Maximum likelihood parameter estimation and sampling from Bayesian posterior distributions are problematic when the probability distribution for the parameter of interest involves an intractable normalizing constant which is also a function of that parameter. Most methods to date have used various approximations to estimate or eliminate such normalizing constants. In [4] we present new methodology for drawing samples from such a distribution without approximation. 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 normalizing constant. The method requires the auxiliary variable to be simulated from the distribution which defines the normalizing constant, for which perfect (or exact) simulation as exemplified by the Propp-Wilson algorithm [7] and the dominated coupling from the past (dominating CFTP) algorithm [2, 3] becomes useful.

In [1] we illustrate the method by the following application example. An inhomogeneous point pattern showing the location of cells in a section of the mocous membrane of the stomach of a healthy rat is modelled by a Markov point process, with a location dependent first order term and pairwise interaction only. We consider a flexible non-parametric Bayesian setting, assuming a priori that the first order term is a shot noise process, the interaction function for each pair of points depends only on the distance between the two points, and the interaction function is a piecewise linear function modelled by a marked Poisson process. Since the Markov point process density involves a normalizing constant where no closed form expression is known, we apply the auxiliary variable technique from [4] for posterior simulation. The auxiliary variable in [1] is specified by a partially ordered Markov point process model.

Next, we briefly discuss the dominating CFTP algorithm due to Kendall [2]. In [3] we give a general formulation of dominated CFTP, which applies for certain stochastic models on ordered spaces, and discuss in particular how to make perfect simulation of general locally stable point processes. In the present talk we consider only the case of perfect simulation of pairwise interaction point processes as equilibrium distributions of spatial birth-and-death processes.
For those interested in further details of spatial point process models, simulation and inference, see [5] and [6].

Finally, if time allows, current research on extensions of the auxiliary variable technique will be discussed.

References:

[1] K.K. Berthelsen and J. Møller (2007). Non-parametric Bayesian inference for inhomogeneous Markov point processes. Research Report R-2007-9, Department of Mathematical Sciences, Aalborg University.

[2] W.S. Kendall (1998). Perfect simulation for the area-interaction point process. In L. Accardi and C.C. Heyde, Probability Towards 2000, Springer Lecture Notes in Statistics 128, Springer Verlag, New York.

[3] W.S. Kendall and J. Møller (2000). Perfect simulation using dominating processes on ordered spaces, with application to locally stable point processes. Advances in Applied Probability, 32, 844-865.

[4] J. Møller, A.N. Pettitt, K.K. Berthelsen and R.W. Reeves (2006). An efficient Markov chain Monte Carlo method for distributions with intractable normalising constants. Biometrika, 93, 451-458.

[5] J. Møller and R.P. Waagepetersen (2003). Statistical Inference and Simulation for Spatial Point Processes. Chapman and Hall/CRC, Boca Raton.

[6] J. Møller and R.P. Waagepetersen (2007). Modern statistics for spatial point processes (with discussion). Scandinavian Journal of Statistics (to appear).

[7] J.G. Propp and D.B. Wilson (1996). Exact sampling with coupled Markov chains and applications to statistical mechanics. Random Structures and Algorithms, 9, 223-252.