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Markov Chain Monte Carlo Based on Deterministic Transformations
In this article we propose a novel MCMC method based on deterministic
transformations T: X x D --> X where X is the state-space and D is some set
which may or may not be a subset of X. We refer to our new methodology as
Transformation-based Markov chain Monte Carlo (TMCMC). One of the remarkable
advantages of our proposal is that even if the underlying target distribution
is very high-dimensional, deterministic transformation of a one-dimensional
random variable is sufficient to generate an appropriate Markov chain that is
guaranteed to converge to the high-dimensional target distribution. Apart from
clearly leading to massive computational savings, this idea of
deterministically transforming a single random variable very generally leads to
excellent acceptance rates, even though all the random variables associated
with the high-dimensional target distribution are updated in a single block.
Since it is well-known that joint updating of many random variables using
Metropolis-Hastings (MH) algorithm generally leads to poor acceptance rates,
TMCMC, in this regard, seems to provide a significant advance. We validate our
proposal theoretically, establishing the convergence properties. Furthermore,
we show that TMCMC can be very effectively adopted for simulating from doubly
intractable distributions.
TMCMC is compared with MH using the well-known Challenger data, demonstrating
the effectiveness of of the former in the case of highly correlated variables.
Moreover, we apply our methodology to a challenging posterior simulation
problem associated with the geostatistical model of Diggle et al. (1998),
updating 160 unknown parameters jointly, using a deterministic transformation
of a one-dimensional random variable. Remarkable computational savings as well
as good convergence properties and acceptance rates are the results.Comment: 28 pages, 3 figures; Longer abstract inside articl
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