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From the early developments of machines for reasoning and decision making in higher-level information fusion, there was a need for a systematic and reliable evaluation of their performance. Performance evaluation is important for comparison and assessment of alternative solutions to real-world problems. In this paper we focus on one aspect of performance assessment for reasoning under uncertainty: the accuracy of the resulting belief (prediction or estimate). We propose a framework for assessment based on the assumption that the system under investigation is uncertain only due to stochastic variability (randomness), which is partially known. In this context we formulate a distance measure between the “ground truth” and the output of an automated system for reasoning in the framework of one of the non-additive uncertainty formalisms (such as imprecise probability theory, belief function theory or possibility theory). The proposed assessment framework is demonstrated with a simple numerical example.

Quite frequently, catastrophes impact populated areas of the world, and hence the need for proper risk evaluations that support mitigation and management processes. Because of the uncertain nature of extreme natural hazards and lack of data, forecasts of the potential damage and losses before the event happens are needed. Catastrophe (CAT) models build on scenarios that represent all possible realizations of the hazard in terms of recurrence and intensity. Probabilistic risk models require the characterization of the hazards, the exposure model for the infrastructure, and its vulnerability. The main objective of this research is to compare loss exceedance curves, probable maximum loss curves, and average annual losses using four different available seismic hazard models for Chile. To isolate the effect of changing the hazard model in the risk results, the exposure and vulnerability information is fixed to the one available from the Global Assessment Report, GAR 15, and GAR ATLAS 2017. Imprecise probability theory, logic trees, and frequency and severity blends used by CAT modelers are the approaches applied and compared herein to propose either model blending or an interval of possible realizations. Both types of results have pros and cons. Blended results are point estimates which make them useful in a more traditional way, but their computation necessarily implies assigning weights to the models according to the modeler preferences. On the other hand, raw intervals of variability (without any knowledge of how the variable is distributed inside) are more transparent, as they simply state the bound of what is known without any preference, but their use is less understood among practitioners and could be even impractical.

This paper studies connections between two alternatives to the standard probability calculus for representing and reasoning about uncertainty: imprecise probability and comparative probability. The goal is to identify complete logics for reasoning about uncertainty in a comparative probabilistic language whose semantics is given in terms of imprecise probability. Comparative probability operators are interpreted as quantifying over a set of probability measures. Modal and dynamic operators are added for reasoning about epistemic possibility and updating sets of probability measures.

The classical credibility theory is a cornerstone of experience rating, especially in the field of property and casualty insurance. An obstacle to putting the credibility theory into practice is the conversion of available prior information into a precise choice of crucial hyperparameters. In most real-world applications, the information necessary to justify a precise choice is lacking, so we propose an imprecise credibility estimator that honestly acknowledges the imprecision in the hyperparameter specification. This results in an interval estimator that is doubly robust in the sense that it retains the credibility estimator’s freedom from model specification and fast asymptotic concentration, while simultaneously being insensitive to prior hyperparameter specification.

For the reliability analysis with imprecise probability distributions, the failure probability and its sensitivity are always denoted as the functions of distribution parameters. The computation of those functional relationships remains a challenging task due to the intensive computational burden. To further improve the computational efficiency, this work proposes a new computational method based on the classical active learning Kriging model with U learning function. The Kriging model is constructed with a group of initial point and updated by the best contributing samples selected from different sample spaces corresponding to discrete distribution parameters, which guarantees the prediction ability in the variation range of distribution parameters. Consequently, the functions of failure probability and sensitivity can be estimated by the same Kriging model, which avoids re-constructing Kriging models corresponding to different distribution parameters. Several examples including two numerical examples and three engineering practices are investigated to validate the reasonability and superiority of the proposed method. Finally, the proposed method is applied to the fatigue life reliability and sensitivity analyses of a turbine disk structure.

The time-dependent mechanism reliability and sensitivity analysis with imprecise probability distributions is investigated in this article. First, the interval model is employed to describe the unc...

In this paper we extend the notion of stochastic efficiency and inefficiency in portfolio optimization to the case of incomplete information by means of set-valued probabilities. The notion of set-valued probability models the concept of incomplete information about the underlying probability space and the probability associated with each scenario. Unlike other approaches in literature, our notion of inefficiency is introduced by means of the Monge–Kantorovich metric. We provide some numerical examples to illustrate this approach.

Conditional mutual information (CMI) maximization is a promising criterion for feature selection in a computationally efficient stepwise way, but it is hard to be applied comprehensively because of imprecise probability calculation and heavy computational load. Many dimension-reduced CMI-based and mutual information (MI)-based methods have been reported to achieve state-of-art performances in terms of classification. However, model deviations are introduced into the CMI and MI formulations in these methods during dimension reduction. In this paper, we start with the full-dimensional CMI to deal with the feature selection problem, so as to retain full inter-feature and feature-label mutual information when selecting new features. The cost function is approximated and simplified from a mathematical perspective to overcome the difficulties for maximizing the original full-dimensional CMI. A relationship is established between the proposed feature selection criterion and the one based on Hilbert-Schmidt independence, which explains qualitatively how the new criterion succeeds to achieve relevance maximization and redundance minimization simultaneously. Experiments on real-world datasets demonstrate the predominance of the proposed method over the existing ones.

One important question concerning a risk description is to what extentit should contain information about the uncertainty surrounding an estimatedprobability for an event, for example, by using some form ofprobability range. Presenting a point estimate together with a rangecan inform the decision-maker about both the best judgement as wellas the strength of knowledge regarding the probability in question, sothat the decision-maker can take this information into considerationwhen making decisions. However, communicating uncertainty in riskdescriptions may lead to delays in the decision-making process, due topeople’s general aversion to ambiguity and other forms of uncertainty.This can be devastating in time critical decision-making situations,where delaying decisions leads to substantial costs or losses - as in thecase of a military command and control (C2) situation or in crisisresponse management (CRM). This study investigated how 106 universitystudents handled a fictive time critical military decision-making situationwith imprecise probabilities presented in the form of ranges. Thepurpose was to make a first estimation regarding if presenting uncertaintyin risk descriptions could be a problem in these kinds of situations.The results show that almost half of the participants delayed theirdecisions and that many participants showed little restraint regardingtheir waiting time. These results indicate that communicating uncertaintyin risk descriptions can be a problem in time critical decisionmakingsituations, and that presenting uncertainty in risk descriptionsmay require decision-makers to be educated in the specific problemsassociated with decision-making in these types of situations.

This paper considers a linear optimisation problem under uncertainty with at least one element modelled as a non-probabilistic uncertainty. The uncertainty is expressed in the coefficient matrices of constraints and/or coefficients of goal function. Previous work converts such problems to classical (linear) optimisation problems and eliminates uncertainty by converting the linear programming under uncertainty problem to a decision problem using imprecise probability and imprecise decision theory. Our aim here is to generalise this approach numerically and present three methods to calculate the solution. We investigate what numerical results can be obtained for interval and fuzzy types of uncertainty models and compare them to classical probabilistic cases — for two different optimality criteria: maximinity and maximality. We also provide an efficient method to calculate the maximal solutions in the fuzzy set model. A numerical example is considered for illustration of the results.

This paper proposes a novel approach to the solution of optimal control problems under uncertainty (OCPUUs). OCPUUs are first cast in a general formulation that allows the treatment of uncertainties of different nature, and then solved with a new direct transcription method that combines multiple shooting with generalised polynomial algebra to model and propagate extended sets. The continuity conditions on extended sets at the boundary of two adjacent segments are directly satisfied by a bounding approach. The Intrusive Polynomial Algebra aNd Multiple shooting Approach (IPANeMA) developed in this work can handle optimal control problems under a wide range of uncertainty models, including nonparametric, epistemic, and imprecise probability ones. In this paper, the approach is applied to the design of a robust low-thrust trajectory to a Near-Earth Object with uncertain initial conditions. It is shown that the new method provides more robust and reliable trajectories than the solution of an analogous deterministic optimal control problem.

This paper is dedicated to exploring the NASA Langley Challenge on Optimization under Uncertainty by proposing a series of approaches for both forward and inverse treatment of uncertainty propagation and quantification. The primary effort is placed on the categorization of the subproblems as to be forward or inverse procedures, such that dedicated techniques are proposed for the two directions, respectively. The sensitivity analysis and reliability analysis are categorized as forward procedures, while modal calibration & uncertainty reduction, reliability-based optimization, and risk-based design are regarded as inverse procedures. For both directions, the overall approach is based on imprecise probability characterization where both aleatory and epistemic uncertainties are investigated for the inputs, and consequently, the output is described as the probability-box (P-box). Theoretic development is focused on the definition of comprehensive uncertainty quantification criteria from limited and irregular time-domain observations to extract as much as possible uncertainty information, which will be significant for the inverse procedure to refine uncertainty models. Furthermore, a decoupling approach is proposed to investigate the P-box along two directions such that the epistemic and aleatory uncertainties are decoupled, and thus a two-loop procedure is designed to propagate both epistemic and aleatory uncertainties through the systematic model. The key for successfully addressing this challenge is in obtaining on the balance among an appropriate hypothesis of the input uncertainty model, a comprehensive criterion of output uncertainty quantification, and a computational viable approach for both forward and inverse uncertainty treatment.

Despite the increasing importance of multiple priors in various domains of economics and the significant theoretical advances concerning them, choice-based incentive-compatible multiple-prior elicitation largely remains an open problem. This paper develops a solution, comprising a preference-based identification of a subject’s probability interval for an event, and two procedures for eliciting it. The method does not rely on specific assumptions about subjects’ ambiguity attitudes or probabilistic sophistication. To demonstrate its feasibility, we implement it in two incentivized experiments to elicit the multiple-prior equivalent of subjects’ cumulative distribution functions over continuous-valued sources of uncertainty. We find a predominance of non-degenerate probability intervals among subjects for all explored sources, with intervals being wider for less familiar sources. Finally, we use our method to undertake the first elicitation of the mixture coefficient in the Hurwicz α-maxmin EU model that fully controls for beliefs.

Imprecise probabilities have gained increasing popularity for quantitatively modeling uncertainty under incomplete information in various fields. However, it is still a computationally chal...

The consideration of imprecise probability in engineering analysis to account for missing, vague or incomplete data in the description of model uncertainties is a fast-growing field of research. Probability-boxes (p-boxes) are of particular interest in an engineering context, since they offer a mathematically straightforward description of imprecise probabilities, as well as allow for an intuitive visualisation. In essence, p-boxes are defined via lower and upper bounds on the cumulative distribution function of a random variable whose exact probability distribution is unknown. However, the propagation of p-boxes on model inputs towards bounds on probabilistic measures describing the uncertainty on the model responses is numerically still very demanding, and hence is subject of intensive research. In order to provide an overview on the available methods, this paper gives a state-of-the art review for the modelling and propagation of p-boxes with a special focus on structural reliability analysis. ispartof: Structural Safety vol:93 status: published

The fast development of technologies in the smart grids provides new opportunities such as co-optimization of multi-energy systems. One of the new concepts that can utilize multiple energy sources is a hybrid fuel station (HFS). For instance, an HFS can benefit from energy hubs, renewable energies, and natural gas sources to supply electric vehicles along with natural gas vehicles. However, the optimal operation of an HFS deals with uncertainties from different sources that do not have similar natures. Some may lack in term of historical data, and some may have very random and unpredictable behavior. In this study, we present a stochastic mathematical framework to address both types of these uncertainties according to the innate nature of each uncertain variable, namely: epistemic uncertainty variables (EUVs) and aleatory uncertainty variables (AUVs). Also, the imprecise probability approach is introduced for EUVs utilizing the copula theory in the process, and a scenario-based approach combining Monte Carlo simulation with Latin Hypercube sampling is applied for AUVs. The proposed framework is employed to address the daily operation of a novel HFS, leading to a two-stage mixed-integer linear programming problem. The proposed approach and its applicability are verified using various numerical simulations.

In this paper, we propose two generalized Bayesian imprecise probability approaches for estimation of proportions under potentially nonignorable nonresponse using data from small domains. Our approaches produce reliable inference, refraining from strong assumptions on the response process that are typically not testable. Hence, our estimates avoid the possibly severe bias arising from fallaciously imposing such assumptions. Specifically, we generalize the imprecise Beta model to the small area estimation setting, first treating the missing values in a radically cautious way and then deriving a method that allows incorporating powerfully weak knowledge on the missingness process. Additionally, we extend the empirical Bayes small area estimation approach applied by Stasny [27] through considering a set of priors arising from neighborhood of maximum likelihood estimates of the hyperparameters. As an illustration, we reanalyze data from the American National Crime Survey to estimate the probability of victimization in domains formed by cross-classification of certain characteristics.

Recent advances in decision making have incorporated both risk and ambiguity in decision theory and optimization methods. These methods implement a variety of uncertainty representations from probabilistic and non-probabilistic foundations, including traditional probability theory, sets of probability measures, uncertainty sets, ambiguity sets, possibility theory, evidence theory, fuzzy measures, and imprecise probability. The choice of uncertainty representation impacts the expressiveness and tractability of the decision models. We survey recent approaches for representing uncertainty in both decision making and optimization to clarify the trade-offs among the alternative representations. Robust and distributionally robust optimization are surveyed, with particular attention to standard form ambiguity sets. Applications of uncertainty and decision models are also reviewed, with a focus on recent optimization applications. These applications highlight common practices and potential research gaps. The intersection of behavioral decision making and robust optimization is a promising area for future research and there is also opportunity for further advances in distributionally robust optimization in sequential and multi-agent settings.

We present a new nonparametric predictive inference (NPI) method using a power-normal model for accelerated life testing (ALT). Combined with the accelerating link function and imprecise probability theory, the proposed method is a feasible way to predict the life of the product using ALT failure data. To validate the method, we run a series of simulations and conduct accelerated life tests with real products. The NPI lower and upper survival functions show the robustness of our method for life prediction. This is a continuous research, and some progresses have been made by updating the link function between different stress levels. We also explain how to renew and apply our model. Moreover, discussions have been made about the performance.

The factors that influence the inherent reliability of machine tool spindle are a mixture of various uncertainties and it leads that the reliability modeling and analysis of machine tool spindle can’t be dealt with by one mathematical theory. Meanwhile, the reliability data of the machine tool spindle for reliability modeling and analysis is often insufficient, and data of different types such as accumulated historical data, expert opinions, simulation data, etc. are used to make up for the lack of data. Thus, the unified quantification of mixed uncertainties and the data characterization of different types are the major premises for reliability modeling and analysis of machine tool spindle. By considering this, this paper makes use of the advantage of imprecise probability theory in quantizing the multiple types of data and the advantage of the Bayes theory in data fusion, and proposes a new inherent reliability modeling and analysis method based on imprecise Dirichlet model. In the proposed method, imprecise probability theory is used to quantify mixed uncertainties, imprecise Dirichlet model is built to characterize the different types of reliability data. After analyzing the inherent reliability variation regularity, an inherent reliability model is built, and the proposed method is verified by the inherent reliability calculation of a certain heavy-duty CNC machine tool’s milling spindle. This study can provide new method, theory and reference for reliability modeling and analysis when there are various uncertainties mixed and multiple data existed.

This paper introduces a novel method for asset and option trading in a binomial scenario. This method uses nonparametric predictive inference (NPI), a statistical methodology within im- precise probability theory. Instead of inducing a single probability distribution from the existing observations, the imprecise method used here induces a set of probability distributions. Based on the induced imprecise probability, one could form a set of conservative trading strategies for assets and options. By integrating NPI imprecise probability and expectation with the classical nancial binomial tree model, two rational decision routes for asset trading and for European option trading are suggested. The performances of these trading routes are investigated by com- puter simulations. The simulation results indicate that the NPI based trading routes presented in this paper have good predictive properties.

With the tremendous development of financial institutions, credit risk prediction (CRP) plays an essential role in granting loans to customers and helps them to minimize their loss because credit approval sometimes results in massive financial loss. So extra attention is needed to identify risky customer. Researchers have designed complex CRP models using artificial intelligence (AI) and statistical techniques to support the financial institutions to take correct business decisions. Though there are various statistical and AI methods available, the recent literature shows that the ensemble-based CRP model provides improved prediction results than single classifier system. The small increase in the performance of CRP model could result in a significant improvement in the profit of financial institutions and banks. This work proposes a weight-adjusted boosting ensemble method (WABEM) using rough set (RS)-based feature selection (FS) technique with the balancing and regression-based preprocessing called RS$$\_$$RFS-WABEM. Regression is used to fill missing value in the records to improve the performance of CRP. Three credit datasets (Australia, German and Japanese) are chosen to validate the feasibility and effectiveness of the proposed ensemble method. The trade-off between the uncertainty and imprecise probability of the proposed classifier model is evaluated using the performance measures such as accuracy and area under the curve. Experimental results show that the proposed ensemble method performs better than other base and ensemble classifier methods.

© 2019 Elsevier Ltd Uncertainty propagation through the simulation models is critical for computational mechanics engineering to provide robust and reliable design in the presence of polymorphic uncertainty. This set of companion papers present a general framework, termed as non-intrusive imprecise stochastic simulation, for uncertainty propagation under the background of imprecise probability. This framework is composed of a set of methods developed for meeting different goals. In this paper, the performance estimation is concerned. The local extended Monte Carlo simulation (EMCS) is firstly reviewed, and then the global EMCS is devised to improve the global performance. Secondly, the cut-HDMR (High-Dimensional Model Representation) is introduced for decomposing the probabilistic response functions, and the local EMCS method is used for estimating the cut-HDMR component functions. Thirdly, the RS (Random Sampling)-HDMR is introduced to decompose the probabilistic response functions, and the global EMCS is applied for estimating the RS-HDMR component functions. The statistical errors of all estimators are derived, and the truncation errors are estimated by two global sensitivity indices, which can also be used for identifying the influential HDMR components. In the companion paper, the reliability and rare event analysis are treated. The effectiveness of the proposed methods are demonstrated by numerical and engineering examples.

Transformer fault diagnosis is important for improving the operational reliability of power systems. Despite great efforts to enhance the accuracy of fault diagnosis, the precise detection of transformer faults remains difficult. The major barriers are attributed to the lack of transformer fault records and related data, which may produce unconvincing diagnostic results, and to the fact that multiple latent faults in a transformer are difficult to detect distinctly. In this paper, an imprecise probability-based approach to transformer fault diagnosis by using the imprecise Dirichlet model and naive credal classifier is proposed. Instead of providing a single-valued probability, the approach calculates the interval probability for each possible type of transformer faults to explicitly indicate the uncertainty of the diagnosis caused by the insufficiency of historical fault records. Meanwhile, according to whether an overlap exists among the probability intervals of different types of faults, the proposed approach can output either one explicit fault or multiple probable latent faults that occurred in the transformer. In the proposed approach, the imprecise Dirichlet model (IDM) is used to evaluate the imprecise probabilistic relationships between each type of transformer fault and its corresponding symptom attributes. A naive credal classifier (NCC) is established to integrate the IDM estimation results and calculate the interval-valued probabilities of all types of possible faults. Then, the transformer fault diagnosis results can be obtained by the classification criteria of the NCC. The approach can compensate for the defects of conventional methods, including their inability to perform reliable diagnosis with insufficient fault samples or distinctly indicate the multiple latent transformer faults. The accuracy of diagnosis and the efficiency of transformer maintenance can be significantly improved by implementing the proposed approach. The effectiveness of the approach is verified through case studies.

Imprecise probability allows quantifying the level of safety of a system taking into account the effect of both aleatory and epistemic uncertainty. The practical estimation of an imprecise probability is usually quite demanding from a numerical viewpoint, as it is necessary to propagate separately both types of uncertainty, leading in practical cases to a nested implementation in the so-called double loop approach. In view of this issue, this contribution presents an alternative approach that avoids the double loop by replacing the imprecise probability problem by an augmented, purely aleatory reliability analysis. Then, with the help of Bayes’ theorem, it is possible to recover an expression for the failure probability as an explicit function of the imprecise parameters from the augmented reliability problem, which ultimately allows calculating the imprecise probability. The implementation of the proposed framework is investigated within the context of imprecise first excursion probability estimation of uncertain linear structures subject to imprecisely defined stochastic quantities and crisp stochastic loads. The associated augmented reliability problem is solved within the context of Directional Importance Sampling, leading to an improved accuracy at reduced numerical costs. The application of the proposed approach is investigated by means of two examples. The results obtained indicate that the proposed approach can be highly efficient and accurate.

Imprecise probabilities alleviate the need for high-resolution and unwarranted assumptions in statistical modeling. They present an alternative strategy to reduce irreplicable findings. However, updating imprecise models requires the user to choose among alternative updating rules. Competing rules can result in incompatible inferences, and exhibit dilation, contraction and sure loss, unsettling phenomena that cannot occur with precise probabilities and the regular Bayes rule. We revisit some famous statistical paradoxes and show that the logical fallacy stems from a set of marginally plausible yet jointly incommensurable model assumptions akin to the trio of phenomena above. Discrepancies between the generalized Bayes (B) rule, Dempster’s (D) rule and the Geometric (G) rule as competing updating rules for Choquet capacities of order 2 are discussed. We note that (1) B-rule cannot contract nor induce sure loss, but is the most prone to dilation due to “overfitting” in a certain sense; (2) in absence of prior information, both B- and G-rules are incapable to learn from data however informative they may be; (3) D- and G-rules can mathematically contradict each other by contracting while the other dilating. These findings highlight the invaluable role of judicious judgment in handling low-resolution information, and the care that needs to be take when applying updating rules to imprecise probability models.

Gong and Meng show that we can gain insights into classical paradoxes about conditional probability by acknowledging that apparently precise probabilities live within a larger world of imprecise probability. They also show that the notion of updating becomes problematic in this larger world. A closer look at the historical development of the notion of updating can give us further insights into its limitations.

The theory of lower previsions is designed around the principles of coherence and sure-loss avoidance, thus steers clear of all the updating anomalies highlighted in Gong and Meng’s “Judicious Judgment Meets Unsettling Updating: Dilation, Sure Loss and Simpson’s Paradox” except dilation. In fact, the traditional problem with the theory of imprecise probability is that coherent inference is too complicated rather than unsettling. Progress has been made simplifying coherent inference by demoting sets of probabilities from fundamental building blocks to secondary representations that are derived or discarded as needed.

Efficient propagation of imprecise probability models is one of the most important, yet challenging tasks, for uncertainty quantification in many areas and engineering practices, especially when the involved epistemic uncertainty is substantial due to the extreme lack of information. In this work, a new methodology framework, named as “Non-intrusive Imprecise Probabilistic Integration (NIPI)”, is developed for achieving the above target, and specifically, the distributional probability-box model and the estimation of the corresponding probabilistic moments of model responses are of concern. The NIPI owns two attractive characters. First, the spatial correlation information in both aleatory and epistemic uncertainty spaces, revealed by the Gaussian Process Regression (GPR) model, is fully integrated for deriving NIPI estimations of high accuracy by using Bayesian inference. Second, the numerical errors are regarded as a kind of epistemic uncertainty, by analytically propagating them, the posterior variances are derived for indicating the errors of the NIPI estimations. Further, an adaptive experiment design strategy is developed to accelerate the convergence of NIPI by making full use of the information of “contribution to posterior variance” revealed by the GPR model. The performance of the proposed methods is demonstrated by numerical and engineering examples.

This article proposes a confidence interval based distributionally robust real-time economic dispatch (CI-DRED) approach, which considers the risk related to accommodating wind power. In this article, only the wind power curtailment and load shedding due to wind power disturbances are evaluated in the operational risk. The proposed approach can strike a balance between the operational costs and risk even when the wind power probability distribution cannot be precisely estimated. A novel ambiguity set is developed based on the imprecise probability theory, which can be constructed based on the point-wise or family-wise confidence intervals. The worst pair of distributions in the established ambiguity set is then identified, and the original CI-DRED problem is transformed into a determined nonlinear dispatch problem accordingly. By using the sequential convex optimization method and piecewise linear approximation method, the nonlinear dispatch model is reformulated as a mixed integer linear programming problem, for which off-the-shelf solvers are available. A fast inactive constraint filtration method is also applied to further relieve the computational burden. Numerical results on the IEEE 118-bus system and a real 445-bus system verify the effectiveness and efficiency of the proposed approach.

Knowledge-first epistemology includes a knowledge norm of action: roughly, act only on what you know. This norm has been criticized, especially from the perspective of so-called standard decision theory. Mueller and Ross provide example decision problems which seem to show that acting properly cannot require knowledge. I argue that this conclusion depends on applying a particular decision theory (namely, Savage-style Expected Utility Theory) which is ill-motivated in this context. Agents’ knowledge is often most plausibly formalized as an ambiguous epistemic state, and the theory of decision under ambiguity is then the appropriate modeling tool. I show how to model agents as acting rationally on the basis of their knowledge according to such a theory. I conclude that the tension between the knowledge norm of action and formal decision theory is illusory; the knowledge-first paradigm should be used to actively select the decision-theoretical tools that can best capture the knowledge-based decisions in any given situation.

International audience; We present two different strategies to extend the classical multi-label chaining approach to handle imprecise probability estimates. These estimates use convex sets of distributions (or credal sets) in order to describe our uncertainty rather than a precise one. The main reasons one could have for using such estimations are (1) to make cautious predictions (or no decision at all) when a high uncertainty is detected in the chaining and (2) to make better precise predictions by avoiding biases caused in early decisions in the chaining. We adapt both strategies to the case of the naive credal classifier, showing that this adaptations are computationally efficient. Our experimental results on missing labels, which investigate how reliable these predictions are in both approaches, indicate that our approaches produce relevant cautiousness on those hard-to-predict instances where the precise models fail.

Since uncertainty is persistent in engineering analyses, this chapter aimed to introduce methods to describe and reason with under uncertainty in various scenarios. Probability theory is the most widely used methodology for uncertainty quantification for a long time and has proven to be a powerful tool for this task. Nevertheless, the construction of stochastic models relies on very fine information, such as large amount of observations, which is not always available. Without it, the constructed models are only very rough approximations of the real laws and may cause incorrect decisions. In this chapter, we introduce other types of models, based on the theory of imprecise probability, which we are able to construct and reason with under situations with limited available knowledge.

Stochastic processes in general provide a popular framework for modelling uncertainty about the evolution of dynamical systems. The theory of Markov chains uses a number of crucial assumptions about the (in)dependence of such a process on its history that make their analysis tractable. In practice however, the parameters of a Markov chain may not be known exactly, or there may exist doubt as to the applicability of these assumptions to the system under study. This chapter presents an introduction to imprecise Markov chains, which are a robust generalisation of these models that may be used when parameters are not known exactly or when such assumptions could be violated. Their treatment is grounded in the theory of imprecise probabilities. The generalised model can be interpreted as a set of (traditional) stochastic processes, which may or may not be Markovian and which may have different and varying parameter values. Inferences are then performed to ensure robustness with respect to variations within this set. This chapter assumes no advanced familiarity with Markov chains or imprecise probability theory. It aims to develop an intuitive and graphical understanding of (imprecise) Markov chains in discrete and in continuous time.

In this paper, an imprecise probability multi-fault diagnosis approach was proposed based on imprecise probability, which could simultaneously diagnose multi-fault events of the wind turbine. The approach excavated the dependence relationships between fault events and variables using the greedy search algorithm and constructed a credal network structure to express the relationships in an abstract way. The expanded imprecise Dirichlet model is used to obtain the imprecise conditional probability table at each evidence node in the network, thus to realize the parameter estimation of the credal network It was shown that based on the constructed model, features of different targets can adaptively be extracted from the same signal, and then these features are used to perform multi-fault diagnosis. The test results showed that the proposed approach realizes the correct diagnosis of faults under multiple working conditions.

The non-intrusive imprecise stochastic simulation (NISS) is a general framework for the propagation of imprecise probability models and analysis of reliability. The most appealing character of this methodology framework is that, being a pure simulation method, only one precise stochastic simulation is needed for implementing the method, and the requirement of performing optimization analysis on the response functions can be elegantly avoided. However, for rare failure events, the current NISS methods are still computationally expensive. In this paper, the classical line sampling developed for precise stochastic simulation is injected into the NISS framework, and two different imprecise line sampling (ILS) methods are developed based on two different interpretations of the classical line sampling procedure. The first strategy is devised based on the set of hyperplanes introduced by the line sampling analysis, while the second strategy is developed based on an integral along each individual line. The truncation errors of both methods are measured by sensitivity indices, and the variances of all estimators are derived for indicating the statistical errors. A test example and three engineering problems of different types are introduced for comparing and demonstrating the effectiveness of the two ILS methods.

There is currently much discussion about how decision making should proceed when an agent’s degrees of belief are imprecise; represented by a set of probability functions. I show that decision rules recently discussed by Sarah Moss, Susanna Rinard and Rohan Sud all suffer from the same defect: they all struggle to rationalise diachronic ambiguity aversion. Since ambiguity aversion is among the motivations for imprecise credence, this suggests that the search for an adequate imprecise decision rule is not yet over.

Recent work has formally linked the traditional axiomatisation of incomplete preferences a la Anscombe-Aumann with the theory of desirability developed in the context of imprecise probability, by showing in particular that they are the very same theory. The equivalence has been established under the constraint that the set of possible prizes is finite. In this paper, we relax such a constraint, thus de facto creating one of the most general theories of rationality and decision making available today. We provide the theory with a sound interpretation and with basic notions, and results, for the separation of beliefs and values, and for the case of complete preferences. Moreover, we discuss the role of conglomerability for the presented theory, arguing that it should be a rationality requirement under very broad conditions.

Uncertainty and sensitivity analysis of building energy has become an active research area in order to consider variations of input variables and identify key variables influencing building energy. When there is only limited information available for uncertainty of building inputs, a specific probability for a given variable cannot be defined. Then, it is necessary to develop alternative approaches to probabilistic uncertainty and sensitivity analysis for building energy. Therefore, this paper explores the application of the Dempster-Shafer theory (DST) of evidence to conduct uncertainty and sensitivity analysis for buildings. The DST method is one of imprecise probability theories to allow combining uncertainty from different sources in terms of interval-valued probabilities in order to construct the belief and plausibility (two uncertainty measures) of system responses. The results indicate that the DST uncertainty analysis in combination with machine learning methods can provide fast and reliable information on uncertainty of building energy. It is recommended that at least two inherently different learning algorithms should be applied to provide robust simulation results of building energy. A spectrum of distributions should be implemented in global sensitivity analysis with the DST method because there are no specific distributions for intervals of input factors. Moreover, the stability of results from uncertainty and sensitivity analysis should be assessed when applying the DST method in building energy analysis.

We consider Aumann's famous result on “agreeing to disagree” in the context of imprecise probabilities. Our primary aim is to reveal a connection between the possibility of agreeing to disagree and the interesting and anomalous phenomenon known as dilation. For such a purpose it is convenient to use Geanakoplos and Polemarchakis' communication setting, where agents repeatedly announce and update credences until no new information is conveyed by the announcements. We show that for agents who share the same set of priors and update by conditioning on every prior, once the procedure of communicating credences stops, it is impossible to agree to disagree on the lower or upper probability of a hypothesis unless a certain dilation occurs. With some common topological assumptions, the result entails that it is impossible to agree not to have the same set of posterior probability values unless dilation is present. This result may be used to generate sufficient conditions for guaranteed full agreement for some important models of imprecise priors, and we illustrate the potential with an agreement result involving density ratio classes. We also provide a formulation of our results in terms of “dilation-averse” agents who ignore information about the value of a dilating partition but otherwise update by full Bayesian conditioning.

Kurt Weichselberger, one of the influential senior members of the imprecise probability community, passed away on February 7, 2016. Almost throughout his entire academic life the major focus of his research interests has been on the foundations of probability and statistics. The present article is a first attempt to trace back chronologically the development of Weichselberger's work on interval probability and his symmetric theory of logical probability based on it, aiming at a new framework for statistical inference. We also try to work out the intellectual background of his different projects together with some close links between them.

This paper presents a ‘possibilistic’ application of the finite element method (FEM) considering a linear elastic body with sparse information of input design variables. The corresponding uncertain...

Safety risk assessment of metro construction is necessary to prevent catastrophic accidents that may cause heavy financial losses and casualties. In this paper, we introduce a comprehensive risk assessment framework that incorporates credal networks (CNs) and an improved evaluation based on distance from average solution (EDAS) method. First, the CN model can intuitively demonstrate dependencies between risk factors, capture the epistemic uncertainty of expert judgement in the form of imprecise probability, and propagate the uncertainty quickly and accurately with advanced inference algorithms. In addition, the improved EDAS method can be used to comprehensively identify critical risk factors with multiple indicators, including the nature of the risk itself and company capabilities, being taken into account. Moreover, to effectively aggregate expert opinions, a weighting method that integrates subjective weights and similarity-based objective weights is provided. A case study on the safety risk analysis of metro construction in China verified the feasibility of the framework. The proposed approach can be extended to other similar projects to help engineers systematically assess risks during the life cycle of the project to ensure the safety of metro construction.

In almost all the realistic circumstances, such as atmospheric dispersion and health risk assessment analysis, it is very essential to include all the information into modelling. The parameters associated with a particular model may include different kinds of variability, imprecision and uncertainty. More often, it is seen that available information is interpreted in probabilistic sense. Probability theory is a well-established theory to measure such kind of variability. However, not all available information, data or model parameters affected by variability, imprecision and uncertainty can be handled by traditional probability theory. Uncertainty or imprecision may occur due to incomplete information or data, measurement error or data obtained from expert judgment or subjective interpretation of available data or information. Thus, model parameters’ data may be affected by subjective uncertainty. Traditional probability theory is inappropriate to represent subjective uncertainty. Possibility theory is another branch of mathematics which is used as a tool to describe the parameters with insufficient knowledge. It may also happen that the probabilistic information about the parameters may be imprecise; that is, it might be associated with some probability bounds. Such types of uncertainty are incorporated into imprecise probability, a theory that extends traditional probability theory by allowing for intervals or sets of probabilities. In this paper, an attempt has been made to combine imprecise probability knowledge and possibility knowledge and draw the uncertainty. The article describes a combination of imprecise probability and fuzzy knowledge and applied to atmospheric dispersion with a case study. Further, fuzziness measure has been utilized in quantifying the uncertainty.

Probabilistic information about random variables describing equipment's reliability is not complete when there is a lack of statistical data about failure. The traditional reliability assessment cannot deal with the incomplete probabilistic information. Interval probability is an efficient method to address the incomplete probabilistic information. The interval value of reliability indices can reflect the degree of completeness of probabilistic information. In this study, the optimisation model of generating systems’ imprecise reliability assessment (IRA) is established and the efficient unit-adding algorithm is proposed to obtain the upper and lower bounds of reliability indices. The probability density and the expectation of reliability indices are also calculated by the Monte Carlo simulation method. In the process of IRA, massive calculations of the traditional reliability are needed, therefore the recursive convolution algorithm, which is based on the outage capacity table and has the merit of high-computation efficiency, is adopted. A case study on a revised IEEE-RTS79 system shows the rationality and equity of the presented method.

We introduce the idea of maxitive belief structures, MBS, as a framework for modeling imprecise possibility distributions in a manner analogous to the way Dempster–Shafer belief structures allows the modeling of imprecise probability distributions. We study various properties of these MBS. We consider the problem of obtaining possibility distributions compatible with these belief structures. We consider the problem of fusing multiple MBS and the performance of various arithmetic operations with these objects.

This paper concentrates on modeling probabilistic events with fuzzy probability measures in relational databases and XML (Extensible Markup Language). Instead of crisp probability degrees or interval probability degrees, fuzzy sets are applied to represent imprecise probability degrees in relational databases and XML. A probabilistic XML model with fuzzy probability measures is introduced, which incorporates fuzzy probability measures to handle imprecision and uncertainty. In particular, the formal approach to reengineering the relational database model with fuzzy probability measures into the DTD (document type definition) model with fuzzy probability measures is developed in the paper.

Many tools for thinking about the future employ probability. For example, Delphi studies often ask expert participants to assign probabilities to particular future outcomes. Similarly, while some scenario planners reject probability, others insist that assigning probabilities to scenarios is required to make them meaningful. Formal modelling and forecasting methods often also employ probability in one way or another. The paper questions this widespread use of probability as a device for considering the future, firstly showing that objective probability, based on empirically-observed frequencies, has some well-known drawbacks when used for this purpose. However, what is less-widely acknowledged is that this is also true of the subjective probability used in, for example, Delphi. Subjective probability is less distinct from objective probability than proponents of its use might imply, meaning it therefore suffers from similar problems. The paper draws on the foundations of probability theory as set out by Kolmogorov, as-well-as the work of Keynes, Shackle, Aumann, Tversky and Kahneman, and others, to reassert the essential distinction between risk and uncertainty, and to warn about the dangers of inappropriate use of probability for considering the future. The paper sets out some criteria for appropriate use.

Prediction of future observations is an important and challenging problem. The two mainstream approaches for quantifying prediction uncertainty use prediction regions and predictive distributions, respectively, with the latter believed to be more informative because it can perform other prediction-related tasks. The standard notion of validity, what we refer to here as Type-1 validity, focuses on coverage probability of prediction regions, while a notion of validity relevant to the other prediction-related tasks performed by predictive distributions is lacking. Here we present a new notion, called Type-2 validity, relevant to these other prediction tasks. We establish connections between Type-2 validity and coherence properties, and show that imprecise probability considerations are required in order to achieve it. We go on to show that both types of prediction validity can be achieved by interpreting the conformal prediction output as the contour function of a consonant plausibility measure. We also offer an alternative characterization of conformal prediction, based on a new nonparametric inferential model construction, wherein the appearance of consonance is natural, and prove its validity., 34 pages, 2 figures, 2 tables. Comments welcome at https://www.researchers.one/article/2020-01-12