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The title of this article is (essentially) the same as the famous paper Basu (2011b). Basu often opined that counterexamples were the best way to learn limitations of theories or methods and I have followed his directive in my own teaching. A number of counterexamples I use extensively in teaching are collected here. [ABSTRACT FROM AUTHOR]

Basu's via media is what he referred to as the middle road between the Bayesian and frequentist poles. He seemed skeptical that a suitable via media could be found, but I disagree. My basic claim is that the likelihood alone can't reliably support probabilistic inference, and I justify this by considering a technical trap that Basu stepped in concerning interpretation of the likelihood. While reliable probabilistic inference is out of reach, it turns out that reliable possibilistic inference is not. I lay out my proposed possibility-theoretic solution to Basu's via media and I investigate how the flexibility afforded by my imprecise-probabilistic solution can be leveraged to achieve the likelihood principle (or something close to it). [ABSTRACT FROM AUTHOR]

Evidence theory is a useful tool for modeling and reasoning uncertain information inherent in experts' evaluations, which is not handled efficiently in traditional failure mode and effects analysis (FMEA). This study proposes an integrated FMEA method that incorporates evidence theory and is applied to human reliability assessment. The human error information of a human-machine system in FMEA is described as a directed graph by a Bayesian network (BN) to assess the dependence among potential human-related failure modes. The BN is extended to propagate the epistemic uncertainty of FMEA team members, where belief mass is applied to model uncertainties in team members' knowledge and to convert their subjective cognition into varying levels of uncertainty. Risk indexes for occurrence, severity and detection from multiple sources are defined as a special assessment state. The combination of the belief mass of different failure modes is performed using extended Dempster's rules to avoid the influence of conflicting evidence. Finally, an application in the healthcare system is provided to verify the effectiveness of our model. A comparison with other fuzzy FMEA methods is also conducted, demonstrating the advantages of the proposed model in dealing with decision-makers' epistemic uncertainty and potential failure mode interdependencies. [ABSTRACT FROM AUTHOR]

This paper studies connections between two alternatives to the standard probability calculus for representing and reasoning about uncertainty: imprecise probability andcomparative 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.

In the literature on imprecise probability, little attention is paid to the fact that imprecise probabilities are precise on a set of events. We call these sets systems of precision. We show that, under mild assumptions, the system of precision of a lower and upper probability form a so-called (pre-)Dynkin system. Interestingly, there are several settings, ranging from machine learning on partial data over frequential probability theory to quantum probability theory and decision making under uncertainty, in which, a priori, the probabilities are only desired to be precise on a specific underlying set system. Here, (pre-)Dynkin systems have been adopted as systems of precision, too. We show that, under extendability conditions, those pre-Dynkin systems equipped with probabilities can be embedded into algebras of sets. Surprisingly, the extendability conditions elaborated in a strand of work in quantum probability are equivalent to coherence from the imprecise probability literature. On this basis, we spell out a lattice duality which relates systems of precision to credal sets of probabilities. We conclude the presentation with a generalization of the framework to expectation-type counterparts of imprecise probabilities. The analogue of pre-Dynkin systems turns out to be (sets of) linear subspaces in the space of bounded, real-valued functions. We introduce partial expectations, natural generalizations of probabilities defined on pre-Dynkin systems. Again, coherence and extendability are equivalent. A related but more general lattice duality preserves the relation between systems of precision and credal sets of probabilities. [ABSTRACT FROM AUTHOR]

Sklar's theorem represents a single multivariate distribution function through its univariate marginal distributions and a copula. However, it fails in general when it comes to the representation of the envelopes of a family F of multivariate distribution functions (i.e. its point-wise infimum and supremum), even if we allow more general representing functions than copulas. In this paper we develop an alternative representation which describes the envelopes of a family F in terms of the copulas that correspond to the members of F. Our representation is reminiscent of Sklar's representation but is based on the idea of corner patches of copulas. We prove a series of four representation theorems; (1) a general one, which holds even if the envelopes of the family do not possess a Sklar type representation, (2) a version tailored for the case when the envelopes do possess a Sklar type representation, (3) a theorem for families of continuous distribution functions, and (4) a variant appropriate for modeling dependencies of absolutely continuous random variables. In addition, we demonstrate how our results can be applied to give a Sklar type theorem in the imprecise probability setting, which offers a new approach to multivariate probability boxes. [ABSTRACT FROM AUTHOR]

The logrank test is a well-known nonparametric test which is often used to compare the survival distributions of two samples including right-censored observations, it is also known as the Mantel-Haenszel test. The G ρ family of tests, generalizes the logrank test by using weights assigned to observations. In this paper, we present a switch monotonicity property for the G ρ family of tests, which was motivated by the need to derive bounds for the test statistic in case of imprecise data observations. This property states that, when all observations from two independent groups are ranked together, the value of the z-test statistic is monotonically increasing after switching a pair of adjacent values from the two groups. Two examples are provided to motivate and illustrate the result presented in this paper. [ABSTRACT FROM AUTHOR]

Uncertainty is widely present in composite structures, and the probabilistic model is one of the most common ways to describe uncertainty. In some engineering problems, incomplete knowledge leads to uncertainty in the distribution parameters of probabilistic models. In this paper, the uncertainty of the distribution parameters is described by intervals, and the uncertainty analysis of a radome made by honeycomb sandwich composite is performed under imprecise probability. To reduce computational costs, a back propagation analysis neural network (BPA-NN) based on data from finite element analysis (FEA) is constructed. A variance-based global sensitivity analysis (GSA) is conducted to identify input variables that have a large influence on the output characteristics of the composite structure. A buckling failure model is established to evaluate the safety of the composite structure, and then GSA based on failure probability is carried out to identify input variables that have a large influence on the failure probability of the structure. The uncertainty analysis under imprecise probability in this paper provides a framework for the reliability assessment and design of composites. [ABSTRACT FROM AUTHOR]

Wilcox proposed an argument against imprecise probabilities and for the principle of indifference based on a thought experiment where he argues that it is very intuitive to feel that one's confidence in drawing a ball of a given colour out of an unknown urn should decrease while the number of potential colours in the urn increases. In my response to him, I argue that one's intuitions may be unreliable because it is very hard to truly feel completely ignorant in such a situation. I further argue that Wilcox must also account for the conflicting intuition that it is absurd to have to feel completely convinced that a specific claim about reality is true in the absence of any evidence in order to avoid being irrational. It is dubious that this intuition is considerably less universal and strongly-held than Wilcox's own intuition. Finally, I point out that even if Wilcox's intuition were to be universally shared among members of our biological species, it is far from being clear that someone refusing to let that intuition dictate his or her beliefs would be irrational. For all these reasons, I believe that Wilcox was not successful in proving that philosophers and scientists representing uncertainty through imprecise probabilities are violating the principles of rationality. [ABSTRACT FROM AUTHOR]

The received model of degrees of belief represents them as probabilities. Over the last half century, many philosophers have been convinced that this model fails because it cannot cmake room for the idea that an agent’s degrees of belief should respect the available evidence. In its place they have advocated a model that represents degrees of belief using imprecise probabilities (sets of probability functions). This paper presents a model of degrees of belief based on Dempster–Shafer belief functions and then presents arguments for belief functions over imprecise probabilities as a model of evidence-respecting degrees of belief. The arguments cover three kinds of issue: theoretical virtues (simplicity, interpretability and flexibility); motivations; and problem cases (dilation and belief inertia). [ABSTRACT FROM AUTHOR]

In this paper, we prove that Neutrosophic Statistics is more general than Interval Statistics, since it may deal with all types of indeterminacies (with respect to the data, inferential procedures, probability distributions, graphical representations, etc.), it allows the reduction of indeterminacy, and it uses the neutrosophic probability that is more general than imprecise and classical probabilities and has more detailed corresponding probability density functions. While Interval Statistics only deals with indeterminacy that can be represented by intervals. And we respond to the arguments by Woodall et al. [1]. We show that not all indeterminacies (uncertainties) may be represented by intervals. Also, in some cases, we should better use hesitant sets (that have less indeterminacy) instead of intervals. We redirect the authors to the Plithogenic Probability and Plithogenic Statistics which are the most general forms of MultiVariate Probability and Multivariate Statistics respectively (including, of course, the Imprecise Probability and Interval Statistics as subclasses). [ABSTRACT FROM AUTHOR]

This article considers a novel exotic option pricing method for incomplete markets. Nonparametric predictive inference (NPI) is applied to the option pricing procedure based on the binomial tree model allowing the method to evaluate exotic options with limited information and few assumptions. As the implementation of the NPI method is greatly simplified by the monotonicity of the option payoff in the tree, we categorize exotic options by their payoff monotonicity and study a typical type of exotic option in each category, the barrier option and the look‐back option. By comparison with the classic binomial tree model, we investigate the performance of our method either with different moneyness or varying maturity. All outcomes show that our model offers a feasible approach to price the exotic options with limited information, which makes it can be utilized for both complete and incomplete markets. [ABSTRACT FROM AUTHOR]

Qualitative and quantitative approaches to reasoning about uncertainty can lead to different logical systems for formalizing such reasoning, even when the language for expressing uncertainty is the same. In the case of reasoning about relative likelihood, with statements of the form φ ≥ ψ expressing that φ is at least as likely as ψ, a standard qualitative approach using preordered preferential structures yields a dramatically different logical system than a quantitative ap- proach using probability measures. In fact, the standard pref- erential approach validates principles of reasoning that are incorrect from a probabilistic point of view. However, in this paper we show that a natural modification of the preferential approach yields exactly the same logical system as a probabilistic approach—not using single probability measures, but rather sets of probability measures. Thus, the same preferential structures used in the study of non-monotonic logics and belief revision may be used in the study of comparative probabilistic reasoning based on imprecise probabilities.

Although wind power ramp events (WPREs) are relatively scarce, they can inevitably deteriorate the stability of power system operation and bring risks to the trading of electricity market. In this paper, an imprecise conditional probability estimation method for WPREs is proposed based on the Bayesian network (BN) theory. The method uses the maximum weight spanning tree (MWST) and greedy search (GS) to build a BN that has the highest fitting degree with the observed data. Meanwhile, an extended imprecise Dirichlet model (IDM) is developed to estimate the parameters of the BN, which quantificationally reflect the ambiguous dependencies among the random ramp event and various meteorological variables. The BN is then applied to predict the interval probability of each possible ramp state under the given meteorological conditions, which is expected to cover the target probability at a specified confidence level. The proposed method can quantify the uncertainty of the probabilistic ramp event estimation. Meanwhile, by using the extracted dependencies and Bayesian rules, the method can simplify the conditional probability estimation and perform reliable prediction even with scarce samples. Test results on a real wind farm with three-year operation data illustrate the effectiveness of the proposed method.

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. [ABSTRACT FROM AUTHOR]

Standard accuracy‐based approaches to imprecise credences have the consequence that it is rational to move between precise and imprecise credences arbitrarily, without gaining any new evidence. Building on the Educated Guessing Framework of Horowitz (2019), we develop an alternative accuracy‐based approach to imprecise credences that does not have this shortcoming. We argue that it is always irrational to move from a precise state to an imprecise state arbitrarily, however it can be rational to move from an imprecise state to a precise state arbitrarily. [ABSTRACT FROM AUTHOR]

Imprecise probabilities (IP) are an increasingly popular way of reasoning about rational credence. However they are subject to an apparent failure to display convincing inductive learning. This paper demonstrates that a small modification to the update rule for IP allows us to overcome this problem, albeit at the cost of satisfying only a weaker concept of coherence. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Although bivariate imprecise copulas have recently attracted substantial attention, the multivariate case seems still to be open. So, it is natural to test it first on shock model induced copulas, a family which might be the most useful in various applications. We investigate a model in which some of the shocks are assumed imprecise and develop the corresponding set of copulas. In the Marshall's case we get a coherent set of distributions and a coherent set of copulas, where the bounds are naturally corresponding to each other. The situation in the other two groups of multivariate imprecise shock model induced copulas, i.e., the maxmin and the reflected maxmin (RMM) copulas, is substantially more involved, but we are still able to exhibit their properties. These are the main results of the paper that serves as the first step into a theory that should develop in this direction. Also, the theory of imprecise RMM copulas seems to be new even in the bivariate case. It turns out that all imprecise copulas under consideration are coherent. [ABSTRACT FROM AUTHOR]

Epistemic uncertainty in seismic hazard analysis is traditionally addressed by utilizing a logic-tree structure with subjective probabilities for branches. However, many studies have argued that probability is not a suitable choice for addressing epistemic uncertainties; in particular in addressing the background knowledge supporting the probabilities. In this regard, the application of imprecise probability (IP) is investigated. It is discussed that IP could provide a flexible tool for a more objective presentation of experts' knowledge. Moreover, the importance of addressing the strength of knowledge and surprises relative to knowledge in seismic hazard analysis along with methods to do so, are discussed. Then, a method is proposed for providing seismic hazard curves with consideration of IPs for logic-tree branches and addressing the knowledge dimension. It is suggested to consider the worst possible combination of branch probabilities for the expected intensity measures at all return-periods in order to construct the seismic hazard curve. A process is also suggested to demonstrate how the proposed seismic hazard analysis method could be properly used in the seismic design of buildings. The performance of the suggested method was investigated on Uniform California Earthquake Rupture Forecast, Version 2 (UCERF2) logic-tree for two sites in Los Angeles and Oakland, California, US. The results indicate that even with a similar logic-tree, the effects of imprecision in logic-tree weights could be different at different sites. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Computational simulation provides an efficient means to predict the behavior of customized hybrid material configurations using validated, physics-based models. One limitation to this approach is the quality and quantity of available data to characterize the many constituent input properties. Therefore, a systematic approach to identify the most influential parameters on the hybrid behavior and quantify the corresponding uncertainty in predictive capabilities is required. In this work, an approach using Bayesian multimodel inference and imprecise global sensitivity analysis is presented to investigate the effects of sparse constituent data on the prediction of composite material properties. The methodology allows the identification, using quantified uncertainties, of the most influential constituent material parameters for specified homogenized properties. This sensitivity analysis further enables a dimension reduction when assessing the influence of uncertainties on material properties and can be used to inform testing programs of the constituent properties that require additional testing/data collection in order to minimize uncertainty in macro-scale composite properties. The methodology is specifically demonstrated on the prediction and sensitivity analysis of out-of-plane mechanical properties of a unidirectional lamina. [ABSTRACT FROM AUTHOR]

Recent work has formally linked the traditional axiomatisation of incomplete preferences à 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. [ABSTRACT FROM AUTHOR]

In this article, we present the American option pricing procedure based on the binomial tree from an imprecise statistical aspect. Nonparametric Predictive Inference (NPI) is implemented to infer imprecise probabilities of underlying asset movements, reflecting uncertainty while learning from data, which is superior to the constant risk-neutral probability. In a recent article, we applied the NPI method to the European option pricing procedure that gives good results when the investor has non-perfect information. We now investigate the NPI method for American option pricing, of which imprecise probabilities are considered and updated for every one-time-step path. Different from the classic models, this method is shown that it may be optimal to early exercise an American non-dividend call option because our method considers all information that occurs in the future steps. We also study the performance of the NPI pricing method for American options via simulations in two different scenarios compared to the Cox, Ross and Rubinstein binomial tree model (CRR), first where the CRR assumptions are right, and second where the CRR model uses wrong assumptions. Through the performance study, we conclude that the investor using the NPI method tends to achieve good results in the second scenario. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]