Bayesian methods OpenBUGS

The course is motivated and emphasis is made on using MCMC methods and OpenBUGS to transform deterministic models into random ones or performing a Bayesian analysis of already random models. With this tool, answers to problems can be given as density functions instead of simple point estimates or confidence intervals.

In this block we introduce Bayesian methods, first, by giving the definitions of its main elements, and second, by means of some simple but illustrative examples. The concept of prior and posterior distributions are introduced and the formula to obtain the posterior, in terms of the prior and the sample likelihood is given. Prior and posterior predictive distributions are also defined and obtained. Some graphical examples are used to illustrate the new and deep concepts. The Doodle tool, to build the Bayesian network defining the models and their parameters is described in detail and some examples are given. Finally, an initial view of OpenBUGS is given using the BPR traffic model, which is converted from deterministic to random.

We dedicate this second block to some statistical applications. We start with the problem of estimating a rate considering two cases: (a) the sample size is known, and (b) it is unknown. As already known, if the parameters are assumed to be random we get a Bayesian method to analyze this problem. Some interesting dependencies of the parameters will be discovered. The second example is the Pearson correlation example, in which this statistic is considered as a random variable, instead of a deterministic value. This allows us to answer the problem of finding the Pearson correlation of a model by means of a density function instead of a point or confidence estimate, which goes further beyond than the classical statistics.

This block is dedicated to some programming tools that facilitate the use of Bayesian MCMC methods. We start with scripts, that allow us to use the software without the mouse, i. e., using program commands called scripts. In this way the speed in getting the results increases substantially. Next, we describe the JAGS tool, an alternative to the standard OpenBUGS, that allows as to compare results with a small extra work. The call to JAGS from Matlab is described. Finally, we explain how to use OpenBUGS from Matlab, by explaining how to generate the commands, the data and the initial files and how to generate scripts automatically and call then OpenBUGS later to initiate the simulation process.

In this part we describe some engineering applications. We start with a heteroscedastic parabolic regression model, in which the parameters are considered as random instead of deterministic. Since in the deterministic version we can plot percentile regression curves, with this model each percentile curve has its own percentiles curves. The second example is a powerful non-linear random fatigue example, which is selected to satisfy some engineering, physical and statistical conditions and is enriched by assuming that its five parameters are random variables. In this way, each of the S-N percentile curves of the model can be given as stochastic processes, whose percentiles are obtained. Several deal data sets are used to fit the models.

This block is dedicated to provide the readers with some of the above examples implemented in matlab and/or JAGS. The whole listing of the programs are given, so that they can immediately run the programs and used them as templates to write their own examples. The complete results, figures, tables and files are automatically saved in a folder, whose name is selected by the user. I hope they enjoy with this experience that can change the way they work with deterministic and statistical models in the furure. If this is so, I will be happy of the change and progress this attitude and orientation implies.

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