Estimation Of Rare Event Probabilities Research Articles

Uncertainty quantification and confidence intervals for naive rare-event estimators

Abstract We consider the estimation of rare-event probabilities using sample proportions output by naive Monte Carlo or collected data. Unlike using variance reduction techniques, this naive estimator does not have an a priori relative efficiency guarantee. On the other hand, due to the recent surge of sophisticated rare-event problems arising in safety evaluations of intelligent systems, efficiency-guaranteed variance reduction may face implementation challenges which, coupled with the availability of computation or data collection power, motivate the use of such a naive estimator. In this paper we study the uncertainty quantification, namely the construction, coverage validity, and tightness of confidence intervals, for rare-event probabilities using only sample proportions. In addition to the known normality, Wilson, and exact intervals, we investigate and compare them with two new intervals derived from Chernoff’s inequality and the Berry–Esseen theorem. Moreover, we generalize our results to the natural situation where sampling stops by reaching a target number of rare-event hits. Our findings show that the normality and Wilson intervals are not always valid, but they are close to the newly developed valid intervals in terms of half-width. In contrast, the exact interval is conservative, but safely guarantees the attainment of the nominal confidence level. Our new intervals, while being more conservative than the exact interval, provide useful insights into understanding the tightness of the considered intervals.

Bayesian active learning line sampling with log-normal process for rare-event probability estimation

Line sampling (LS) stands as a powerful stochastic simulation method for structural reliability analysis, especially for assessing small failure probabilities. To further improve the performance of traditional LS, a Bayesian active learning idea has recently been pursued. This work presents another Bayesian active learning alternative, called ‘Bayesian active learning line sampling with log-normal process’ (BAL-LS-LP), to traditional LS. In this method, we assign an LP prior instead of a Gaussian process prior over the distance function so as to account for its non-negativity constraint. Besides, the approximation error between the logarithmic approximate distance function and the logarithmic true distance function is assumed to follow a zero-mean normal distribution. The approximate posterior mean and variance of the failure probability are derived accordingly. Based on the posterior statistics of the failure probability, a learning function and a stopping criterion are developed to enable Bayesian active learning. In the numerical implementation of the proposed BAL-LS-LP method, the important direction can be updated on the fly without re-evaluating the distance function. Four numerical examples are studied to demonstrate the proposed method. Numerical results show that the proposed method can estimate extremely small failure probabilities with desired efficiency and accuracy.

Rare Event Probability Estimation for Groundwater Inverse Problems With a Two‐Stage Sequential Monte Carlo Approach

AbstractBayesian inversions followed by estimations of rare event probabilities are often needed to analyze groundwater hazards. Instead of focusing on the posterior distribution of model parameters, the main interest lies then in the distribution of a specific quantity of interest contingent upon these parameters. To address the associated methodological challenges, we introduce a two‐stage Sequential Monte Carlo approach. In the first stage, it generates particles that approximate the posterior distribution; in the second stage, it employs subset sampling techniques to assess the probability of the rare event of interest. By considering two hydrogeological problems of increasing complexity, we showcase the efficiency and accuracy of the resulting PostRisk‐SMC method for rare event probability estimation related to groundwater hazards. We compare the performance of the PostRisk‐SMC method with a traditional Monte Carlo approach that relies on Markov chain Monte Carlo samples. We showcase that our estimates align with those of the traditional method, but the coefficients of variation are notably lower for the same computational budget when targeting more rare events. Furthermore, we highlight that the PostRisk‐SMC method allows estimating rare event probabilities approaching one in a billion using less than one hundred thousand forward simulations. Even if the presented examples are related to groundwater hazards, the methodology is well‐suited for addressing a wide range of topics in the geosciences and beyond.

Estimation of small failure probabilities using the Accelerated Weight Histogram method

Simulation of rare events, such as small failure probabilities, is a common problem in several scientific and engineering fields. This paper presents a novel Monte-Carlo-based simulation approach for this purpose, called the Accelerated Weight Histogram (AWH) method. The method was originally developed to solve challenging sampling problems in statistical and biological physics, but its algorithm has here been reformulated for estimation of rare event probabilities. The applicability of the method is investigated for a couple of simpler computational examples and for a more advanced practical case, which consists of a rock tunnel stability problem. To estimate the probability of failure of the latter in a realistic manner, a boolean indicator function was used to describe failure, based on a mechanical concept known as unbalanced force ratio. The investigated cases indicate that the AWH method performs well for both simpler limit states and more complex failure definitions.

Reliability analysis of discrete-state performance functions via adaptive sequential sampling with detection of failure surfaces

The paper presents a new efficient and robust method for rare event probability estimation for computational models of an engineering product or a process returning categorical information only, for example, either success or failure. For such models, most of the methods designed for the estimation of failure probability, which use the numerical value of the outcome to compute gradients or to estimate the proximity to the failure surface, cannot be applied. Even if the performance function provides more than just binary output, the state of the system may be a non-smooth or even a discontinuous function defined in the domain of continuous input variables. This often happens because the mathematical model features non-smooth components or discontinuities (e.g., in the constitutive laws), bifurcations, or even domains in which no reasonable model response is obtained. In these cases, the classical gradient-based methods usually fail. We propose a simple yet efficient algorithm, which performs a sequential adaptive selection of points from the input domain of random variables to extend and refine a simple distance-based surrogate model. Two different tasks can be accomplished at any stage of sequential sampling: (i) estimation of the failure probability, and (ii) selection of the best possible candidate for the subsequent model evaluation if further improvement is necessary. The proposed criterion for selecting the next point for model evaluation maximizes the expected probability classified by using the candidate. Therefore, the perfect balance between global exploration and local exploitation is maintained automatically. If there are more rare events such as failure modes, the method can be generalized to estimate the probabilities of all these event types. Moreover, when the numerical value of model evaluation can be used to build a smooth surrogate, the algorithm can accommodate this information to increase the accuracy of the estimated probabilities. Lastly, we define a new simple yet general geometrical measure of the global sensitivity of the rare-event probability to individual variables, which is obtained as a by-product of the proposed refinement algorithm.

Adaptive sequential sampling for reliability estimation of binary functions

A novel method for estimation of rare event probability is proposed, which works also for computational models returning categorical information only: success or failure. It combines the robustness of simulation methods (counting failure events) with the strength of approximation methods which refine the boundary between the failure and safe sets. Two basic tasks are identified: (i) extension of the experimental design (ED) and (ii) estimation of probabilities. The new extension algorithm adds points for limit state evaluation to the ED by balancing the global exploration and local exploitation, and the estimation uses the pointwise information to build a simple surrogate and perform a novel optimized importance sampling. No connection is presumed between the limit function value at point and its proximity to the failure surface. A new global sensitivity measure of the failure probability to individual variables is proposed and obtained as a by-product of the proposed methods.

Importance Sampling for a Simple Markovian Intensity Model Using Subsolutions

This article considers importance sampling for estimation of rare-event probabilities in a specific collection of Markovian jump processes used for, e.g., modeling of credit risk. Previous attempts at designing importance sampling algorithms have resulted in poor performance and the main contribution of the article is the design of efficient importance sampling algorithms using subsolutions. The dynamics of the jump processes cause the corresponding Hamilton-Jacobi equations to have an intricate state-dependence, which makes the design of efficient algorithms difficult. We provide theoretical results that quantify the performance of importance sampling algorithms in general and construct asymptotically optimal algorithms for some examples. The computational gain compared to standard Monte Carlo is illustrated by numerical examples.

Efficient subset simulation for rare-event integrating point-evolution kernel density and adaptive polynomial chaos kriging

Rare-event probability estimation has a wide range of applications, including the design and manufacture of precision equipment, aerospace systems, and critical industrial and civil structures. However, traditional simulation-based reliability calculation methods, such as brute Monte Carlo simulation (MCS) and subset simulation (SS), face challenges in efficiently evaluating small-failure probabilities due to the need for a large number of simulations, especially for non-linear and complex scenarios. Thus, to efficiently assess the probability of rare failure events in structural engineering, this paper develops a novel method for assessing the small-failure probability by integrating the point-evolution kernel density (PKDE), SS, and polynomial chaos kriging (PCK). The proposed PKDE-Adaptive PCK-based SS (PAPS) method aims to reduce the implementation of the original performance function by PCK and enrich the training set using an adaptive strategy. Moreover, the initial cumulative distribution function (CDF) of the performance function estimated by PKDE is modified gradually to facilitate the estimation of small-failure probability. Four numerical examples of small-failure probability estimation involving classical analytical cases, time-variant cases, and non-linear stochastic structures are used to illustrate the accuracy and efficiency of the proposed method. The computational results show that the proposed method can provide accurate computational results with a smaller computational burden than traditional methods (e.g., MCS, SS, LHS-PCK-SS).

Improvement of the cross-entropy method in high dimension for failure probability estimation through a one-dimensional projection without gradient estimation

Rare event probability estimation is an important topic in reliability analysis. Stochastic methods, such as importance sampling, have been developed to estimate such probabilities but they often fail in high dimension. In this paper, we propose a new cross-entropy-based importance sampling algorithm to improve rare event probability estimation in high dimension. We focus on the cross-entropy method with Gaussian auxiliary distributions and we suggest to update the Gaussian covariance matrix only in a one-dimensional subspace. For that purpose, the main idea is to consider the projection in the one-dimensional subspace spanned by the sample mean vector, which gives an influential direction for the variance estimation. This approach does not require any additional simulation budget compared to the basic cross-entropy algorithm and we show on different numerical test cases that it greatly improves its performance in high dimension.

Variance based sensitivity analysis for Monte Carlo and importance sampling reliability assessment with Gaussian processes

Running a reliability analysis on engineering problems involving complex numerical models can be computationally very expensive, requiring advanced simulation methods to reduce the overall numerical cost. Gaussian process based active learning methods for reliability analysis have emerged as a promising way for reducing this computational cost. In this paper, we propose a methodology to quantify the sensitivity of the failure probability estimator to uncertainties generated by the Gaussian process and the sampling strategy. This quantification also enables to control the whole error associated to the failure probability estimate and thus provides an accuracy criterion on the estimation. Thus, an active learning approach integrating this analysis to reduce the main source of error and stopping when the global variability is sufficiently low is introduced. The approach is proposed for both a Monte Carlo based method as well as an importance sampling based method, seeking to improve the estimation of rare event probabilities. Performance of the proposed strategy is then assessed on several examples.

Sequential stratified splitting for efficient Monte Carlo integration

The efficient evaluation of high-dimensional integrals is important from both theoretical and practical points of view. In particular, multidimensional integration plays a central role in Bayesian inference, statistical physics, data science, and machine learning. However, due to the curse of dimensionality, deterministic numerical methods are inefficient in the high-dimensional setting. Consequentially, for many practical problems one must resort to approximate estimation techniques such as Monte Carlo methods. In this article, we introduce a novel sequential Monte Carlo algorithm called stratified splitting. The method provides unbiased estimates and can handle various integrand types including indicator functions, which are important for rare-event probability estimation problems. We provide rigorous analysis of the efficiency of the proposed method and present a numerical demonstration of the algorithmic performance when applied to practical application domains. Our numerical experiments suggest that the stratified splitting method is capable of delivering accurate results for a variety of integration problems while requiring reasonable computational effort.

Cross-Entropy-Based Importance Sampling with Failure-Informed Dimension Reduction for Rare Event Simulation

The estimation of rare event or failure probabilities in high dimensions is of interest in many areas of science and technology. We consider problems where the rare event is expressed in terms of a...

Pricing discretely-monitored double barrier options with small probabilities of execution

In this paper, we propose a new stochastic simulation-based methodology for pricing discretely-monitored double barrier options and estimating the corresponding probabilities of execution. We develop our framework by employing a versatile tool for the estimation of rare event probabilities known as subset simulation algorithm. In this regard, considering plausible dynamics for the price evolution of the underlying asset, we are able to compare and demonstrate clearly that our treatment always outperforms the standard Monte Carlo approach and becomes substantially more efficient (measured in terms of the sample coefficient of variation) when the underlying asset has high volatility and the barriers are set close to the spot price of the underlying asset. In addition, we test and report that our approach performs better when it is compared to the multilevel Monte Carlo method for special cases of barrier options and underlying assets that make the pricing problem a rare event estimation. These theoretical findings are confirmed by numerous simulation results.

Selecting Reduced Models in the Cross-Entropy Method

This paper deals with the estimation of rare event probabilities using importance sampling (IS), where an {\it optimal} proposal distribution is computed with the cross-entropy (CE) method. Although, IS optimised with the CE method leads to an efficient reduction of the estimator variance, this approach remains unaffordable for problems where the repeated evaluation of the score function represents a too intensive computational effort. This is often the case for score functions related to the solution of a partial differential equation (PDE) with random inputs. This work proposes to alleviate computation by adapting a score function approximation along the CE optimisation process. The score function approximation is obtained by selecting the surrogate of lowest dimensionality, whose accuracy guarantees to pass the current CE optimisation stage. The adaptation of the surrogate relies on certified upper bounds on the error norm. An asymptotic analysis provides some theoretical guarantees on the efficiency and convergence of the proposed algorithm. Numerical results demonstrate the gain brought by the adaptive method in the context of pollution alerts and a system modelled by a PDE.

Bayesian inference of random fields represented with the Karhunen–Loève expansion

The integration of data into engineering models involving uncertain and spatially varying parameters is oftentimes key to obtaining accurate predictions. Bayesian inference is effective in achieving such an integration. Uncertainties related to spatially varying parameters are typically represented through random fields discretized into a finite number of random variables. The prior correlation length and variance of the field, as well as the number of terms in the random field discretization, have a considerable impact on the outcome of the Bayesian inference, which has received little attention in the literature. Here, we investigate the implications of different choices in the prior random field model on the solution of Bayesian inverse problems. We employ the Karhunen–Loève expansion for the representation of random fields. We show that a higher-order Karhunen–Loève discretization is required in Bayesian inverse problems as compared to standard prior random field representations, since the updated fields are non-homogeneous. Furthermore, the smoothing effect of the forward operator has a large influence on the posterior solution, particularly if we are analyzing quantities of interest that are sensitive to local random fluctuations of the inverse quantity. This is also reflected in posterior predictions, such as the estimation of rare event probabilities. We illustrate these effects analytically through a 1D cantilever beam with spatially varying flexibility, and numerically using a 2D linear elasticity example where the Young’s modulus is spatially variable.

Rare event simulation for steady-state probabilities via recurrency cycles.

We develop a new algorithm for the estimation of rare event probabilities associated with the steady-state of a Markov stochastic process with continuous state space Rd and discrete time steps (i.e., a discrete-time Rd-valued Markov chain). The algorithm, which we coin Recurrent Multilevel Splitting (RMS), relies on the Markov chain's underlying recurrent structure, in combination with the Multilevel Splitting method. Extensive simulation experiments are performed, including experiments with a nonlinear stochastic model that has some characteristics of complex climate models. The numerical experiments show that RMS can boost the computational efficiency by several orders of magnitude compared to the Monte Carlo method.

Kriging-Aided Cross-Entropy-Based Adaptive Importance Sampling Using Gaussian Mixture

In the reliability analysis, rare-event probability estimation imposes serious difficulties on conventional simulation methods like crude Monte Carlo. Several advanced variance reduction techniques such as importance sampling and importance splitting are presented to address this issue. Even though these methods require a low number of samples compared with crude Monte Carlo of the same accuracy, their implementation on time-consuming simulation codes is still very demanding. A joint employment of sampling methods and surrogate models is a proper solution. In this study, we integrate the Kriging surrogate model with a cross-entropy-based adaptive importance sampling, which uses the Gaussian mixture as the auxiliary sampling function. The novelty resides in efficient integration of the importance sampling and Kriging model. Results of various presented tests show a remarkable improvement with respect to importance sampling without surrogate. The statistical analysis results reveal that this advantage is achievable without any significant loss of accuracy.

An Adaptive Importance Sampling Method for Spinning Reserve Risk Evaluation of Generating Systems Incorporating Virtual Power Plants

The concept of virtual power plant (VPP) has been proposed to manage distributed renewable energy sources as packaging to engage in the energy and reserve planning on contemporary generating portfolios. In this context, an efficient tool is needed to support analysis of generating flexibility of conventional units in combination with VPPs to cooperatively counterbalance the fluctuation of net demand. In this paper, an adaptive importance sampling method is proposed, intentionally for efficiently evaluating specific indices capturing the possibility and severity of rare inadequate spinning reserve events of a deployed unit comment schedule for a generating system incorporating VPPs, in terms of short-term stochastic unit failures and power fluctuation of VPPs. The proposed method is based on the standard cross-entropy (CE) method, which newly introduces a mathematical transformation aiming at diverting evaluations of customized risk indices to a generic rare-event probability estimation problem. A Markov chain Monte Carlo method is employed to train the proposal density, to efficiently gain on the best, owing to avoiding the iterative parameter-updating mechanism of the standard CE method. The efficacy of the proposed method is tested on a modified RTS-96 generating system emulating a portfolio of conventional and renewable generating sources modeled as VPPs.

A multi-level splitting algorithm based on differential evolution

Multi-level splitting algorithm is a famous rare event simulation (RES) method which reaches rare set through splitting samples during simulation. Since choosing sample paths is a key factor of the method, this paper embeds differential evolution in multi-level splitting mechanism to improve the sampling strategy and precision, so as to improve the algorithm efficiency. Examples of rare event probability estimation demonstrate that the new proposed algorithm performs well in convergence rate and precision for a set of benchmark cases.

Efficient Pricing of Barrier Options on High Volatility Assets Using Subset Simulation

In this paper, we propose a new stochastic simulation-based methodology for pricing discretely-monitored double barrier options and estimating the corresponding probabilities of execution. We develop our framework by employing a versatile tool for the estimation of rare event probabilities known as subset simulation algorithm. In this regard, considering plausible dynamics for the price evolution of the underlying asset, we are able to compare and demonstrate clearly that our treatment always outperforms the standard Monte Carlo approach and becomes substantially more efficient (measured in terms of the sample coefficient of variation) when the underlying asset has high volatility and the barriers are set close to the spot price of the underlying asset. In addition, we test and report that our approach performs better when it is compared to the multilevel Monte Carlo method for special cases of barrier options and underlying assets that make the pricing problem a rare event estimation. These theoretical findings are confirmed by numerous simulation results.