Bayesian theorem

One parametric routine implements a quadratic discriminant function using the Bayesian theorem. The equation for the discriminants are as follows ... 

Based on the above axioms one can define the fiizzy Bayesian theorem. 

According to Equation (40), the revised value of instantaneous survival probabilities of slabs the analysis of whose is based on Bayesian theorem is ... 

According to Equation (1) and (2), the posterior distribution of the failure rate X can be obtained based on Bayesian theorem,... 

The Local Structure Operator By the Kolmogorov consistency theorem, we can use the Bayesian extension of Pn to define a measure on F. This measure -called the finite-block measure, /i f, where N denotes the order of the block probability function from which it is derived by Bayesian extension - is defined by assigning t.o each cylinder c Bj) = 5 G F cti = 6i, 0 2 = 62, , ( j — bj a value equal to the probability of its associated block ... 

Three algorithms have been implemented in both single and multiperspective environments. In this way any bias introduced by a single algorithm should be removed. The first is the statistical Naive Bayesian Classifier, ft reduces the decision-making problem to simple calculations of feature probabilities, ft is based on Bayes theorem and calculates the posterior probability of classes conditioned on the given unknown feature... 

Maximum entropy (ME) is a tool of Bayesian Statistics, and thus is built around Bayes Theorem. Since, as diffractionists, we are interested in maps and particularly in obtaining an optimum map from measured data, we can state this theorem in the following way... 

While the use of Bayes Theorem in this context is not generally controversial its use more generally in medical and clinical research has not always been positively received." It is not the scope of the present chapter to illustrate the use of Bayesian statistics in a more general context and interested readers should read the excellent introduction to the use of Bayesian methods in health-care evaluation provided by Spiegelhalter et alP... 

Equation (5.10) is a statement of Bayes theorem. Since the theorem is proved using results or axioms valid for both frequentist and Bayesian views, its use is not limited to Bayesian applications. Note that it relates 2 conditional probabilities where the events A and B are interchanged. 

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 ... 

Confidence intervals nsing freqnentist and Bayesian approaches have been compared for the normal distribntion with mean p and standard deviation o (Aldenberg and Jaworska 2000). In particnlar, data on species sensitivity to a toxicant was fitted to a normal distribntion to form the species sensitivity distribution (SSD). Fraction affected (FA) and the hazardons concentration (HC), i.e., percentiles and their confidence intervals, were analyzed. Lower and npper confidence limits were developed from t statistics to form 90% 2-sided classical confidence intervals. Bayesian treatment of the uncertainty of p and a of a presupposed normal distribution followed the approach of Box and Tiao (1973, chapter 2, section 2.4). Noninformative prior distributions for the parameters p and o specify the initial state of knowledge. These were constant c and l/o, respectively. Bayes theorem transforms the prior into the posterior distribution by the multiplication of the classic likelihood fnnction of the data and the joint prior distribution of the parameters, in this case p and o (Fignre 5.4). 

Bias The systematic or persistent distortion of an estimate from the true value. From sampling theory, bias is a characteristic of the sample estimator of the sufficient statistics for the distribution of interest. Therefore, bias is not a function of the data, but of the method for estimating the population statistics. For example, the method for calculating the sample mean of a normal distribution is an unbiased estimator of the true but unknown population mean. Statistical bias is not a Bayesian concept, because Bayes theorem does not relay on the long-term frequency expections of sample estimators. 

The cornerstone of Bayesian methods is Bayes Theorem, which was first published in 1763 (Box Tiao, 1973). Bayes Theorem provides a method for statistical inference in which a prior distribution, based upon subjective judgement, can be updated with empirical data, to create a posterior distribution that combines both judgement and data. As the sample size of the data becomes large, the posterior distribution will tend to converge to the same result that would be obtained with frequentist methods. In situations in which there are no relevant sample data, the analysis can be conducted based upon the prior distribution, without any updating. 

Bayesian procedures are important not only for estimating parameters and states, but also for decision making in various fields. Chapters 6 and 7 include applications to model discrimination and design of experiments further applications appear in Appendix C. The theorem also gives useful guidance in economic planning and in games of chance (Meeden, 1981). 

Bayesian networks are based on Bayes Theorem, which gives a mathematical framework for describing the probability of an event that may have been the result of any of two or more causes [37]. The questions is this What is the probability that the event was the result of a particular cause, and how does it change if the cause is changing ... 

Bayesian networks are statistic models for describing probabilistic dependencies for a set of variables. They trace back to a theorem in the eighteenth century found by Thomas Bayes, who first established a mathematical base for probability inference [38]. Bayes theorem is based on two different states ... 

Bayesian Networks are statistic models for describing probabilistic dependencies for a set of variables based on Bayes theorem. 

Other approaches, based on the probability analysis of the profile of the diffraction peaks, were developed over the past decade. Three methods should be mentioned the maximum entropy method, the maximum likelihood method and the Bayesian techniques, based on Bayes theorem. 

The basic tool that Bayesians use derives from Bayes theorem, from which it is known that the... 

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