithm is shown using experiments conducted on a semi-real InPhase data set constructed using human face images and medical imaging applications involving real magnetic resonance imaging (MRI) data. It is observed that, in most of the experiments, the SIS estimator shows better results compared to the state-of-the-art algorithms, yielding a minimum improvement of 1 dB in peak signal-to-noise ratio (PSNR) for low to high noise levels.

On a Metropolis–Hastings importance sampling estimator

A classical approach for approximating expectations of functions w.r.t. partially known distributions is to compute the average of function values along a trajectory of a Metropolis–Hastings (MH) Markov chain. A key part in the MH algorithm is a suitable acceptance/rejection of a proposed state, which ensures the correct stationary distribution of the resulting Markov chain. However, the rejection of proposals causes highly correlated samples. In particular, when a state is rejected it is not taken any further into account. In contrast to that we consider a MH importance sampling estimator which explicitly incorporates all proposed states generated by the MH algorithm. The estimator satisfies a strong law of large numbers as well as a central limit theorem, and, in addition to that, we provide an explicit mean squared error bound. Remarkably, the asymptotic variance of the MH importance sampling estimator does not involve any correlation term in contrast to its classical counterpart. Moreover, although the analyzed estimator uses the same amount of information as the classical MH estimator, it can outperform the latter in scenarios of moderate dimensions as indicated by numerical experiments.

Keeping the balance—Bridge sampling for marginal likelihood estimation in finite mixture, mixture of experts and Markov mixture models

Finite mixture models and their extensions to Markov mixture and mixture of experts models are very popular in analysing data of various kind. A challenge for these models is choosing the number of components based on marginal likelihoods. The present paper suggests two innovative, generic bridge sampling estimators of the marginal likelihood that are based on constructing balanced importance densities from the conditional densities arising during Gibbs sampling. The full permutation bridge sampling estimator is derived from considering all possible permutations of the mixture labels for a subset of these densities. For the double random permutation bridge sampling estimator, two levels of random permutations are applied, first to permute the labels of the MCMC draws and second to randomly permute the labels of the conditional densities arising during Gibbs sampling. Various applications show very good performance of these estimators in comparison to importance and to reciprocal importance sampling estimators derived from the same importance densities.

Fast Search and Estimation of Bayesian Nonparametric Mixture Models Using a Classification Annealing EM Algorithm

Bayesian nonparametric (BNP) infinite-mixture models provide flexible and accurate density estimation, cluster analysis, and regression. However, for the posterior inference of such a model, MCMC algorithms are complex, often need to be tailor-made for different BNP priors, and are intractable for large data sets. We introduce a BNP classification annealing EM algorithm which employs importance sampling estimation. This new fast-search algorithm, for virtually any given BNP mixture model, can quickly and accurately calculate the posterior predictive density estimate (by posterior averaging) and the maximum a-posteriori clustering estimate (by simulated annealing), even for data sets containing millions of observations. The algorithm can handle a wide range of BNP priors because it primarily relies on the ability to generate prior samples. The algorithm can be fast because in each iteration, it performs a sampling step for the (missing) clustering of the data points, instead of a costly E-step; and then performs direct posterior calculations in the M-step, given the sampled (imputed) clustering. The new algorithm is illustrated and evaluated through BNP Gaussian mixture model analyses of benchmark simulated data and real data sets. MATLAB code for the new algorithm is provided in Supplementary Materials.

Structural reliability and reliability sensitivity analysis of extremely rare failure events by combining sampling and surrogate model methods

The aim of this article is to study the reliability analysis, parametric reliability sensitivity analysis and global reliability sensitivity analysis of structures with extremely rare failure events. First, the global reliability sensitivity indices are restudied, and we show that the total effect index can also be interpreted as the effect of randomly copying each individual input variable on the failure surface. Second, a new method, denoted as Active learning Kriging Markov Chain Monte Carlo (AK-MCMC), is developed for adaptively approximating the failure surface with active learning Kriging surrogate model as well as dynamically updated Monte Carlo or Markov chain Monte Carlo populations. Third, the AK-MCMC procedure combined with the quasi-optimal importance sampling procedure is extended for estimating the failure probability and the parametric reliability sensitivity and global reliability sensitivity indices. For estimating the global reliability sensitivity indices, two new importance sampling estimators are derived. The AK-MCMC procedure can be regarded as a combination of the classical Monte Carlo Simulation (AK-MCS) and subset simulation procedures, but it is much more effective when applied to extremely rare failure events. Results of test examples show that the proposed method can accurately and robustly estimate the extremely small failure probability (e.g. 1e–9) as well as the related parametric reliability sensitivity and global reliability sensitivity indices with several dozens of function calls.

Multifidelity probability estimation via fusion of estimators

This paper develops a multifidelity method that enables estimation of failure probabilities for expensive-to-evaluate models via information fusion and importance sampling. The presented general fusion method combines multiple probability estimators with the goal of variance reduction. We use low-fidelity models to derive biasing densities for importance sampling and then fuse the importance sampling estimators such that the fused multifidelity estimator is unbiased and has mean-squared error lower than or equal to that of any of the importance sampling estimators alone. By fusing all available estimators, the method circumvents the challenging problem of selecting the best biasing density and using only that density for sampling. A rigorous analysis shows that the fused estimator is optimal in the sense that it has minimal variance amongst all possible combinations of the estimators. The asymptotic behavior of the proposed method is demonstrated on a convection-diffusion-reaction partial differential equation model for which 105 samples can be afforded. To illustrate the proposed method at scale, we consider a model of a free plane jet and quantify how uncertainties at the flow inlet propagate to a quantity of interest related to turbulent mixing. Compared to an importance sampling estimator that uses the high-fidelity model alone, our multifidelity estimator reduces the required CPU time by 65% while achieving a similar coefficient of variation.

Estimation and prediction for spatial generalized linear mixed models with parametric links via reparameterized importance sampling

Spatial generalized linear mixed models (SGLMMs) are popular for analyzing non-Gaussian spatial data. These models assume a prescribed link function that relates the underlying spatial field with the mean response. There are circumstances, such as when the data contain outlying observations, where the use of a prescribed link function can result in poor fit, which can be improved by using a parametric link function. Some popular link functions, such as the Box–Cox, are unsuitable because they are inconsistent with the Gaussian assumption of the spatial field. We present sensible choices of parametric link functions which possess desirable properties. It is important to estimate the parameters of the link function, rather than assume a known value. To that end, we present a generalized importance sampling (GIS) estimator based on multiple Markov chains for empirical Bayes analysis of SGLMMs. The GIS estimator, although more efficient than the simple importance sampling, can be highly variable when used to estimate the parameters of certain link functions. Using suitable reparameterizations of the Monte Carlo samples, we propose modified GIS estimators that do not suffer from high variability. We use Laplace approximation for choosing the multiple importance densities in the GIS estimator. Finally, we develop a methodology for selecting models with appropriate link function family, which extends to choosing a spatial correlation function as well. We present an ensemble prediction of the mean response by appropriately weighting the estimates from different models. The proposed methodology is illustrated using simulated and real data examples.

Stochastic Security Assessment for Power Systems With High Renewable Energy Penetration Considering Frequency Regulation

With the deepening penetration of renewable resources worldwide, power system operators are faced with emerging challenges, e.g., the increase of operating risks due to the volatility and uncertainty of wind and solar power. To efficiently identify the operational limit violations, a switch from deterministic to stochastic framework for assessing the system security, which could manage various types of uncertainties, has been advocated in this paper. The established model is based on an improved probabilistic load flow, which is adapted to incorporate the steady-state behavior of frequency regulation. An efficient importance sampling (IS) technique is also developed to speed up the crude Monte Carlo (MC) simulation in estimating the low probability of violations of security constraints. Extensive computational experiments on both the IEEE 14-bus test case and a simplified regional system show that the proposed IS estimator makes significant enhancement to the crude MC in the computational efficiency and has better numerical performance as compared with other IS schemes.

Importance sampling and its optimality for stochastic simulation models

We consider the problem of estimating an expected outcome from a stochastic simulation model. Our goal is to develop a theoretical framework on importance sampling for such estimation. By investigating the variance of an importance sampling estimator, we propose a two-stage procedure that involves a regression stage and a sampling stage to construct the final estimator. We introduce a parametric and a nonparametric regression estimator in the first stage and study how the allocation between the two stages affects the performance of the final estimator. We analyze the variance reduction rates and derive oracle properties of both methods. We evaluate the empirical performances of the methods using two numerical examples and a case study on wind turbine reliability evaluation.

Optimal importance sampling for Lévy processes

We develop importance sampling estimators for Monte Carlo pricing of European and path-dependent options in models driven by Lévy processes. Using results from the theory of large deviations for processes with independent increments, we compute an explicit asymptotic approximation for the variance of the pay-off under a time-dependent Esscher-style change of measure. Minimizing this asymptotic variance using convex duality, we then obtain an importance sampling estimator of the option price. We show that our estimator is logarithmically optimal among all importance sampling estimators. Numerical tests in the variance gamma model show consistent variance reduction with a small computational overhead.

On the Sum of Order Statistics and Applications to Wireless Communication Systems Performances

We consider the problem of evaluating the cumulative distribution function (CDF) of the sum of order statistics, which serves to compute outage probability (OP) values at the output of generalized selection combining receivers. Generally, closed-form expressions of the CDF of the sum of order statistics are unavailable for many practical distributions. Moreover, the naive Monte Carlo (MC) method requires a substantial computational effort when the probability of interest is sufficiently small. In the region of small OP values, we instead propose two effective variance reduction techniques that yield a reliable estimate of the CDF with small computing cost. The first estimator, which can be viewed as an importance sampling estimator, has bounded relative error under a certain assumption that is shown to hold for most of the challenging distributions. A possible improvement of this estimator is then proposed for the Pareto and the Weibull cases. The second is a conditional MC estimator that achieves the bounded relative error property for the generalized Gamma case and the logarithmic efficiency for the Log-normal case. Finally, the efficiency of these estimators is compared via various numerical simulations.

Importance sampling of heavy-tailed iterated random functions

AbstractWe consider the stationary solutionZof the Markov chain {Zn}n∈ℕdefined byZn+1=ψn+1(Zn), where {ψn}n∈ℕis a sequence of independent and identically distributed random Lipschitz functions. We estimate the probability of the event {Z>x} whenxis large, and develop a state-dependent importance sampling estimator under a set of assumptions on ψnsuch that, for largex, the event {Z>x} is governed by a single large jump. Under natural conditions, we show that our estimator is strongly efficient. Special attention is paid to a class of perpetuities with heavy tails.

Scalable importance sampling estimation of Gaussian mixture posteriors in Bayesian networks

In this paper we propose a scalable importance sampling algorithm for computing Gaussian mixture posteriors in conditional linear Gaussian Bayesian networks. Our contribution is based on using a stochastic gradient ascent procedure taking as input a stream of importance sampling weights, so that a mixture of Gaussians is dynamically updated with no need to store the full sample. The algorithm has been designed following a Map/Reduce approach and is therefore scalable with respect to computing resources. The implementation of the proposed algorithm is available as part of the AMIDST open-source toolbox for scalable probabilistic machine learning (http://www.amidsttoolbox.com).

Convergence and Efficiency of Adaptive Importance Sampling Techniques with Partial Biasing

We propose a new Monte Carlo method to efficiently sample a multimodal distribution (known up to a normalization constant). We consider a generalization of the discrete-time Self Healing Umbrella Sampling method, which can also be seen as a generalization of well-tempered metadynamics. The dynamics is based on an adaptive importance technique. The importance function relies on the weights (namely the relative probabilities) of disjoint sets which form a partition of the space. These weights are unknown but are learnt on the fly yielding an adaptive algorithm. In the context of computational statistical physics, the logarithm of these weights is, up to an additive constant, the free-energy, and the discrete valued function defining the partition is called the collective variable. The algorithm falls into the general class of Wang–Landau type methods, and is a generalization of the original Self Healing Umbrella Sampling method in two ways: (i) the updating strategy leads to a larger penalization strength of already visited sets in order to escape more quickly from metastable states, and (ii) the target distribution is biased using only a fraction of the free-energy, in order to increase the effective sample size and reduce the variance of importance sampling estimators. We prove the convergence of the algorithm and analyze numerically its efficiency on a toy example.

Multiple importance sampling revisited: breaking the bounds

We revisit the multiple importance sampling (MIS) estimator and investigate the bound on the efficiency improvement over balance heuristic estimator with equal count of samples established in Veach’s thesis. We revise the proof for this and come to the conclusion that there is no such bound and henceforth it makes sense to look for new estimators that improve on balance heuristic estimator with equal count of samples. Next, we examine a recently introduced non-balance heuristic MIS estimator that is provably better than balance heuristic with equal count of samples, and we improve it both in variance and efficiency. We then obtain an equally provably better one-sample balance heuristic estimator, and finally, we introduce a heuristic for the count of samples that can be used when the individual techniques are biased. All in all, we present three new sampling strategies to improve on both variance and efficiency on the balance heuristic using non-equal count of samples.Our scheme requires the previous knowledge of several quantities, but those can be obtained in an adaptive way. The results also show that by a careful examination of the variance and properties of the estimators, even better estimators could be discovered in the future. We present examples that support our theoretical findings.

Estimating standard errors for importance sampling estimators with multiple Markov chains

The naive importance sampling estimator based on the samples from a single importance density can be extremely numerically unstable. We consider multiple distributions importance sampling estimators where samples from more than one probability distributions are combined to consistently estimate means with respect to given target distributions. These generalized importance sampling estimators provide more stable estimators than the naive importance sampling estimators. Importance sampling estimators can also be used in the Markov chain Monte Carlo (MCMC) context, that is, where iid samples are replaced with positive Harris Markov chains with invariant importance distributions. If these Markov chains converge to their respective target distributions at a geometric rate, then under two finite moment conditions a central limit theorem (CLT) holds for the importance sampling estimators. In order to calculate valid asymptotic standard errors, it is required to consistently estimate the asymptotic variance in the CLT. Recently Tan and Doss and Hobert (2015) developed an approach based on regenerative simulation for obtaining consistent estimators of the asymptotic variance. It is well-known that in practice it is often difficult to construct a useful minorization condition that is required in Tan and Doss and Hobert ’s (2015) regenerative simulation method. We provide an alternative estimator for these standard errors based on the easy to implement batch means methods. The multi-chain importance sampling estimators depend on Geyer’s (1994) reverse logistic estimator (of ratios of normalizing constants) which has wide applications, in its own right, in both frequentist and Bayesian inference. We also provide batch means estimator for calculating asymptotically valid standard errors of Geyer’s (1994) reverse logistic estimator. We illustrate the method with an application in Bayesian variable selection in linear regression. In particular, the multi-chain importance sampling estimator is used to perform empirical Bayes variable selection and the batch means estimator is used to obtain standard errors in the large p situation where regenerative method is not applicable.