Belief, degree

The lower and upper boundary of the function of belief degree (m) of the truth of a hypothesis H with regard to X (or its components) define the measure (Kacprzyk Fedrizzi 1994, Shafer 1976, Yager 1987, Luo Caselton 1997) ... 

Quantitative risk analysis is a forecast concerning the degree of belief associated with the occurrence of future events. It normally focuses on those classes of events that are rarely expected to occur at a facility. However, because the potential consequences of such events may be so great, the possibility that the events could occur at all gives rise to concern. When a QRA generates results that reflect a very small likelihood of an event and confirm the suspicion that the event could have a severe impact, these questions inevitably arise What does it all mean What should I do about it ... 

Probability in Bayesian inference is interpreted as the degree of belief in the truth of a statement. The belief must be predicated on whatever knowledge of the system we possess. That is, probability is always conditional, p(X l), where X is a hypothesis, a statement, the result of an experiment, etc., and I is any information we have on the system. Bayesian probability statements are constructed to be consistent with common sense. This can often be expressed in tenns of a fair bet. As an example, I might say that the probability that it will rain tomorrow is 75%. This can be expressed as a bet I will bet 3 that it will rain tomorrow, if you give me 4 if it does and nothing if it does not. (If I bet 3 on 4 such days, I have spent 12 I expect to win back 4 on 3 of those days, or 12). 

Bayes s methods aim to satisfy two needs the concept of probability as degree of belief, and the need to use all available information in a probability cstimaie. Cia icists reject all... 

Figure 2.5-1 illustrates the fact that probabilities are not precisely known but may be represented by a "bell-like" distribution the amplitude of which expresses the degree of belief. The probability that a system will fail is calculated by combining component probabilities as unions (addition) and intersection (multiplication) according to the system logic. Instead of point values for these probabilities, distributions are used which results in a distributed probabilitv of system fadure. This section discusses several methods for combining distributions, namely 1) con olution, 2i moments method, 3) Taylor s series, 4) Monte Carlo, and 5) discrete probability distributions (DPD). 

For a long time, this finding was correlated with the observation that substituents at a radical center tend to enhance its stability (Section 1.1.2). This in turn led to the belief that the degree of stabilization conferred on the product radical by the substituents was the prime factor determining the orientation and rate of radical addition to olefins. That steric, polar, or other factors might favor the same outcome was either considered to be of secondary importance or simply ignored. ... 

There is often a feeling in the young that most of the important things in chemistry have already been discovered and explored. I felt that Way in 1938, when I received my own degree. As a result of my own experiences, I have altered my position. Perhaps this discussion of our research program over the past 35 years will help persuade others of the validity of my present belief that we are surrounded by major new areas awaiting discovery. 

In terms of downstream processes, the flow-rates, compositions, and so on, dictate the size and number of each unit operation for example, while a batch distillation may be used to separate a single feed into a number of different product streams, a continuous distillation train would in general require N columns for N different product streams. The fact that a high degree of modeling is used in the design of each MPI, results in the generally held belief that continuous processes... 

Many drug users without antisocial features will feel some degree of guilt as a result of their drug use and associated behaviors. Frequently, clients will have behaved while under the influence of drugs in ways that may have violated their principles, or may have taken advantage of others. Clients may have hurt other people, conned other people, violated their own belief systems, or violated the... 

As the analysis progresses, evidence is accumulated supporting the presence or absence of defined substructures. The evidence is combined by the Reasoner module to form a belief function, which describes the degree to which each substructure is currently believed. This information is stored in the chemical database, where it is available to the Expert modules and to the Controller as it decides the course of the analysis. As the belief function evolves, the current state is displayed graphically to the user, who may halt the analysis, query the current state, and redirect the course of the analysis by supplying evidence for or against a substructure. 

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