Hierarchical Models and More on Convergence Assessment
So far, all of the Bayesian models that we have encountered have had only two components—the likelihood, which describes the data as draws from a probability distribution, and the prior, which specifies a probability distribution on the unknown parameters
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Hierarchical Models and More on Convergence Assessment
So far, all of the Bayesian models that we have encountered have had only two components—the likelihood, which describes the data as draws from a probability distribution, and the prior, which specifies a probability distribution on the unknown parameters in the likelihood. Such a simple model is inadequate for many (probably most) real-world applications. As a result, more complex Bayesian models with additional levels are very commonly used. Such models are called hierarchical models. Sometimes hierarchical models are needed because the structure of the data itself is a hierarchy. For example, consider children’s scores on a standardized test taken by some third graders across the USA. The individual children are in classrooms; the classrooms are in schools; the schools are in school districts; the school districts are in states, etc. In this case, a hierarchical model would enable us to estimate parameters at each level of the hierarchy so as to address questions such as: How variable are average test scores in different schools in the same school district? How variable are average test scores in different school districts? More generally, hierarchical models are appropriate when there are natural groupings of observations in the data or of parameters in the model. Early references on Bayesian hierarchical models are Box and Tiao (1973); Good (1965). Chapter 5 of Gelman et al. (2004) offers an up-to-date discussion, including computational aspects.
9.1 Specifying Bayesian Hierarchical Models Example: A Better Model for the College Softball Player’s Batting Average In problems at the end of Chap. 3, you estimated a college softball player’s collegecareer batting average from her number of hits in 30 at bats that occurred during eight games. In answering Problem 3.3, you probably noted that considering each at M.K. Cowles, Applied Bayesian Statistics: With R and OpenBUGS Examples, Springer Texts in Statistics 98, DOI 10.1007/978-1-4614-5696-4 9, © Springer Science+Business Media New York 2013
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9 Hierarchical Models and More on Convergence Assessment
bat an independent Bernoulli trial was not justified, because the outcomes of at bats within a single game were likely to be correlated (since within the same game, the player was facing the same opposing team, playing in the same stadium, etc.) Now we will use a hierarchical model to estimate the player’s overall batting average while appropriately accounting for the correlation structure of the data. Specifically, we will assume that the at bats within each game are exchangeable— that is, that their outcomes are conditionally independent given the player’s success probability for that game. Furthermore, we will assume that the player’s probability πi of getting a hit could be different in different games, i = 1, . . . , 8, but that all the πi ’s are random draws from the same probability distribution, with mean the true overall batting average that is of primary interest. We also have secondary inte
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