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Bayesian Statistics: An Introduction, 4th Edition by Peter M. Lee

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9.4 The Gibbs sampler

9.4.1 Chained data augmentation

We will now restrict our attention to cases where m=1 and the augmented data  consists of the original data  augmented by a single scalar z (as in the linkage example). The algorithm can then be expressed as follows: Start from a value  generated from the prior distribution for η and then iterate as follows:

(a1) Choose  of η from the density  ;
(a2) Choose z(i+1) of z from the density  .

(There is, of course, a symmetry between η and z and the notation is used simply because it arose in connection with the first example we considered in connection with the data augmentation algorithm.)   This version of the algorithm can be referred to as chained data augmentation, since it is easy to see that the distribution of the next pair of values given the values up to now depends only on the present pair and so these pairs move as a Markov chain. ...

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