Bayesian approach

This paper is structured as follows in section 2, we recall the statement of the forward problem. We remind the numerical model which relates the contrast function with the observed data. Then, we compare the measurements performed with the experimental probe with predictive data which come from the model. This comparison is used, firstly, to validate the forward problem. In section 4, the solution of the associated inverse problem is described through a Bayesian approach. We derive, in particular, an appropriate criteria which must be optimized in order to reconstruct simulated flaws. Some results of flaw reconstructions from simulated data are presented. These results confirm the capability of the inversion method. The section 5 ends with giving some tasks we have already thought of. 

O. Venard. Eddy current tomography a bayesian approach with a compound weak membrane-beta prior model. In Advances in Signal Processing for Non Destructive Evaluation of Materials, 1997. 

S Greenland. Probability logic and probability induction. Epidemiology 9 322-332, 1998. GM Petersen, G Parmigiam, D Thomas. Missense mutations in disease genes A Bayesian approach to evaluate causality. Am J Hum Genet. 62 1516-1524, 1998. 

CE Buck, WG Cavanaugh, CD Litton. The Bayesian Approach to Interpreting Archaeological Data. New York Wiley, 1996. 

ME Ochs, RS Stoyanova, E Arias-Mendoza, TR Brown. A new method for spectral decomposition using a bilinear Bayesian approach. I Magn Reson 137 161-176, 1999. 

A more general approach to finding a solution than simply assuming some coefficients are zero is provided by a Bayesian approach. Since the displacement d is a linear function of two unknowns, 02 and 07, we can write... 

As a simple rule of thumb if a simple least squares estimate is employed the number of modes estimated should be half the number of measurements. If a Bayesian approach is employed the number of modes estimated should be at least the number of measurements. 

The method for estimating parameters from Monte Carlo simulation, described in mathematical detail by Reilly and Duever (in preparation), uses a Bayesian approach to establish the posterior distribution for the parameters based on a Monte Carlo model. The numerical nature of the solution requires that the posterior distribution be handled in discretised form as an array in computer storage using the method of Reilly 2). The stochastic nature of Monte Carlo methods implies that output responses are predicted by the model with some amount of uncertainty for which the term "shimmer" as suggested by Andres (D.B. Chambers, SENES Consultants Limited, personal communication, 1985) has been adopted. The model for the uth of n experiments can be expressed by... 

We take a Bayesian approach to research process modeling, which encourages explicit statements about the prior degree of uncertainty, expressed as a probability distribution over possible outcomes. Simulation that builds in such uncertainty will be of a what-if nature, helping managers to explore different scenarios, to understand problem structure, and to see where the future is likely to be most sensitive to current choices, or indeed where outcomes are relatively indifferent to such choices. This determines where better information could best help improve decisions and how much to invest in internal research (research about process performance, and in particular, prediction reliability) that yields such information. 

Friedman [12] introduced a Bayesian approach the Bayes equation is given in Chapter 16. In the present context, a Bayesian approach can be described as finding a classification rule that minimizes the risk of misclassification, given the prior probabilities of belonging to a given class. These prior probabilities are estimated from the fraction of each class in the pooled sample ... 

In brief, the Bayesian approach uses PDFs of pattern classes to establish class membership. As shown in Fig. 22, feature extraction corresponds to calculation of the a posteriori conditional probability or joint probability using the Bayes formula that expresses the probability that a particular pattern label can be associated with a particular pattern. 

The knowledge required to implement Bayes formula is daunting in that a priori as well as class conditional probabilities must be known. Some reduction in requirements can be accomplished by using joint probability distributions in place of the a priori and class conditional probabilities. Even with this simplification, few interpretation problems are so well posed that the information needed is available. It is possible to employ the Bayesian approach by estimating the unknown probabilities and probability density functions from exemplar patterns that are believed to be representative of the problem under investigation. This approach, however, implies supervised learning where the correct class label for each exemplar is known. The ability to perform data interpretation is determined by the quality of the estimates of the underlying probability distributions. 

In the last section of the paper, we discuss a Bayesian approach to the treatment of experimental error variances, and its first limited implementation to obtain MaxEnt distributions from a fit to noisy data. 

We have described in this paper the first implementation of this Bayesian approach to charge density studies, making joint use of structural models for the atomic cores substructure, and MaxEnt distributions of scatterers for the valence part. Used in this way, the MaxEnt method is safe and can usefully complement the traditional modelling based on finite multipolar expansions. This supports our initial proposal that accurate charge density studies should be viewed as the late stages of the structure determination process. 

An alternative method, which uses the concept of maximum entropy (MaxEnt), appeared to be a formidable improvement in the treatment of diffraction data. This method is based on a Bayesian approach among all the maps compatible with the experimental data, it selects that one which has the highest prior (intrinsic) probability. Considering that all the points of the map are equally probable, this probability (flat prior) is expressed via the Boltzman entropy of the distribution, with the entropy defined as... 

Kelly PC, Horlick G (1974) Bayesian approach to resolution with comparisons to conventional resolution techniques. Anal Chem 46 2130... 

Finally, approaches are emerging within the data reconciliation problem, such as Bayesian approaches and robust estimation techniques, as well as strategies that use Principal Component Analysis. They offer viable alternatives to traditional methods and provide new grounds for further improvement. 

Monte Carlo data for y were generated according to with mean x, to simulate process sampling data. A window size of 25 was used here and to demonstrate the performance of the Bayesian approach. 

There are now common practices in the analysis of safety data, though they are not necessarily the best. These are discussed in the remainder of this chapter, which seeks to review statistical methods on a use-by-use basis and to provide a foundation for the selection of alternatives in specific situations. Some of the newer available methodologies (meta-analysis and Bayesian approaches) should be kept in mind, however. 

The commercially available software (Maximum Entropy Data Consultant Ltd, Cambridge, UK) allows reconstruction of the distribution a.(z) (or f(z)) which has the maximal entropy S subject to the constraint of the chi-squared value. The quantified version of this software has a full Bayesian approach and includes a precise statement of the accuracy of quantities of interest, i.e. position, surface and broadness of peaks in the distribution. The distributions are recovered by using an automatic stopping criterion for successive iterates, which is based on a Gaussian approximation of the likelihood. 

Mckeigue, P.M., Carpenter, J., Parra, E.J., and Shriver, M.D. (2000) Estimation of admixture and detection of linkage in admixed populations by a Bayesian approach application to African-American populations. Ann. Hum. Genet. 64, 171-186. 

Peterson has presented a Bayesian approach to defining the DS, which provides design space reliability as well, as it takes into account both model parameter uncertainty and the correlation structure of data. To aid in use of this approach, he proposes a means of organizing information about the process in a sortable spreadsheet, which can be used by manufacturing engineers to aid them in making informed process changes as needed, and continue to operate in the DS. 

Spiegelhalter DJ, Abrams KR, Myles JR Bayesian Approaches to Clinical Trials Health-Care Evaluation. Chichester John Wiley and Sons, 2003. 

Lane DA, Hutchinson TA, Jones JK, et al. A Bayesian Approach to Causality Assessment. University of Minnesota School of Statistics Tech Reps No 472 (no date available). 

Lane DA, Kramer MS, Hutchinson TA, et al. The causality assessment of adverse drug reactions using a Bayesian approach. Pharm Med 1987 2 265-83. 

Model uncertainty can be represented by formulating 2 or more different models to represent alternative hypotheses or viewpoints and then combining the model outputs by assigning weights representing their relative probability or credibility, using either Bayesian and non-Bayesian approaches. 

Another approach is to develop a global model that contains plausible models as special cases, defined by alternative values of particular parameters. This converts model uncertainty into uncertainty about the model parameters. Again this can be done using either Bayesian or non-Bayesian approaches. This approach is favored by Morgan and Henrion (1990), who describe how it can be applied to uncertainty about dose-response functions (threshold versus nonthreshold, linear versus exponential). 

Bayesian approaches are discussed throughout this book. Unfortunately, because frequentist methods are typically presented in introductory statistics courses, most environmental scientists do not clearly understand the basic premises of Bayesian methods. This lack of understanding could hamper appreciation for Bayesian approaches and delay the adaptation of these valuable methods for analyzing uncertainty in risk assessments. 

The basic premise of the Bayesian approach is that observations change the state of knowledge of a system. Let us suppose for simplicity that the item of interest is some parameter, 0, describing a state of nature (as in the above example, where 0 was a property of the coin and the conditions under which it was tossed). Figure 5.1 indicates symbolically the development of knowledge. 

The Bayesian approach reverses the role of the sample and model the sample is fixed and unique, and the model itself is uncertain. This viewpoint corresponds more closely to the practical situation facing the individual researcher there is only 1 sample, and there are doubts either what model to use, or, for a specified model, what parameter values to assign. The model uncertainty is addressed by considering that the model parameters are distributed. In other words Bayesian interpretation of a confidence interval is that it indicates the level of belief warranted by the data the... 

The classical or frequentist approach to probability is the one most taught in university conrses. That may change, however, becanse the Bayesian approach is the more easily nnderstood statistical philosophy, both conceptually as well as numerically. Many scientists have difficnlty in articnlating correctly the meaning of a confidence interval within the classical frequentist framework. The common misinterpretation the probability that a parameter lies between certain limits is exactly the correct one from the Bayesian standpoint. 

Apart from this pedagogical aspect (cf Lee 1989, preface), there is a more technical reason to prefer the Bayesian approach to the confidence approach. The Bayesian approach is the more powerfnl one eventnally, for extending a model into directions necessary to deal with its weaknesses. These are various relaxations of distribntional assnmptions. The conceptnal device of an infinite repetition of samples, as in the freqnentist viewpoint, does not yield enongh power to accomplish these extensions. 

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). 

For the normal distribution there are analytical solutions allowing the assessment of both FA and HC using frequentist statistics. In contrast, Bayesian solutions are numerical. This highlights the flexibility of the Bayesian approach since it can easily deal with any distribution, which is not always possible with the frequentist approach. 

Aldenberg and Jaworska (2000) demonstrate that frequentist statistics and the Bayesian approach with noninformative prior results in identical confidence intervals for the normal distribution. Generally speaking, this is more the exception than the rule. 

For those who feel more confident with the frequentist approach and find the Bayesian approach controversial to some extent, it is advantageous that both approaches yield the same answers in this simplest case. This might add confidence in the Bayesian approach for some practitioners. 

Howson C, Urbach P. 1989. Scientific reasoning. The Bayesian approach. La SaUe (fL) Open Court. 

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