22.5. v 15.40 hod. v seminární místnosti KPMS se koná přednáška


Jesper Mřller (Aalborg University):


MCMC algorithms for distributions with intractable normalizing constants, with a view to perfect simulation and non-parametric Bayesian inference for inhomogeneous Markov point processes


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 [3] we present new methodology for drawing samples from such a distribution without approximationThe 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 [6] and the
dominated coupling from the past (dominating CFTP) [2] becomes useful. We illustrate the method  by producing posterior samples for the following application example.

With reference to a specific data set, we consider in [1] how to perform a flexible non-parametric Bayesian analysis of an inhomogeneous point pattern modelled by a Markov point process, with a location dependent first order term and pairwise interaction only. A priori we assume that the first order term is a shot noise process, and the interaction function for a pair of points depends only on the distance between the two points and is a piecewise linear function modelled by a marked Poisson process. Simulation of the resulting posterior using a Metropolis-Hastings algorithm in the ``conventional'' way involves evaluating ratios of unknown normalizing constants. We avoid this problem by applying the new auxiliary variable technique [3], where the auxiliary variable is an example of  a partially ordered Markov point process model.

Finally, we review the dominating CFTP algorithm. In [2] we give a general formulation
of the method of dominated CFTP which apply for various stochastic models. In particular
we apply it in [2] to the problem of perfect simulation of general locally stable point
processes as equilibrium distributions of spatial birth-and-death processes. In the present
talk we consider for simplicity and specificity only the case of perfect simulation of pairwise
interaction point processes. For those interested in further details of spatial point process
models, simulation and inference, see [4] and [5].


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

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

[3] J. Moller, 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.

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

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

[6] 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.