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Unformatted text preview: Chapter 6 Beyond the Normal. Other Bivariate Posteriors using the grid method. No conjugacy and no easy conditional distributions situation (I) 6.1 Introduction Beyond the normal distribution, few multiparameter sampling models allow simple explicit calculation of posterior distributions. To illustrate that practical analysis is possible for some problems using simple simulation techniques, we present here an example of nonconjugate models. This is an example of a nonconjugate model for a bioassay experiment, drawn from literature on applied bayesian statistics. This case was analyzed in Gelman et al’s book (1995, p. 82), and we only add the programming to it, along with some interpretations. The model is a twoparameter example from the broad class of generalized linear models to be considered more thoroughly later in the course. Required reading for this topic, in addition to this lesson, is chapter 4 of Ho ff ’s book and the discussions on grid in chapter 6 of Ho ff . 6.2 Preliminary knowledge required The parameter θ that will appear in this problem is between 0 and 1. The logistic function applied to a variable z is given by f ( z ) = 1 1 + e z ∞ < z < ∞ ≤ f ( z ) ≤ 1 Now let’s go from the logistic function to the logistic model, which is usually the primary focus. To obtain the logistic model from the logistic function we make z = α + β x . We will also call θ = f ( z ) and, then we have that θ = 1 1 + e ( α + β x ) The parameters α and β are unknown and need to be estimated from the data. When a probability parameter θ is written in this form, we say that the model is a logistic model. There is an alternative way of writing the logistic model, called the logit form of the model. To get the logit from the logistic model, we make a transformation of the model. The logistic transformation of θ is defined as logit ( θ ) = log θ 1 θ 37 Stats C180 / C236 Introduction to Bayesian Statistics Juana Sanchez UCLA Department of Statistics where the log is the natural log (base e). Substituting for θ , we get logit ( θ ) = 1 1 + e ( α + β x ) 1 1 1 + e ( α + β x ) = log 1 1 + e ( α + β x ) 1 ! = log 1 e ( α + β x ) ! = α + β x This last expression is called a logistic regression model. As we will see below, a count of deaths are binomial random variables P ( x  θ ) = n x ! θ x (1 θ ) n x So we do the inverse transformation of the logit to put it into this probability distribution. Let’s go through the steps: logit ( θ ) = log θ 1 θ This implies that e logit ( θ ) = θ 1 θ e α + β x = θ 1 θ e α + β x θ e α + β x = θ θ (1 + e α + β x ) = e α + β x θ = e α + β x 1 + e α + β x With this background on the logistic model (which, by the way, is more interesting when you have more covariates), we can investigate the data that follows....
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 Spring '10
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 Normal Distribution, Probability theory, Cumulative distribution function, Sanchez UCLA Department, Juana Sanchez UCLA Department of Statistics, Juana Sanchez UCLA

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