An Efficient Markov Chain Monte Carlo Method for Distributions with Intractable Normalising Constants

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.