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Unformatted text preview: Lecture 11 Bayesian Data Analysis Until now, we have been fitting logistic (logit) and loglinear regression models to cate- gorical data. The utility of these models is that certain parameters have interpretations as odds ratios or relative risks on the log scale. When you look at the output produced by either R or SAS, you see that the parameter estimates and their standard errors are calculated using the Fisher scoring algorithm . Based on the data that you input and the model assumptions that you specify (eg. binomial, Poisson), the Fisher scoring algorithm maximizes a likelihood function . The likelihood function is a probability-based function of the data and parameters in your model. Fisher scoring maximizes this function with respect to the parameters. This is why the parameter estimates are called maximum likelihood estimates (MLEs). Based on the probability model, the MLEs represent the most likely estimates, given the data. For the logit and loglinear models we have considered, the likelihood function is nonlinear in the parameters, and the Fisher scoring algorithm iterates until convergence to the MLEs. For example, you may have noticed that R displays the number of iterations until convergence of the Fisher scoring algorithm. From our applied perspective, the estimation procedure is transparent to us since R or SAS does all the work. However, its important for us to understand, at least in some sense, technical aspects of the fitting process so that we are aware of the differences between classical or frequentist data analysis, which we have studied so far, and Bayesian data analysis, which we will study now. Instead of using R or SAS, we will now be using WinBugs (Bayesian Inference Using Gibbs Sampling) to perform estimation....
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- Two '09