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**Identification of Regeneration Times
in MCMC Simulation, with Application to Adaptive Schemes**

**Anthony E. Brockwell and Joseph B. Kadane**

### Abstract:

Regeneration is a useful tool in Markov chain Monte Carlo
simulation, since it can be used to side-step the burn-in
problem and to construct better estimates of the variance of
parameter estimates themselves. It also provides a simple
way to introduce adaptive behaviour into a Markov chain,
and to use parallel processors to build a single chain.
Regeneration is often difficult to
take advantage of, since for most
chains, no recurrent proper atom exists, and it is not
always easy to use Nummelin's splitting method
to identify regeneration times.
This paper describes a constructive method for
generating a Markov chain with a specified
target distribution and identifying regeneration times.
As a special case of the method, an
algorithm which can be ``wrapped'' around an
existing Markov transition kernel is given.
In addition, a specific rule for adapting the transition
kernel at regeneration times is introduced,
which gradually replaces the original
transition kernel with an independence-sampling Metropolis-Hastings
kernel using a mixture normal approximation to the target density
as its proposal density. Computational
gains for the regenerative adaptive algorithm
are demonstrated in examples.

*Heidi Sestrich*

*12/13/2002*
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