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Lecture notes; Video tutorials: Tutorial talks available online as streaming videos. Further reading: References on various topics in Bayesian nonparametrics.
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All of Bayesian Nonparametrics
In a previous post, we showed how to use nonparametric mixture models with different kernels for density estimation. In this post, we will take a parametric generalized linear mixed model and show how to switch to a nonparametric representation of the random effects that avoids the assumption of normally-distributed random effects.
We will illustrate the use of nonparametric mixture models for modeling random effects distributions in the context of a meta-analysis of the side effects of a formerly very popular drug for diabetes called Avandia. The data we analyze played a role in raising serious questions about the safety of this drug. The question is whether Avandia use increases the risk of myocardial infarction heart attack. There are 48 studies the 49th study in the data file is different in some ways and excluded here , each with treatment and control arms.
We begin with a standard generalized linear mixed model GLMM -based meta analysis. The vectors and contain the total number of patients in the control and the number of patients suffering from myocardial infarctions in the control group of each study, respectively.
Similarly, the vectors and contain similar information for patients receiving the drug Avandia. The model takes the form.
Nonparametric Bayes Tutorial
The parameter quantifies the difference in risk between the control and treatment arms, while the quantify study-specific variation. The results suggests there is an overall difference in risk between the control and treatment arms. But what about the normality assumption? Are our conclusions robust to that assumption?
Perhaps the random effects distribution are skewed. And recall that the estimates above of the random effects are generated under the normality assumption, which pushes the estimated effects to look more normal…. Now, we use a nonparametric distribution for the s. More specifically, we assume that each is generated from a location-scale mixture of normal distributions:.
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This specification induces clustering among the random effects. As in the case of density estimation problems, the DP prior allows the data to determine the number of components, from as few as one component i. This allows the distribution of the random effects to be multimodal if the data supports such behavior, greatly increasing its flexibility.