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Rough sets generalize set membership in some sense beyond the uncertainty implicit in fuzzy sets, but with similar objectives and consequences. They generalize sets using a third truth value uncertain or undecideable. Rough sets differ from fuzzy sets in that no degree of membership, probability, or degree of belief in membership is ascribed to an uncertain member. Therefore, a rough set consists of known members and possible members. A union may either include or not include the possible members, yielding upper and lower sets, respectively, with corresponding constructs on union and intersection operators. Machine learning algorithms have been implemented using rough sets [306]. The approach expresses uncertainty about set membership with a degree of generality from the predicate calculus and requires no a priori assignment of degree of membership as is necessary to effectively use fuzzy sets. This comes at the expense of introducing virtual states and returning uncertain results that can be combinatorially explosive. The CRA <Self/> includes na ve, fuzzy, and rough sets in RXML as canonical templates for representation and reasoning. The methods of reasoning with uncertainty may be organized into systematic mathematical systems called certainty calculi.
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Not every CR need have a computer model of probability or fuzzy sets, but each CR needs some way of reasoning with uncertainty. A systematic method that includes the representation of positive and negative reinforcement along with at least one method for the aggregation of reinforcement across multiple stimuli (e.g., via a logic or rule-based process) may be called a certainty calculus. The Bayes community of interest in UAI [307], Java Bayes [308], and recent texts [13] offer strong support for Bayes reasoning with uncertainty. <User/>-domain driven approaches to evidential reasoning include methods for the nonlinear combination of evidence in such classic reasoning systems like Mycin, Dendral and meta-Dendral, and Tieresias. 11.4.9.1 Uncertainty in the <User/> Domain Suppose the set X = {X, O} represents a closed world of states such as the channel symbols of a BPSK modem or the decision to turn the CWPDA on or off. One can t add another element to X without rede ning the probability system. The world is rarely closed. Just when you think you have accounted for all the possibilities, a new one is discovered experimentally. Instead of implementing a binary power system {On, Off}, the manufacturer implements {On, Off, Pause}, where {Pause} conserves power while preserving state. The <User/> says, Pause, will you The AACR trained for {On, Off} now needs a new symbol for pause, for example, Y. To mark which user paused the
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UNCERTAINTY
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system it might remember <User> X </User> or <User> O </User>. The AACR could introduce Y as a metalevel construct where the states of play are {Run, Pause [Y | x {X, O}]}. Notionally, there could be a probability of Y, so in the re ned X = {X, O, Y} there is a metalevel probability of Pause. The original designers of the notional AACR didn t envision pause, but the user thinks it is a good idea. The <User/>-domain AACR must accommodate an open-world setting: Users are continually moving out of the box that the manufacturer would like them to stay in. Therefore, in order to employ probability to enhance QoI, the AACR designer must restrict the use of probability to those closed domains that are accurately modeled by probability, such as noise and other stochastic processes. Theories of evidence other than Bayes Law offer insights for other approaches to uncertainty in the AACR <User/> domain. 11.4.9.2 Theories of Evidence The Dempster Schaffer (DS) theory of evidence does not need the total probability space of Bayes. In addition, the Dempster Schaffer theory of evidence generalized Bayes notions of a priori and a posteriori probability to the more general problem of evidential reasoning. Although theoretically powerful, Bayes theory requires one to estimate the prior probabilities underlying all possible events. The dif culty of this requirement, among other things, has led to a proliferation of ad hoc techniques for representing uncertainty. To perform consistent logic in uncertain domains requires a calculus that manipulates numerical representations of uncertainty with associated Boolean logic or assertions in rule-based systems. Some powerful uncertainty calculi are nonlinear [318]. There is also much relevant technology from probability and statistics literature. Mixture modeling, for example, is the process of representing a statistical distribution in terms of a mixture or weighted sum of other distributions [309]. AACR exhibits statistical mixtures of uncertainty in RF, in sensory perception, and in interpreting user interactions. The CRA embeds general facilities for reasoning under uncertainty by prescribing reinforcement and expressing <Uncertainty/> tags as schema-schema for application-speci c certainty calculi, such as that of Tieresias. 11.4.9.3 Tieresias Certainty Calculus Tieresias employed the Mycin calculus for reasoning under uncertainty. Medical doctors dealing with bacterial infections of the blood were known to consider a weight of evidence for and against a causative agent. The Mycin certainty calculus therefore independently aggregated weight of evidence for and against a given diagnosis. Subsequently, these aggregates were evaluated to determine whether positive indications outweighed negative indications and conversely, preferring the explanation that most positively endorsed a given diagnosis. Thresholds for weight of evidence were also employed to defer decisions until suf cient knowledge was applied.
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