Degrees of belief

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). 

As practiced, Monte Carlo is not only a statistical method, but also a process that involves numerous cascading decisions involving statisticians, toxicologists, and risk assessors. The degree of belief inherent in the Monte Carlo outputs is as much a function of the numerous decisions the investigator makes during the course of the analysis as it is the correct selection of the sampling distributions. 

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... 

The numerical properties of probability and degrees of belief can be defined effectively and sensibly using a few axioms. 

Bayesian interpretation and application of the theorem quantifies the development of information. Suppose that A is a statement or hypothesis, and let p A) stand for the degree of belief in the statement or hypothesis A, based on prior knowledge, it is called the prior probability. Let B represent a set of observations, then p(B A) is the probability that those observations occur given that A is true. This is called the likelihood of the data and is a function of the hypothesis. The left side, p(A B), is the new degree of belief in A, taking into account the observations B, it is called the posterior probability. Thus Bayes theorem tracks the effect that the observations have upon the changing knowledge about the hypothesis. The theorem can be expressed thus ... 

Credible interval In a Bayesian analysis, the area under the posterior distribution. Represents the degree of belief, including all past and current information,... 

Expert judgment A critical source of information based npon the collective experience of a scientist or expert in a particular field of study. For Bayesians, expert judgement is frequently used to form the prior distribution, thns formally incorporating an expert s degree of belief into statistical procednres. 

Probability The Bayesian or subjective view is that the probability of an event is the degree of belief that a person has, given some state of knowledge, that the event will occur. In the classical or frequentist view, the probability of an event is the frequency of an event occurring given a long sequence of identical and independent trials. 

The probabilities of different outcomes can thus be seen as resulting from the causal powers and capacities of the system and their arrangement. This makes probability a function of the nature of the system, not merely a statement of degrees of belief or the frequency with which an outcome occurs. We can account for the observed probability (in a frequency sense) by the interplay of capacities or causal powers, and we can estimate a probability (in the epistemic sense) if we know something about the capacities of the things that may influence the outcome. 

Similar to a confidence interval, except that a credibility interval represents the degree of belief regarding the true value of a statistical parameter. 

Probability. The likelihood of the occurrence of an event or a measure of degree of belief, the values of which range from 0 to 1. 

Probability P/A can also be interpreted subjectively as a measure of degree of belief, on a scale from 0 to 1, tliat tlie event A occurs. Tliis interpretation is frequently used in ordinary conversation. For example, if someone says, Tlie probability that I will go to the movies tonight is 90% , tlien 90% is a measure of the person s belief tliat he will go to tlie movies. This interpretation is also used whea in tlie absence of concrete data needed to estimate an unknown probability on the basis of observed relative frequency, Ute personal opinion of an expert is sought. For e.xample, an expert niiglit be asked to estimate tlie probability tliat tlie seals in a newly designed pump will leak at liigh pressures. Tlie estimate would be based on the expert s familiarity with tlie liistory of pumps of similar design. 

The Dempster-Shafer theory, also known as the theory of belief functions, is a generalization of the Bayesian theory of subjective probability [30,42]. Whereas the Bayesian theory requires probabilities for each question of interest, belief functions allow us to base degrees of belief for one question on probabilities for a related question. These degrees of belief may or may not have the mathematical properties of probabilities how much they differ from probabilities will depend on how closely the two questions are related. 

In summary, we obtain degrees of belief for one question (i.e.. Will John reach the bus in time ) from probabilities for another question (i.e.. Is the alarm clock reliable ). Dempster s rule begins with the assumption that the questions for which we have probabilities are independent with respect to our subjective probability judgments, but this independence is only a priori it disappears when conflict is discerned between the different items of evidence. 

Dempster-Shafer Theory of Evidence is a generalization of the Bayesian theory based on degrees of belief rather than probabilities. 

The true placebo effect of treatment with a cough medicine is related to the patient s belief about the efficacy of the medicine (Evans 2003) and the meaning that the patient relates to the treatment (Moerman 2002). The degree of belief in the treatment will depend on many factors, such as the healer-patient interaction, cultural beliefs about traditional treatments, the environment in which the medicine is administered, the properties of the medicine, such as taste, colom and smell, advertising and claims made about the efficacy of the medicine, the brand name of the medicine, and side effects associated with treatment that may reinforce the belief of efficacy. This list of factors that may influence the true placebo effect is not exhaustive and it illustrates how difficult it is to properly control and standardise studies on the true placebo effect. 

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