Imprecise Probability Approach Research Articles

Incorporating ignorance within game theory: An imprecise probability approach

Ignorance within non-cooperative games, reflected as a player's uncertain preferences towards a game's outcome, is examined from a Bayesian point of view. This topic has had scarce treatment in the literature, which emphasises exogenous uncertainties caused by other players or nature and not by players themselves. That is primarily because a player's endogenous uncertainty over an outcome poses significant challenges and complex sequences of reciprocal expectations. Therefore, it is often ignored, and preferences are either assumed from a continuous domain or set using introspection, resulting in non-optimal models. We here explore a solution concept based on recent research in imprecise probabilities and de Finetti's approach to defining subjective probabilities, which utilises bets to assess beliefs. The resulting model allows players to be ignorant about their initial preferences and learn about them in repeated games. Furthermore, it permits improving the value of information in these situations. This model is proposed as a possible solution to the problem of utility inference in game-theoretic settings that include uncertainty over outcomes. We demonstrate it through motivating repeated-game problems modified to have uncertainty and through a simulation over a case of extreme ignorance.

A Mixed Epistemic-Aleatory Stochastic Framework for the Optimal Operation of Hybrid Fuel Stations

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.

Keynes’s Major Result From Part II of the A Treatise on Probability Was That, Given That Numerical Probabilities Are Additive, Then Non Numerical Probabilities Must Be Non Additive: Non Additivity Is a Sufficient Condition for Some Degree of Uncertainty to Exist

Keynes’s major accomplishment in Part II of the A Treatise on Probability (1921),which he also accomplished in the 1907 and 1908 Fellowship Dissertation versions, was to show that the addition property of the purely mathematical calculus of probability could only be operational in certain circumstances where the evidential weight of the argument, V(a/h) =degree w, was equal to a w of 1,where 0≤w≤1, so that the decision maker had complete data/evidence set.All relevant information or evidence had be known before any decision had to be made. Thus, all numerical probabilities are additive, so that they would sum to 1. However, there were also non numerical probabilities, which were non additive because of the existence of missing, relevant data or evidence, that would not sum to 1. By far the most important case was sub additive,as opposed to super additive, probabilities, which would sum to less than 1. Note the obvious fact that ordinal probabilities are not only not non additive, since they can’t sum to less than 1, but they also can’t be multiplied. Keynes then showed that non numerical probabilities, which were non additive because they summed to less than 1, had to be interval valued probabilities with a lower (greatest lower bound) and a upper (least upper bound) bound. This, of course, followed directly from pp.265-268 of Boole’s 1854 The Laws of Thought. The adherents of all Heterodox, Post Keynesian, and Institutionalist schools of economics (Skidelsky, O’Donnell, Carabelli, Dow, Lawson, and Runde, for example. See the references) argue that Keynes’s non numerical probabilities were ordinal probabilities. However, this conclusion is an oxymoron, because ordinal probability can have absolutely nothing to do with questions concerned about additivity of probabilities summing to 1 or non additivity of probabilities not summing to one, but being less than 1, since, by definition, ordinal probabilities can’t be summed, added or multiplied. Numerical and ordinal probability can’t deal with non measurability, non comparability or incommensurability. Such problems require nonlinear and non additive approaches to measurement. Only Keynes’s and Boole’s initial ,imprecise approaches to probability, using interval valued probability, can do this. It should be obvious to a mathematically trained reader that Boole and Keynes are the founders of the imprecise probability approach to decision making. Unfortunately, there are no economists or philosophers who covered Part II of the A Treatise on Probability in the 20th or 21st centuries. All readers, like Emile Borel, the French mathematician, skipped Part II of the A Treatise on Probability. This is why the obviously false claim, that Keynes’s non numerical probabilities had to be ordinal probabilities, which directly contradicts Keynes’s position that they are non additive, continues to be generally accepted nearly one hundred years after the publication of Keynes’s A Treatise on Probability.

A CONSISTENT DATA FUSION APPROACH FOR UNCERTAINTY QUANTIFICATION IN RIGID MUSCULOSKELETAL SIMULATION

Uncertainty quantification in rigid musculoskeletal modeling is essential to analyze the risks related to the simulation outcomes. Data fusion from multiple sources is a potential solution to reduce data uncertainties. This present study aimed at proposing a new data fusion rule leading to a more consistent and coherent data for uncertainty quantification. Moreover, a new uncertainty representation was developed using imprecise probability approach. A biggest maximal coherent subsets (BMCS) operator was defined to fuse interval-valued data ranges from multiple sources. Fusion-based probability-box structure was developed to represent the data uncertainty. Case studies were performed for uncertainty propagation through inverse dynamics and static optimization algorithms. Hip joint moment and muscle force estimation were computed under effect of the uncertainties of thigh mass and muscle properties. Respective p-boxes of these properties were generated. Regarding the uncertainty propagation analysis, correlation coefficients showed a very good value ([Formula: see text]) for the proposed fusion operator according to classical operators. Muscle force variation of the rectus femoris was computed. Peak-to-peak (i.e., difference between maximal values) rectus femoris forces showed deviations of 55[Formula: see text]N and 40[Formula: see text]N for the first and second peaks, respectively. The development of the new fusion operator and fusion-based probability-box leads to a more consistent uncertainty quantification. This allows the estimation of risks associated with the simulation outcomes under input data uncertainties for rigid musculoskeletal modeling and simulation.

An imprecise probability approach for squeal instability analysis based on evidence theory

An imprecise probability approach based on evidence theory is proposed for squeal instability analysis of uncertain disc brakes in this paper. First, the squeal instability of the finite element (FE) model of a disc brake is investigated and its dominant unstable eigenvalue is detected by running two typical numerical simulations, i.e., complex eigenvalue analysis (CEA) and transient dynamical analysis. Next, the uncertainty mainly caused by contact and friction is taken into account and some key parameters of the brake are described as uncertain parameters. All these uncertain parameters are usually involved with imprecise data such as incomplete information and conflict information. Finally, a squeal instability analysis model considering imprecise uncertainty is established by integrating evidence theory, Taylor expansion, subinterval analysis and surrogate model. In the proposed analysis model, the uncertain parameters with imprecise data are treated as evidence variables, and the belief measure and plausibility measure are employed to evaluate system squeal instability. The effectiveness of the proposed approach is demonstrated by numerical examples and some interesting observations and conclusions are summarized from the analyses and discussions. The proposed approach is generally limited to the squeal problems without too many investigated parameters. It can be considered as a potential method for squeal instability analysis, which will act as the first step to reduce squeal noise of uncertain brakes with imprecise information.

Availability analysis of systems using random set theory

In this paper, we introduce the use of random set theory for availability assessment of systems with rare failure events. Instead of using failure probabilities calculated directly from each component's observation, we propose to construct pseudo-system observations directly from components observations. We then applied operations defined in the random set framework (expectations, confidence intervals, etc.) to obtain upper and lower bounds and confidence intervals of system availability without assuming any prior distribution about failures of the system components. Finally, a case study is used to prove the efficiency of the proposed method and a comparison with imprecise probability approach based on Monte-Carlo simulations is also done.

An imprecise probability approach for the detection of over and underused taxonomic groups with the Campania (Italy) and the Sierra Popoluca (Mexico) medicinal flora

Aim of the studyWe use the IDM model to test for over- and underuse of plant taxa as source for medicine. In contrast to the Bayes approach, which only considers the uncertainty around the data of medicinal plant surveys, the IDM model also takes the uncertainty around the inventory of the flora into account, which is used for the comparison between medicinal and local floras. Materials and methodsStatistical analysis of the medicinal flora of Campania (Italy) and of the medicinal flora used by the Sierra Popoluca (Mexico) was performed with the IDM model and the Bayes approach. For Campania 423 medicinal plants and 2237 vascular plant species and for the Sierra Popoluca 605 medicinal plants and 2317 vascular plant species were considered. ResultsThe IDM model (s=4) indicates for Campania the Lamiaceae and Rosaceae as overused, and the Caryophyllaceae, Poaceae, and Orchidaceae as underused. Among the Popoluca the Asteraceae and Piperaceae turn out to be overused, while Cyperaceae, Poaceae, and Orchidaceae are underused. In comparison with the Bayes approach, the IDM approach indicates fewer families as over- or underused. ConclusionsThe IDM model leads to more conservative results compared to the Bayes approach. Only relatively few taxa are indicated as over- or underused. The larger the families (nj's) are, the more similar do the results of the two approaches turn out. In contrast to the Bayes approach, small taxa with most or all species used as medicine (e.g., nj=2, xj=2) tend not to be indicated as overused with the IDM model.

Reliability-Based Decision Making: A Comparison of Statistical Approaches

This paper considers the problem of choosing between an existing component whose reliability is well established and a new component that has an unknown reliability. In some scenarios, the designer may have some initial beliefs about the new component’s reliability. The designer may also have the opportunity to obtain more information and to update these beliefs. Then, based on these updated beliefs, the designer must make a decision between the two components. This paper examines the statistical approaches for updating reliability assessments and the decision policy that the designer uses. We consider four statistical approaches for modeling the uncertainty about the new component and updating assessments of its reliability: a classical approach, a precise Bayesian approach, a robust Bayesian approach, and an imprecise probability approach. The paper investigates the impact of different approaches on the decision between the components and compares them. In particular, given that the test results are random, the paper considers the likelihood of making a correct decision with each statistical approach under different scenarios of available information and true reliability. In this way, the emphasis is on practical comparisons of the policies rather than on philosophical arguments.

A comparison of statistical approaches for assessing reliability

Reliability estimates are useful for making design decisions. We consider the case where a designer must choose between an existing component whose reliability is well-established and a new component that has an unknown reliability. This paper compares the statistical approaches for updating reliability assessments based on additional simulation or experimental data. We consider four statistical approaches for modelling the uncertainty about a new component's failure probability: a classical approach, a precise Bayesian approach, a robust Bayesian approach and an imprecise probability approach. We show that an imprecise beta model is compatible with both the robust Bayesian approach and the imprecise probability approach. The different approaches for forming and updating the designer's beliefs about the product reliability are illustrated and compared under different scenarios of available information. The goal is to gain insight into the relative strengths and weaknesses of the approaches. Examples are presented for illustrating the conclusions.

Imprecise probabilities of climate change: aggregation of fuzzy scenarios and model uncertainties

Whilst the majority of the climate research community is now set upon the objective of generating probabilistic predictions of climate change, disconcerting reservations persist. Attempts to construct probability distributions over socio-economic scenarios are doggedly resisted. Variation between published probability distributions of climate sensitivity attests to incomplete knowledge of the prior distributions of critical parameters and structural uncertainties in climate models. In this paper we address these concerns by adopting an imprecise probability approach. We think of socio-economic scenarios as fuzzy linguistic constructs. Any precise emissions trajectory (which is required for climate modelling) can be thought of as having a degree of membership in a fuzzy scenario. Next, it is demonstrated how fuzzy scenarios can be propagated through a low-dimensional climate model, MAGICC. Fuzzy scenario uncertainties and imprecise probabilistic representation of climate model uncertainties are combined using random set theory to generate lower and upper cumulative probability distributions for Global Mean Temperature anomaly. Finally we illustrate how non-additive measures provide a flexible framework for aggregation of scenarios, which can represent some of the semantics of socio-economic scenarios that defy conventional probabilistic representation.

Dealing with Imprecise Probabilities: Interval-Related Talks at ISIPTA'05

What was the conference? From July 20–24, 2005, the 4th International Symposium on Imprecise Probabilities and Their Applications (ISIPTA’05) was held on the campus of Carnegie Mellon University in Pittsburgh, Pennsylvania, USA. This biennial conference is organized by the Society for Imprecise Probability Theory and Applications (SIPTA) [13]. The main organizers of ISIPTA’05 were Fabio G. Cozman (University of Sao Paulo, Brazil), Robert Nau (Duke University, USA), and Teddy Seidenfeld (Carnegie Mellon University). Papers presented at ISIPTA’05 were authored by researchers from a very diverse list of countries: Belgium, Brazil, the Czech Republic, France, Germany, Israel, Italy, New Zealand, Russia, Slovenia, Spain, Switzerland, the UK, and the US. This ISIPTA conference had an interesting format that had been found valuable in previous conferences. The first day was devoted to tutorials, starting with an introduction to imprecise probabilities given by Gert de Cooman, President of SIPTA. Starting the second day, each regular paper was presented in a plenary session, which was followed by a poster session where each presenter of that session also presented a poster and actively communicated with interested participants. A special session was devoted to open problems, and the conference was immediately followed by a half-day workshop on a key application area, financial risk assessment. Next we review subjective probabilities as traditionally handled. With that as background, we review the imprecise probability approach. In this review, we mainly follow the tutorial given by G. de Cooman. This review is followed by problems, challenges, and steps toward their resolution presented by researchers at the conference.