**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**

Abstract:

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 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 [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].

References:

[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.