Bayesian formulation

In fact it is possible to show in a Bayesian formulation that if some prior doubt is allowed that the trial may not be competent to find a difference, then although, conditional upon a belief in competence, the posterior probability of equivalence will rise the more patients are studied and the more that prognosis is found to be the same in the two groups, the more the posterior belief must increase that the trial is not competent to find a difference (Senn, 1993). In short, the whole field is summed up by the oxymoron equivalence is different . 

One of the most attractive features of the Bayesian formulation in causal models is the ease of making predictions as to future events, such as the outcome of a social episode, the outcome of a given test and the prognosis of a disease. Pearl (1998). 

In this section, a Bayesian probabilistic approach is presented for online estimation of the noise parameters of the process noise and measurement noise. Sectimi ParameterizatiOTi and Bayesian Formulation introduces the parameterization of the noise covariance matrices and the Bayesian formulation. Thereafter, the online identification algorithm is presented in section Online Estimation of Noise Parameters. ... 

In theory, ambient modal identificadmi can be performed in either the time domain or the frequency domain. In practice, however, a frequency domain approach is preferable because it allows a natural partitioning of information in the data for identifying the modes of interest It sigifificantly simplifies the identification model because it only needs to model the modes in the selected band. For well-separated modes, one can select the band to cover one mode only, so that it can be identified independently of other modes. In general, the number of modes in the identification model only needs to be equal to the number of closely spaced modes, which rarely exceeds three. In the Bayesian formulation, this does not require any band-pass filtering because it can be done by simply omitting the FFT data of the excluded bands from the likelihood function. 

The main ingredients of a Bayesian formulation of the inverse problem are (i) the prior probability distribution (ii) the likelihood function (iii) the data (iv) Bayes theorem formulated as Bayes rule for producing the posterior probability distribution (v) a technique for sampling from the posterior distribution (vi) techniques for visualising the posterior distribution and (vii) a technique for summarising the posterior distribution. Summarising the distribution implies, for example, calculating the mean and correlation functions. 

A convenient and popular summary of such a posterior distribution is the maximum probability interpolant (known as the MAP, maximum a posteriori probability estimate). If this is calculated using the calculus of variations, then a minimisation problem, similar to that of the Tikhonov methods is obtained. In the Bayesian formulation however, the free parameters need less ad hoc arguments for their assignment and have a clearer interpretation. 

The Bayesian formulation is not universally used in history matching, but there is a general trend toward thinking in this way. One advantage that the Bayesian formulation provides is that it places aU methods in a common... 

As the Bayesian formulation was described in Section 5 it is sufficient to recall the main uses of the formulation in the maximum a posteriori (MAP) mode or in the stochastic sampling mode. The maximum likelihood estimation method is obtained by setting the prior to unity in the MAP method. The MLE method is essentially the least squares method. Without a suitable choice of prior it may be necessary to introduce further ad hoc regularisation in the case of MLE. A carefully chosen prior should regularise the problem in a satisfactory way. 

In the next subsection, I describe how the basic elements of Bayesian analysis are formulated mathematically. I also describe the methods for deriving posterior distributions from the model, either in terms of conjugate prior likelihood forms or in terms of simulation using Markov chain Monte Carlo (MCMC) methods. The utility of Bayesian methods has expanded greatly in recent years because of the development of MCMC methods and fast computers. I also describe the basics of hierarchical and mixture models. 

If the assumptions (multivariate normal distributions with equal group covariance matrices) are fulfilled, the Fisher rule gives the same result as the Bayesian rule. However, there is an interesting aspect for the Fisher rule in the context of visualization, because this formulation allows for dimension reduction. By projecting the data... 

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. 

Further work is required on methods for searching the space of non-linear models of a particular class (eg. AR-NAR) to determine the required model complexity. This may perhaps be best achieved by extending the Maximum Likelihood approach to a full Bayesian posterior probability formulation and using the concept of model evidence [Pope and Rayner, 1994] to compare models of different complexity. Some... 

This model is augmented by an unobserved indicator vector 8. Each element 8j(j = 1,..., h) of 8 takes the value 0 or 1, indicating whether the corresponding P j belongs to an inactive or an active effect, respectively. Because the intercept Po is always present in the model, it has no corresponding 50 element. An inactive effect has Pj close to 0 and an active effect has Pj far from 0. The precise definition of active and inactive may vary according to the form of prior distribution specified. Under this formulation, the Bayesian subset selection problem becomes one of identifying a posterior distribution on 6. 

Bayesian methods have often proved useful for design of experiments, especially in situations in which the optimal design depends on unknown quantities. Certainly, to identify a design for optimal estimation of /3, the correct subset of active effects must be identified. Bayesian approaches that express uncertainty about the correct subset enable construction of optimality criteria that account for this uncertainty. Such approaches typically find a design that optimizes a criterion which is averaged over many possible subsets. DuMouchel and Jones (1994) exploited this idea with a formulation in which some effects have uncertainty associated with whether they are active. Meyer et al. (1996) extended the prior distributions of Box and Meyer (1993) and constructed a model discrimination design criterion. The criterion is based on a Kullback-Leibler measure of dissimilarity between... 

Mixed effects models under a Bayesian framework have been widely studied and used with the use of Markov chain Monte Carlo methods (10). These methods have gained particular popularity as complex problems became easily formulated using the WinBUGS software (11). See Congdon (12) for an extensive coverage of topics and examples and implementation in WinBUGS. 

In this section the basic principles of the Kalman filter are presented for linear multi-degree-of-freedom (MDOF) systems. More details can be found elsewhere [36,84,128,129], Even though it is not often emphasized and it was not shown explicitly in the original formulation [128,129], the Kalman filter is a Bayesian updating procedure. Consider a second-order... 

However, for a large number of observed data points, repeated evaluations of the factor p(V 0, C) for different values of 0 becomes computationally prohibitive. It is obvious from Equation (4.9) that it requires the computation of the solution X of the algebraic equation F( )X = Y and the determinant of the x matrix F( ). This task is computationally very expensive for large N even though the former can be done efficiently by pre-conditioners [43,49,124]. Repeated evaluations of the likelihood function for thousands of times in the optimization process is computationally prohibitive for large N. Therefore, the exact Bayesian approach described above, based on direct use of the measured data V, becomes practically infeasible. In the next section, the model updating problem will be formulated with a nonsta-tionary response measurement. Standard random vibration analysis will be reviewed. Then, an approximated approach is introduced and it overcomes the computational obstacles and renders the problem practically feasible. 

In this chapter, the Bayesian time-domain approach was introduced for identification of the model parameters and stochastic excitation parameters of linear multi-degree-of-freedom systems using noisy stationary or nonstationary response measurements. The direct exact formulation was presented but it turned out to be computationally prohibited for a large number of data points. Then, an approximated likelihood function expansion was proposed to resolve this obstacle. For a globally identifiable case with a large number of data points, the updated PDF... 

In this section, we present a general methodology for Bayesian meta-design only for the log-linear random effects or fixed effecfs regression models. We assume thaf the hypotheses for "noninferiorify" fesfing can be formulated as follows ... 

In this example, the Bayesian approach provides an appropriate way to conduct the continuous monitoring of this EOI. Data from literature (for control) and the Phase lb data (for Drug X) could be used to formulate priors. Each new event on study changes the posterior probability of the treatment effect and a threshold for stopping the study can be tested on a continuous basis. 

R. Caspeele is a Research Assistant of the FWO Research Foundation of Flanders. The authors thank the FWO for the financial support on the research project Probabilistic formulation of the influence of conformity control on the strength distribution of concrete and the safety level of concrete structures by means of Bayesian updating techniques . 

Gaussian Processes predict the values of a function whose form is not explicitly known by using function observations as evidence. If t = fjjli are values of a function / M" M measured at the points X = x,, with some error, predicting the value r v+i at Xjv+i can be formulated as a Bayesian inference problem. Bayes theorem states that... 

A Bayesian inference to a problem starts with the formulation of a model with hopes that it is adequate to describe the situation of interest initially. A prior probability distribution function (pdf) is then suggested over the unknown model parameters, which is meant to capture one s beliefs about the situation without data. However, with the incoming of data one may then apply Bayesian inference rule to obtain a posterior distribution for the same unknown parameters, which take into account the prior pdf and the data. From the posterior distribution, one can then compute updated predictive pdf for future observations [6]. The Bayesian approach can be simply applied and justified theoretically as the proper approach to uncertain inference by various arguments involving consistency with clear principles of rationality. Even though, a prior pdf selection scans subjective, but it is not arbitrary. It is necessary that the priory pdf should capture one s correct prior information by taking into consideration a combination of prior beliefs. 

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