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Unformatted text preview: Lecture 12 More Practice With Bayesian Data Analysis Lecture 11 was designed to be an introduction to Bayesian data analysis. The purpose was to show what BDA is, and how it differs from classical or frequentist data analysis. In Lecture 11 we saw that inference about model parameters is made by examining charac teristics such as the means or the variances of posterior distributions. Bayesian modeling is characterized by the following three steps: 1) Specify a likelihood for the data and unknown parameter(s), call it p ( y  θ ). 2) Specify a prior distribution for the parameter(s), call it p ( θ ). 3) Calculate the posterior distribution by application of Bayes Formula, call it p ( θ  y ), and use the mean and variance of this distribution to make inference about θ . It is our job to perform Steps 1 and 2 above. That is, we specify the likelihood for the data and the prior distribution for the parameters. Step 3, calculation of posterior distributions of model parameters will be performed by the WinBugs software. Often in practice, direct calculation of these distributions is in tractable, so methods of simulation known as Markov Chain Monte Carlo (MCMC) have been developed, whereby we can simulate samples from the posterior distributions of model parameters. All inference about model parameters is made using simulation. Since we will be using WinBugs to get results, it will be necessary to get some practice with the code that is needed to specify Steps 1 and 2 above. The best way to learn Bayesian Data Analysis is to work examples. We will now conduct a complete analysis from beginning to end using the Pump Data from Lecture 11. As we did in Lecture 11, we will • Fit the standard loglinear model for rates, and show that there is lack of fit. • Formulate the Bayesian model. • Show the WinBugs code. • Show how to use WinBugs. • Get the results and interpret the results. Pump Data Example The data below show number of failures of pumps at a power plant for 10 pumps. Two different kinds of pumps were in use. Also shown is the length of operation in thousands of hours for each pump.of hours for each pump....
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 Two '09
 R
 Statistics, Poisson Distribution, Probability theory, Exponential distribution

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