An Efficient Markov Chain Monte Carlo Method for Distributions with Intractable Normalising Constants
by: Jesper Møller
, A. N. Pettitt, K. K. Berthelsen and R. W. Reeves.
We present new methodology for drawing samples from a posterior
distribution when (i) the likelihood function or (ii) a part of the
prior distribution is only specified up to a normalising constant.
In the case (i), the novelty lies in the introduction of an
auxiliary variable in a
Metropolis-Hastings algorithm and the choice of proposal distribution
so that the algorithm does not depend upon the unknown normalising
constant. In the case (ii), similar ideas apply and the situation is
even simpler as no auxiliary variable is required.
Our method is ``on-line'' as compared with alternative approaches to
the problem which require ``off-line'' computations. Since it is
needed to simulate from the ``unknown distribution'', e.g. the
likelihood function in case (i), perfect simulation such as the
Propp-Wilson algorithm becomes useful. We illustrate the method in
case (i) by producing posterior samples when the likelihood is given
by an Ising model and by a Strauss point process.
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