Bayesian probability theory

Bayesian probability theory and methods that are based on fuzzy-set theory. The principles of both theories are explained in Chapter 16 and Chapter 19, respectively. Both approaches have advantages and disadvantages for the use in expert systems and it must be emphasized that none of the methods, developed up to now are satisfactory [7,11]. 

Fig. 9.10. Flux dependence of the chemical erosion yield for Tmax and an ion energy of 30 eV determined from spectroscopic measurements in different fusion devices and plasma simulators. The solid lines are a fit using Bayesian probability theory and its confidence intervals [58,59]. The dashed line is a prediction from an earlier analytic description [44]...

M. Meier, R. Preuss, V. Dose Interaction of CH3 and H with amorphous hydrocarbon surfaces Estimation of reaction cross-sections using Bayesian probability theory. New J. Phys. 5, 133 (2003)... 

The residual difference after a successful DDM refinement or/and decomposition can be considered as a scattering component of the powder pattern free of Bragg diffraction. The separation of this component would facilitate the analysis of the amorphous fraction of the sample, the radial distribution function of the non-crystalline scatterers, the thermal diffuse scattering properties and other non-Bragg features of powder patterns. The background-independent profile treatment can be especially desirable in quantitative phase analysis when amorphous admixtures must be accounted for. Further extensions of DDM may involve Bayesian probability theory, which has been utilized efficiently in background estimation procedures and Rietveld refinement in the presence of impurities.DDM will also be useful at the initial steps of powder diffraction structure determination when the structure model is absent and the background line cannot be determined correctly. The direct space search methods of structure solution, in particular, may efficiently utilize DDM. 

Bayesian probability theory, and is an iterative technique designed to produce an object function which, when convoluted with the microscope probe, gives the best fit to the experimental image. The iterative process starts with an object function of uniform intensity, which makes no prior assumptions about the structure of the specimen, and moves towards the best fit to the experimental image by calculating the maximum entropy while minimizing the fit with the experimental image (Fig. 11.3). This process allows the coordinates for the atom column positions to be determined to an accuracy of 0.1 A... 

After a first paper in 1988, " Bretthorst paved the way for the application of Bayesian probability theory (BPT) in a series of three papers.- The first presents the connection between the theory and the case of NMR phenomena and discusses parameter estimation and detection of quadrature signals. The second " shows the ability of Bayesian methods to measure the quality of a model. The third " provides examples of applications to experimental data where decaying sinusoids are assumed. A fourth paper, produced during the following two years, discusses computer time requirements and noise, and it is shown to be important to include knowledge in the analysis," " while a fifth publication is devoted to amplitude estimations for multiplets of well-separated resonances. 

Modern theory is often called Bayesian probability theory after Thomas Bayes, F.R.S. (1702-1761) who was a minister of the Presbyterian church. The theorem attributed to his name is central to the modern interpretation, but according to Maistrov, it appears nowhere in his writings, and was first mentioned by Laplace though it was only expressed in words. The theorem enables an updating of a probability estimate, in the light of new information. For a set of mutually exclusive collectively exhaustive events Bi, B. ., B then P A) can be expressed. Fig. 5.4, as... 

The term P(D H) represents the likelihood function and provides the probability of the observed data arising from the hypothesis. Normally, this is known because it expresses one s knowledge of how to look at the data, given that the hypothesis is tme. The term P(H) or P(0) is called the prior distribution, as it reflects one s prior knowledge before the data are considered. The advantages of Bayesian probability theory is that one s assumptions are made up front, and any element of subjectivity in the reasoning process is directly exposed [2]. 

B.A. Olshausen, Bayesian Probability Theory, March 2004. http //redwood.berkeley.edu/ bruno/npbl63/bayes.pdf. 

Bayesian probability theory The approach to probability theory which views a probability as a measure of our uncertainty of knowledge rather than as relative frequency of occurrence. 

Bayesian probability theory shares some similarities with the problem of image restoration they both are required to make some choice in the presence of insufficient data or information. It is not surprising, then, that Bayesian techniques have been applied in image restoration. Applying Bayesian principles, of the possible solutions to an image restoration problem (i.e., of all the images that are consistent with the data), we choose that image which maximizes the entropy. 

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