Bayesian Modeling
The basic tenets of Bayesian inference are introduced. This includes the construction of a prior and the use of the posterior distribution to perform inferences. Simulation is helpful in summarizing posterior distributions and Markov chain Monte Carlo alg
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Bayesian Modeling
12.1 Introduction There are two general approaches in statistical inference. In the inference and regression chapters, we have discussed the familiar frequentist inferential methods such as the t confidence interval, the chi-square test, and the ANOVA test of equality of means. These are called frequentist methods since one evaluates the goodness of these methods by their average performance in repeated sampling. For example a 90% confidence interval has the property that the random interval will cover the unknown parameter 90% of the time in repeated sampling. This chapter introduces the second general approach to inference, the Bayesian method. Bolstad [4] and Hoff [23] present respectively elementary and intermediate level introductions to Bayesian thinking. Albert [1] illustrates the use of R to perform Bayesian computations. Example 12.1 (Deciding authorship). This chapter discusses the basic components of a Bayesian analysis using data from a famous Bayesian analysis from Mosteller and Wallace [36] about literary style. The Federalist Papers were a series of essays, written in 17871788, to persuade the New York state citizens to ratify the U.S. Constitution. Of the essays, it is known that Alexander Hamilton was the sole author of 51, James Madison was the sole author of 14, and the two collaborated on another three. The authorship of the remaining 12 papers has been disputed, and the main problem addressed by Mosteller and Wallace was to determine the author of the disputed papers.
J. Albert and M. Rizzo, R by Example, Use R, DOI 10.1007/978-1-4614-1365-3__12, © Springer Science+Business Media, LLC 2012
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12 Bayesian Modeling
12.2 Learning about a Poisson Rate A first step in the authorship study is to examine the frequency of word use of the authors Alexander Hamilton and James Madison. One useful class of words in this study are so-called function words – these are words such as a, an, by, to, and than that are used to connect or amplify nouns and verbs. Mosteller and Wallace divided a large amount of the writings of James Madison into blocks of 1000 words, and the number of occurrences of the word from was observed in each block. Table 12.1 summarizes the frequency of use of the word from from 262 blocks of writing. From the table, we see that there were 90 blocks where the word did not appear, 93 blocks where the word appeared exactly one time, 42 blocks where the word appeared two times, and so on. Table 12.1 Frequency table of the occurrences of the word from in 262 blocks of text from the writings of James Madison. Occurrences 0 1 2 3456 observed 90 93 42 17 8 9 3
Suppose y represents the number of occurrences of the word from in a randomly selected block of text from Madison’s writing. A popular probability model for y is the Poisson distribution with rate λ with probability function given by exp(−λ)λy f (y|λ) = , y = 0, 1, 2, ... y! If this Poisson model is a suitable fit to these data, then one can measure Madison’s tendency to use this particular word by the rate paramet
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