Glenn Shafer is an American mathematician and statistician. He is the co-creator of Dempster–Shafer theory. He is a University Professor and Board of Governors Professor at Rutgers University.
The theory of belief functions, also referred to as evidence theory or Dempster–Shafer theory (DST), is a general framework for reasoning with uncertainty, with understood connections to other frameworks such as probability, possibility and imprecise probability theories. First introduced by Arthur P. Dempster in the context of statistical inference, the theory was later developed by Glenn Shafer into a general framework for modeling epistemic uncertainty—a mathematical theory of evidence. The theory allows one to combine evidence from different sources and arrive at a degree of belief that takes into account all the available evidence.
In psychology, decision-making is regarded as the cognitive process resulting in the selection of a belief or a course of action among several possible alternative options. It could be either rational or irrational. The decision-making process is a reasoning process based on assumptions of values, preferences and beliefs of the decision-maker. Every decision-making process produces a final choice, which may or may not prompt action.
Decision theory is a branch of applied probability theory and analytic philosophy concerned with the theory of making decisions based on assigning probabilities to various factors and assigning numerical consequences to the outcome.
Multiple-criteria decision-making (MCDM) or multiple-criteria decision analysis (MCDA) is a sub-discipline of operations research that explicitly evaluates multiple conflicting criteria in decision making. Conflicting criteria are typical in evaluating options: cost or price is usually one of the main criteria, and some measure of quality is typically another criterion, easily in conflict with the cost. In purchasing a car, cost, comfort, safety, and fuel economy may be some of the main criteria we consider – it is unusual that the cheapest car is the most comfortable and the safest one. In portfolio management, managers are interested in getting high returns while simultaneously reducing risks; however, the stocks that have the potential of bringing high returns typically carry high risk of losing money. In a service industry, customer satisfaction and the cost of providing service are fundamental conflicting criteria.
An influence diagram (ID) is a compact graphical and mathematical representation of a decision situation. It is a generalization of a Bayesian network, in which not only probabilistic inference problems but also decision making problems can be modeled and solved.
Upper and lower probabilities are representations of imprecise probability. Whereas probability theory uses a single number, the probability, to describe how likely an event is to occur, this method uses two numbers: the upper probability of the event and the lower probability of the event.
Imprecise probability generalizes probability theory to allow for partial probability specifications, and is applicable when information is scarce, vague, or conflicting, in which case a unique probability distribution may be hard to identify. Thereby, the theory aims to represent the available knowledge more accurately. Imprecision is useful for dealing with expert elicitation, because:
Probabilistic logic involves the use of probability and logic to deal with uncertain situations. Probabilistic logic extends traditional logic truth tables with probabilistic expressions. A difficulty of probabilistic logics is their tendency to multiply the computational complexities of their probabilistic and logical components. Other difficulties include the possibility of counter-intuitive results, such as in case of belief fusion in Dempster–Shafer theory. Source trust and epistemic uncertainty about the probabilities they provide, such as defined in subjective logic, are additional elements to consider. The need to deal with a broad variety of contexts and issues has led to many different proposals.
Evidential reason or evidential reasoning may refer to:
The transferable belief model (TBM) is an elaboration on the Dempster–Shafer theory (DST), which is a mathematical model used to evaluate the probability that a given proposition is true from other propositions that are assigned probabilities. It was developed by Philippe Smets who proposed his approach as a response to Zadeh’s example against Dempster's rule of combination. In contrast to the original DST the TBM propagates the open-world assumption that relaxes the assumption that all possible outcomes are known. Under the open world assumption Dempster's rule of combination is adapted such that there is no normalization. The underlying idea is that the probability mass pertaining to the empty set is taken to indicate an unexpected outcome, e.g. the belief in a hypothesis outside the frame of discernment. This adaptation violates the probabilistic character of the original DST and also Bayesian inference. Therefore, the authors substituted notation such as probability masses and probability update with terms such as degrees of belief and transfer giving rise to the name of the method: The transferable belief model.
In decision theory, the evidential reasoning approach (ER) is a generic evidence-based multi-criteria decision analysis (MCDA) approach for dealing with problems having both quantitative and qualitative criteria under various uncertainties including ignorance and randomness. It has been used to support various decision analysis, assessment and evaluation activities such as environmental impact assessment and organizational self-assessment based on a range of quality models.
A belief structure is a distributed assessment with beliefs.
A decision matrix is a list of values in rows and columns that allows an analyst to systematically identify, analyze, and rate the performance of relationships between sets of values and information. Elements of a decision matrix show decisions based on certain decision criteria. The matrix is useful for looking at large masses of decision factors and assessing each factor's relative significance by weighting them by importance.
In applied mathematics and decision making, the aggregated indices randomization method (AIRM) is a modification of a well-known aggregated indices method, targeting complex objects subjected to multi-criteria estimation under uncertainty. AIRM was first developed by the Russian naval applied mathematician Aleksey Krylov around 1908.
Grey relational analysis (GRA) was developed by Deng Julong of Huazhong University of Science and Technology. It is one of the most widely used models of grey system theory. GRA uses a specific concept of information. It defines situations with no information as black, and those with perfect information as white. However, neither of these idealized situations ever occurs in real world problems. In fact, situations between these extremes, which contain partial information, are described as being grey, hazy or fuzzy. A variant of GRA model, Taguchi-based GRA model, is a popular optimization method in manufacturing engineering.
Decision-making software is software for computer applications that help individuals and organisations make choices and take decisions, typically by ranking, prioritizing or choosing from a number of options.
A probability box is a characterization of uncertain numbers consisting of both aleatoric and epistemic uncertainties that is often used in risk analysis or quantitative uncertainty modeling where numerical calculations must be performed. Probability bounds analysis is used to make arithmetic and logical calculations with p-boxes.
Probability bounds analysis (PBA) is a collection of methods of uncertainty propagation for making qualitative and quantitative calculations in the face of uncertainties of various kinds. It is used to project partial information about random variables and other quantities through mathematical expressions. For instance, it computes sure bounds on the distribution of a sum, product, or more complex function, given only sure bounds on the distributions of the inputs. Such bounds are called probability boxes, and constrain cumulative probability distributions.
DecideIT is a decision-making software for the Microsoft Windows operating system. It is based on multi-criteria decision making (MCDM) and the multi-attribute value theory (MAVT). It supports both value tree analysis for multi-attribute decision problems as well as decision tree analysis for evaluating decisions under risk and can combine these structures in a common model.
