Likelihood inference with hierarchical models is often complicated by the fact that the likelihood function involves intractable integrals. Numerical integration (e.g. quadrature) is an option if the dimension of the integral is low but quickly becomes unreliable as the dimension grows. An alternative approach is to approximate the intractable integrals using Monte Carlo averages. Several dierent algorithms based on this idea have been proposed. In this paper we discuss the relative merits of simulated maximum likelihood, Monte Carlo EM, Monte Carlo Newton-Raphson and stochastic approximation. Key words and phrases : Eciency, Monte Carlo EM, Monte Carlo Newton-Raphson, Rate of convergence, Simulated maximum likelihood, Stochastic approximation All three authors partially supported by NSF Grant DMS-00-72827. 1 1

Introduction Spatial point processes play a fundamental role in spatial statistics. In the simplest case they model \small" objects that may be identied by a map of points showing stores, towns, plants, nests, galaxies or cases of a disease observed in a two or three dimensional region. The points may be decorated with marks (such as sizes or types) whereby marked point processes are obtained. The areas of applications are manifold: astronomy, geography, ecology, forestry, spatial epidemiology, image analysis, and many more. Currently spatial point processes is an active area of research, which probably will be of increasing importance for many new applications, as new technology such as geographical information systems makes huge amounts of spatial point process data available. Textbooks and review articles on dierent aspects of spatial point processes include Matheron (1975), Ripley (1977), Ripley (1981), Diggle (1983), Penttinen (1984), Daley &Vere-Jones (1988),

A new class of Gibbsian models with potentials associated to the connected components or homogeneous parts of images is introduced. For these models the neighbourhood of a pixel is not fixed as for Markov random fields, but given by the components which are adjacent to the pixel. The relationship to Markov random fields and marked point processes is explored and spatial Markov properties are established. Also extensions to infinite lattices are studied, and statistical inference problems including geostatistical applications and statistical image analysis are discussed. Finally, simulation studies are presented which show that the models may be appropiate for a variety of interesting patterns including images exhibiting intermediate degrees of spatial continuity and images of objects against background.

Abbreviated title. Approximating an intractable distribution

We review simulation based methods in optimal design. Expected utility maximization, i.e., optimal design, is concerned with maximizing an mtegra! expression representing expected utility with respect to some design parameter. Except in special cases neither the maximization nor the integration can be solved analytically and approximations and/or simulation based methods are needed. On one hand the integration problem is easier to solve than the integration appearing in posterior inference problems. This is because: the expectation is with respect to the joint distribution of parameters and data, which typically allows efficient random variate generation. On the other hand, the problem is difficult because the integration is embedded in the maximization and has to possibly be evaluated many times for different design parameters. We discuss four related strategies: prior simulation; smoothing of Monte Carlo simulations; Markov chain Monte Carlo (MCMC) simulation in an augmented probability model; a simulated annealing type approach.

. A simulation method based on importance sampling, Gibbs and Metropolis-Hastings techniques allows to approximate the ratio between the likelihood function computed for two different parameter values. Thus it is possible to approximate the maximum likelihood estimator in the general framework of dynamic latent variable models. Some examples of this class of models are factor models, switching regime models, dynamic limited dependent variable models, stochastic volatility models and discretised continuous time models. 1 University Ca' Foscari of Venice 2 CREST-INSEE 3 CREST-INSEE 1 INTRODUCTION The class of parametric dynamic latent variable models (also called factor models or hidden variable models or hierarchical models) is becoming increasingly popular, because of the flexibility they offer in the modelling of complex phenomena. They jointly specify a sequence (y t ) of time dependent variables and a second sequence (y t ) of partially unobserved variables in such a way ...

normalizing constants

The purpose of this overview paper is 1) to give a brief presentation of the statistical problems...

We study a Markov switching stochastic volatility model with heavy tail innovations in the observable process. Due to the economic interpretation of the hidden volatility regimes, these models have many financial applications like asset allocation, option pricing and risk management. The Markov switching process is able to capture clustering effects and jumps in volatility. Heavy tail innovations account for extreme variations in the observed process. Accurate modelling of the tails is important when estimating quantiles is the major interest like in risk management applications. Moreover we follow a Bayesian approach to filtering and estimation, focusing on recently developed simulation based filtering techniques, called Particle Filters. Simulation based filters are recursive techniques, which are useful when assuming non-linear and non-Gaussian latent variable models and when processing data sequentially. They allow to update parameter estimates and state filtering as new observations become available.

We study a Markov switching stochastic volatility model with heavy tail innovations in the observable process. Due to the economic interpretation of the hidden volatility regimes, these models have many financial applications like asset allocation, option pricing and risk management. The Markov switching process is able to capture clustering effects and jumps in volatility. Heavy tail innovations account for extreme variations in the observed process. Accurate modelling of the tails is important when estimating quantiles is the major interest like in risk management applications. Moreover we follow a Bayesian approach to filtering and estimation, focusing on recently developed simulation based filtering techniques, called Particle Filters. Simulation based filters are recursive techniques, which are useful when assuming non-linear and non-Gaussian latent variable models and when processing data sequentially. They allow to update parameter estimates and state filtering as new observations become available.