Rare Event Probabilities Research Articles

Symmetries and Geometrical Properties of Dynamical Fluctuations in Molecular Dynamics

We describe some general results that constrain the dynamical fluctuations that can occur in non-equilibrium steady states, with a focus on molecular dynamics. That is, we consider Hamiltonian systems, coupled to external heat baths, and driven out of equilibrium by non-conservative forces. We focus on the probabilities of rare events (large deviations). First, we discuss a PT (parity-time) symmetry that appears in ensembles of trajectories where a current is constrained to have a large (non-typical) value. We analyse the heat flow in such ensembles, and compare it with non-equilibrium steady states. Second, we consider pathwise large deviations that are defined by considering many copies of a system. We show how the probability currents in such systems can be decomposed into orthogonal contributions that are related to convergence to equilibrium and to dissipation. We discuss the implications of these results for modelling non-equilibrium steady states.

How inelastic scattering stimulates nonlinear reactor core parameter behaviour

Nonlinear and skewed distributions for safety-related parameters of a realistic Pressurized Water Reactor (PWR) are observed due to the current limited knowledge of the inelastic scattering of neutrons on 238U. The current knowledge of this cross section, about 20% uncertainty, does not lead to normally distributed core parameters, thus increasing the probability of rare events. In this paper, it is demonstrated how the simulations of 14 years of PWR operation with realistic history and fuel loading pattern induce non-Normal distributions for 3-dimensional peaking quantities and why a reduction of the 238U inelastic cross section uncertainty by a factor 2 is desirable.

A Generic Simulation Approach for the Fast and Accurate Estimation of the Outage Probability of Single Hop and Multihop FSO Links Subject to Generalized Pointing Errors

When assessing the performance of the free space optical (FSO) communication systems, the outage probability encountered is generally very small, and thereby the use of nave Monte Carlo simulations becomes prohibitively expensive. To estimate these rare event probabilities, we propose in this paper an importance sampling approach which is based on the exponential twisting technique to offer fast and accurate results. In fact, we consider a variety of turbulence regimes, and we investigate the outage probability of FSO communication systems, under a generalized pointing error model based on the Beckmann distribution, for both single and multihop scenarios. Selected numerical simulations are presented to show the accuracy and the efficiency of our approach compared with naive Monte Carlo.

An Adaptive Line Sampling Method for Reliability Analysis

Rare event probability estimation imposes serious difficulties to the conventional simulation methods like crude Monte Carlo. Several methods have been put forward to address this issue. Line sampling, directional simulation, importance sampling and subset simulation are more known among other methods. This paper aims at improving the line sampling method. The proposed method iteratively adapts the line sampling procedure to the limit state surface and at the end yields an estimation of the failure probability with the required coefficient of variation. The proposed method is compared to the conventional line sampling via well-known academic problems. The results show considerable improvement over the conventional line sampling method.

A rare event approach to high-dimensional approximate Bayesian computation

Approximate Bayesian computation (ABC) methods permit approximate inference for intractable likelihoods when it is possible to simulate from the model. However, they perform poorly for high-dimensional data and in practice must usually be used in conjunction with dimension reduction methods, resulting in a loss of accuracy which is hard to quantify or control. We propose a new ABC method for high-dimensional data based on rare event methods which we refer to as RE-ABC. This uses a latent variable representation of the model. For a given parameter value, we estimate the probability of the rare event that the latent variables correspond to data roughly consistent with the observations. This is performed using sequential Monte Carlo and slice sampling to systematically search the space of latent variables. In contrast, standard ABC can be viewed as using a more naive Monte Carlo estimate. We use our rare event probability estimator as a likelihood estimate within the pseudo-marginal Metropolis–Hastings algorithm for parameter inference. We provide asymptotics showing that RE-ABC has a lower computational cost for high-dimensional data than standard ABC methods. We also illustrate our approach empirically, on a Gaussian distribution and an application in infectious disease modelling.

The role of information search and its influence on risk preferences

According to the ‘Description–Experience gap’ (DE gap), when people are provided with the descriptions of risky prospects they make choices as if they overweight the probability of rare events; but when making decisions from experience after exploring the prospects’ properties, they behave as if they underweight such probability. This study revisits this discrepancy while focusing on information-search in decisions from experience. We report findings from a lab-experiment with three treatments: a standard version of decisions from description and two versions of decisions from experience: with and without a ‘history table’ recording previously sampled events. We find that people sample more from lotteries with rarer events. The history table proved influential; in its absence search is more responsive to cues such as a lottery’s variance while in its presence the cue that stands out is the table’s maximum capacity. Our analysis of risky choices captures a significant DE gap which is mitigated by the presence of the history table. We elicit probability weighting functions at the individual level and report that subjects overweight rare events in experience but less so than in description. Finally, we report a measure that allows us to compare the type of DE gap found in studies using choice patterns with that inferred through valuation and find that the phenomenon is similar but not identical across the two methods.

Decision-Theoretic Calibration of Option Pricing Models

We give a decision-theoretic procedure for the calibration of option pricing models. The inspiration behind it comes from the Weighted Monte Carlo (WMC) approach. The procedure incorporates rare-event probability distortion in style of Cumulative Prospect Theory (CPT) by Kahneman and Tversky. We also propose an intertemporal interpolation scheme to use with the probability distortion when more than one option maturity is present in the calibration. We show how the procedure significantly improves the out-of-sample accuracy for the arbitrage models we calibrate compared to the original WMC procedure. To our knowledge, this is the first instance where CPT is used to improve the performance of option pricing models in a practical way.

When Experience Meets Description: How Dyads Integrate Experiential and Descriptive Information in Risky Decisions

How do teams make joint decisions under risk when some team members learn about a prospect from description and others learn from experience? In a series of experiments, we find that two-person teams composed of one participant who learns from description and a second participant who learns from experience arrive at shared decisions via mutual concessions. In doing so, they attenuate individual biases, such as the overweighting and underweighting of the probability of rare events. The social interaction thus leads dyads to make shared decisions that follow normative standards more closely than the decisions made by individual decision makers. Finally, in processing experiential information, dyads appear to be sensitive to the reliability of the experience: the more reliable the experiential information, the larger its influence on the dyad’s decision. Data, as supplemental material, are available at http://dx.doi.org/10.1287/mnsc.2016.2428 . This paper was accepted by Teck-Hua Ho, judgment and decision making.

On the Efficient Simulation of Outage Probability in a Log-Normal Fading Environment

The outage probability (OP) of the signal-to-interference-plus-noise ratio (SINR) is an important metric that is used to evaluate the performance of wireless systems. One difficulty toward assessing the OP is that, in realistic scenarios, closed-form expressions cannot be derived. This is, for instance, the case of the Log-normal environment, in which evaluating the OP of the SINR amounts to computing the probability that a sum of correlated Log-normal variates exceeds a given threshold. Since such a probability does not admit a closed-form expression, it has thus far been evaluated by several approximation techniques, the accuracies of which are not guaranteed in the region of small OPs. For these regions, simulation techniques based on variance reduction algorithms are a good alternative, being quick and highly accurate for estimating rare event probabilities. This constitutes the major motivation behind this paper. More specifically, we propose a generalized hybrid importance sampling scheme, based on a combination of a mean shifting and a covariance matrix scaling, to evaluate the OP of the SINR in a Log-normal environment. We further our analysis by providing a detailed study of two particular cases. Finally, the performance of these techniques is performed both theoretically and through various simulation results.

Study of new rare event simulation schemes and their application to extreme scenario generation

This is a companion paper based on our previous work on rare event simulation methods. In this paper, we provide an alternative proof for the ergodicity of shaking transformation in the Gaussian case and propose two variants of the existing methods with comparisons of numerical performance. In numerical tests, we also illustrate the idea of extreme scenario generation based on the convergence of marginal distributions of the underlying Markov chains and show the impact of the discretization of continuous time models on rare event probability estimation.

The Multilevel Splitting algorithm for graph colouring with application to the Potts model

Calculating the partition function of the zero-temperature antiferromagnetic model is an important problem in statistical physics. However, an exact calculation is hard since it is strongly connected to a fundamental combinatorial problem of counting proper vertex colourings in undirected graphs, for which an efficient algorithm is not known to exist. Thus, one has to rely on approximation techniques. In this paper, we formulate the problem of the partition function approximation in terms of rare-event probability estimation and investigate the performance of a particle-based algorithm, called Multilevel Splitting, for handling this setting. The proposed method enjoys a provable probabilistic performance guarantee and our numerical study indicates that this algorithm is capable of delivering accurate results using a relatively modest amount of computational resources.

Does a better model yield a better argument? An info-gap analysis

Theories, models and computations underlie reasoned argumentation in many areas. The possibility of error in these arguments, though of low probability, may be highly significant when the argument is used in predicting the probability of rare high-consequence events. This implies that the choice of a theory, model or computational method for predicting rare high-consequence events must account for the probability of error in these components. However, error may result from lack of knowledge or surprises of various sorts, and predicting the probability of error is highly uncertain. We show that the putatively best, most innovative and sophisticated argument may not actually have the lowest probability of error. Innovative arguments may entail greater uncertainty than more standard but less sophisticated methods, creating an innovation dilemma in formulating the argument. We employ info-gap decision theory to characterize and support the resolution of this problem and present several examples.

Structure of the optimal path to a fluctuation

Macroscopic fluctuations have become an essential tool to understand physics far from equilibrium due to the link between their statistics and nonequilibrium ensembles. The optimal path leading to a fluctuation encodes key information on this problem, shedding light on, e.g., the physics behind the enhanced probability of rare events out of equilibrium, the possibility of dynamic phase transitions, and new symmetries. This makes the understanding of the properties of these optimal paths a central issue. Here we derive a fundamental relation which strongly constrains the architecture of these optimal paths for general d-dimensional nonequilibrium diffusive systems, and implies a nontrivial structure for the dominant current vector fields. Interestingly, this general relation (which encompasses and explains previous results) makes manifest the spatiotemporal nonlocality of the current statistics and the associated optimal trajectories.

Hybrid Framework for the Estimation of Rare Failure Event Probability

AbstractThis paper presents a novel approach for computing the rare failure event probability. Within the framework of the proposed approach, adaptive hybrid polynomial correlated function expansio...

Estimating rare events in biochemical systems using conditional sampling.

The paper focuses on development of variance reduction strategies to estimate rare events in biochemical systems. Obtaining this probability using brute force Monte Carlo simulations in conjunction with the stochastic simulation algorithm (Gillespie's method) is computationally prohibitive. To circumvent this, important sampling tools such as the weighted stochastic simulation algorithm and the doubly weighted stochastic simulation algorithm have been proposed. However, these strategies require an additional step of determining the important region to sample from, which is not straightforward for most of the problems. In this paper, we apply the subset simulation method, developed as a variance reduction tool in the context of structural engineering, to the problem of rare event estimation in biochemical systems. The main idea is that the rare event probability is expressed as a product of more frequent conditional probabilities. These conditional probabilities are estimated with high accuracy using Monte Carlo simulations, specifically the Markov chain Monte Carlo method with the modified Metropolis-Hastings algorithm. Generating sample realizations of the state vector using the stochastic simulation algorithm is viewed as mapping the discrete-state continuous-time random process to the standard normal random variable vector. This viewpoint opens up the possibility of applying more sophisticated and efficient sampling schemes developed elsewhere to problems in stochastic chemical kinetics. The results obtained using the subset simulation method are compared with existing variance reduction strategies for a few benchmark problems, and a satisfactory improvement in computational time is demonstrated.

EXACT ASYMPTOTICS OF SAMPLE-MEAN-RELATED RARE-EVENT PROBABILITIES

Relying only on the classical Bahadur–Rao approximation for large deviations (LDs) of univariate sample means, we derive strong LD approximations for probabilities involving two sets of sample means. The main result concerns the exact asymptotics (asn→∞) of$$ {\open P}\left({\max_{i\in\{1,\ldots,d_x\}}\bar X_{i,n} \les \min_{i\in\{1,\ldots,d_y\}}\bar Y_{i,n}}\right),$$with the${\bar X}_{i,n}{\rm s}$(${\bar Y}_{i,n}{\rm s}$, respectively) denotingdx(dy) independent copies of sample means associated with the random variableX(Y). Assuming${\open E}X \gt {\open E}Y$, this is a rare event probability that vanishes essentially exponentially, but with an additional polynomial term. We also point out how the probability of interest can be estimated using importance sampling in a logarithmically efficient way. To demonstrate the usefulness of the result, we show how it can be applied to compare the order statistics of the sample means of the two populations. This has various applications, for instance in queuing or packing problems.

Multilevel branching and splitting algorithm for estimating rare event probabilities

We analyze the splitting algorithm performance in the estimation of rare event probabilities in a discrete multidimensional framework. For this we assume that each threshold is partitioned into disjoint subsets and the probability for a particle to reach the next threshold will depend on the starting subset. A straightforward estimator of the rare event probability is given by the proportion of simulated particles for which the rare event occurs. The variance of this estimator is the sum of two parts: with one part resuming the variability due to each threshold, and the second part resuming the variability due to the number of thresholds. This decomposition is analogous to that of the continuous case. The optimal algorithm is then derived by cancelling the first term leading to optimal thresholds. Then we compare this variance with that of the algorithm in which one of the threshold has been deleted. Finally, we investigate the sensitivity of the variance of the estimator with respect to a shape deformation of an optimal threshold. As an example, we consider a two-dimensional Ornstein–Uhlenbeck process with conformal maps for shape deformation.

Flexible Spectrum Management in a Smart City Within Licensed Shared Access Framework

The new generation of communication technologies, named 5G, brings along a variety of emerging applications and services from both human and machine perspectives. The growing demand for bandwidth in 5G may therefore lead to massive deficiency in wireless spectrum availability despite its under-utilization in urban areas. The Smart City paradigm assumes a multitude of communicating machines at high density, which requires improved spectrum management flexibility. The novel licensed shared access (LSA) framework that has attracted recent industrial and academic attention may become a feasible solution to leverage such underutilized spectrum more efficiently. This paper analyzes the effects of applying LSA in the Smart City context by proposing an appropriate mathematical model. Particularly, we focus on the vehicle-to-everything 5G use case where connected devices attempt to distribute their sensed data including occasional video information. The proposed analytical framework allows to capture the probabilities of rare events during such operation by providing with a high level of precision in the resulting performance estimates.

Comparison of the several parameterized estimators for the positive extreme value index

ABSTRACTIn the paper we compare several parameterized estimators for the positive extreme value index, which is a very important parameter appearing in the estimation of the probability of rare events. Firstly, asymptotic comparison at optimal levels of the corresponding tail index estimators is performed. Secondly, the practical validation of asymptotic results for moderate finite samples is done by means of Monte-Carlo simulations. We demonstrate that theoretical domination of the positive extreme value index estimators, which are asymptotically normal with a null asymptotic bias, is not reflected in Monte-Carlo simulations. Moreover, the estimators of such type do not demonstrate stability in the sense of empirical mean-squared error.

Estimation of rare event probabilities in power transmission networks subject to cascading failures

Cascading failures seriously threat the reliability/availability of power transmission networks. In fact, although rare, their consequences may be catastrophic, including large-scale blackouts affecting the economics and the social safety of entire regions. In this context, the quantification of the probability of occurrence of these events, as a consequence of the operating and environmental uncertain conditions, represents a fundamental task. To this aim, the classical simulation-based Monte Carlo (MC) approaches may be impractical, due to the fact that (i) power networks typically have very large reliabilities, so that cascading failures are rare events and (ii) very large computational expenses are required for the resolution of the cascading failure dynamics of real grids. In this work we originally propose to resort to two MC variance reduction techniques based on metamodeling for a fast approximation of the probability of occurrence of cascading failures leading to power losses. A new algorithm for properly initializing the variance reduction methods is also proposed, which is based on a smart Latin Hypercube search of the events of interest in the space of the uncertain inputs. The combined methods are demonstrated with reference to the realistic case study of a modified RTS 96 power transmission network of literature.