After a brief review of the data augmentation (DA) algorithm, I will
introduce a simple modification of that algorithm that results in a
new Markov chain that remains reversible with respect to the target
distribution.  The DA algorithm and the modification are compared in
the context of a toy Bayesian mixture model for which exact
eigen-analysis is possible.  I will also discuss some general results
showing that the modified algorithm is always better than the
corresponding DA algorithm.  (This is joint work with D. Marchev,
C. Robert, J. Rosenthal and V. Roy.)