Measure of belief

It is widely accepted that probabilistic approach was first developed and applied in expert systems MYCIN and PROSPECTOR. In these expert systems the likelihood estimates are calculated for several competitive hypothesis H on the basis of available evidences E. In the expert system MYCIN each hypothesis was estimated by a confidence factor CF(H Ei,E2. . . ) as a difference of estimates for the measure of belief MB P[ Ei,E2,. .) and the measure of distrust MD P[ Ei,E2,...) ... 

Certainty theory is an approach to inexact reasoning that describes uncertain information in a certainty factor [28]. Certainty factors are used as a degree of confirmation of a piece of evidence. Mathematically, a certainty factor is the measure of belief minus the measure of disbelief. Again, using an example with John, an uncertainty could be the following ... 

Certainty Factors are used as a degree of conhrmation of a piece of evidence. Mathematically, a certainty factor is the measure of belief minus the measure of disbelief. 

The highest values of the resulting belief function mE(l-2)(rn), the combination of two experts hypotheses indicates the level of failure risk analysis of water supply network. Measures of beliefs belE(l-2)(rn) and the presumption plE(l-2)(rn) characterized by the so-called interval of uncertainty as to the hypotheses interpretation. The uncertainty factor AF is the difference between belief and presumption in the case of hypothesis. 

Eq. (4.8), which summarizes this second principle, MB is a measure of belief in the occurrence of event P given the occurrence of event E, and MD is a measure of disbelief. And the measure of belief that results from considering two sources of evidence, rule 1 (i i) and rule 2 can be calculated by using Eq. (4.9) when these sources have MBs > 0 or using Eq. (4.10) for MDs > 0. The data and CF values would have to reflect the knowledge of a given component in a given situation and at a particular time. 

Belief measures (also conmiOTly referred to as belief functions) aim to generalize the well-known interpretation of subjective probability theory - i.e., the Bayesian probability for a subjective measure of uncertainty - to a broader concept of evidence. Like possibility and necessity measures, evidence theory is developed from dual measures of belief and plausibility (KJir and Yuan 1995). These measures express beliefs or judgments formulated by available evidence (Yager and Liu 2008). Although Dempster s original woiks were closely linked to classical probability theory, there are some significant distinctions between classical probability theory and evidence theory. 

The only groups in Table 9.5 with negative values of CT/ are the alkyl groups methyl and tert-butyl. There has been some controversy on this point. One opinion is that CT/ values decrease in the series methyl, ethyl, isopropyl, /ert-butyl (respectively, — 0.046, —0.057, —0.065, —0.074). Other evidence, however, has led to the belief that all alkyl groups have approximately the same field effect and that the a/ values are invalid as a measure of the intrinsic field effects of alkyl groups. Another attempt to divide ct values into resonance and field contributions is that of Swain and Lupton, who have shown that the large number of sets of ct values (cTm, <3p, a, a+, CT/, csp etc., as well as others we have not mentioned) are not entirely independent and that linear combinations of two sets of new values F (which expresses the field-effect contribution) and R (the resonance contribution) satisfactorily express 43 sets of values. Each set is expressed as... 

The development and refinement of population balance techniques for the description of the behavior of laboratory and industrial crystallizers led to the belief that with accurate values for the crystal growth and nucleation kinetics, a simple MSMPR type crystallizer could be accurately modelled in terms of its CSD. Unfortunately, accurate measurement of the CSD with laser light scattering particle size analyzers (especially of the small particles) has revealed that this is not true. In mar cases the CSD data obtained from steady state operation of a MSMPR crystallizer is not a straight line as expected but curves upward (1. 32. 33V This indicates more small particles than predicted... 

The extent of knowledge about 0 can be quantified by showing that probability also can be interpreted as degree of belief (Lindley 1965), measure of plausibility (Loredo 1990), or personal probability (O Hagan 2001). Early workers such as... 

Richard Kirwan in 1781 and in 1782 undertook his analyses under the belief that the values could provide a measure of the affinity of an acid for different bases. He reported his analytical values in the form of a table. The following example is offered to illustrate the argument and the difficulty in the conception. 

Note that a useful operational measure of hydrophobicity is the octanol/water log P parameter (Hansch and Leo, 1979). This is a free energy-related parameter and is frequently thought of as synonymous with hydrophobicity, but it undoubtedly incorporates both AGh and AGy w components so is not pure hydrophobicity in the sense above. Lipophilicity is a more appropriate name. Contrary to common belief, log P is mostly enthalpic (Taylor, 1990 Da et al., 1992). 

Many modelers believe that, as far as possible, it is desirable, if not essential, that predictive techniques should based on fundamental physicochemical properties such as the partition coefficient (Cronin and Schultz, 2003 Schultz and Cronin, 2003). The use of physicochemical properties is described in more detail in Chapter 4. Physicochemical properties tend to describe fundamental molecular effects, such as the partition coefficient being a measure of hydrophobicity. There is a belief that such properties may be related to the mechanism of action of a molecule and are thus less susceptible to spurious correlation. 

The idea of hidden variables is fairly common in chemical models such as the kinetic gas model. This theory is formulated in terms of molecular momenta that remain hidden, and evaluated against measurements of macroscopic properties such as pressure, temperature and volume. Electronic motion is the hidden variable in the analysis of electrical conduction. The firm belief that hidden variables were mathematically forbidden in quantum systems was used for a long time to discredit Bohm s ideas. Without joining the debate it can be stated that this proof has finally been falsified. 

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