Seeking Efficient Data Augmentation Schemes Via Conditional
and Marginal Augmentation
Biometrika
Xiao-Li Meng
Department of Statistics, The University of Chicago,
Chicago, IL 60637, U.S.A.
David A. van Dyk
Department of Statistics, Harvard University, Cambridge, MA
02138, U.S.A.
Data augmentation is a powerful tool for constructing
deterministic and stochastic algorithms for likelihood and
Bayesian computation. Constructing an efficient data-augmentation scheme
requires a balance of simple implementation and quick convergence.
Meng & van Dyk (1997) reported the success of
the ``working parameter" approach for constructing fast and simple
EM-type algorithms
for some common models, and suggested that efficient data-augmentation
schemes for EM-type algorithms could also be effective for implementing
the corresponding Gibbs samplers. This paper investigates this possibility
by providing a theoretical study of
two working parameter approaches, the {conditional augmentation} approach
and the {marginal augmentation} approach.
The former was the focus of Meng & van Dyk (1997) and the latter
builds on the parameter-expanded EM approach of Liu et al. (1998).
We use the t model as a running example to illustrate the potential
of these methods.
The t model also turns out to be an excellent springboard for
examining the convergence behavior of the Gibbs sampler
with improper limiting distributions.
A number of open theoretical questions are also discussed, in
particular with respect to marginal augmentation which
in some cases allows one to obtain a fast-mixing positive recurrent
Markov chain by purposely constructing a non-positive recurrent Markov
chain in a larger space.
Some key words: Incomplete data; Markov-chain Monte Carlo;
PXEM Algorithm; Rate of convergence; Working parameter.
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