Rencontre scientifique du laboratoire de statistique
Scientific Meeting of the CRM Statistics Laboratory

MCMC : Theory and Applications

(Date limite pour s'inscrire: 14 octobre ou 57 inscriptions / Deadline for registration : October 14 or 57 registrations)

Vendredi 17 octobre / Friday October 17, 2008 • Salle 0046, sous-sol Pavillon D6, Université de Sherbrooke

Horaire / Schedule

11.00 - 11.50 Arnaud Doucet, University of British Columbia

Recent Advances in Markov Chain Monte Carlo

MCMC methods have been around for more than fifty years. However, there are still many important statistical models where MCMC yield poor results or are not even applicable. I will discuss a few recent promising ideas which have been proposed in the literature to address some of these limitations. This will include MCMC algorithms to handle intractable likelihood functions, adaptive MCMC and population MCMC type algorithms.

11.50 - 12.30 Mylène Bédard, Université de Montréal

Theoretical Aspects of MCMC

In this talk, we shall discuss the optimal construction of MCMC algorithms,and particularly of Metropolis-Hastings (MH) algorithms. We present some classical convergence results, and then focus on the optimal scaling issue for random walk Metropolis (RWM) algorithms with Gaussian proposal distributions, an important class of MH algorithms. In order to implement these algorithms, it is necessary to tune the variance of the Gaussian distribution so that the Markov chain converges rapidly to its stationary distribution. We discuss existing results about the optimal scaling of the proposal distribution and the optimal acceptance rate of the algorithm for the special case of multidimensional target distributions with independent components. We finally propose to extend the theory to the case of multidimensional target distributions with correlated components. To do so, a natural avenue to explore is that of hierarchical targets; these models are widely used in practice and have a simple density function.

12.30 - 13.50 Lunch
Le lunch aura lieu au Pavillon Multifonctionnel (B5 sur le plan) tandis que la pause café aura lieu près de la salle des conférences au d4 - 2019.

Réunion du laboratoire de statistique du CRM / Meeting of the CRM Statistics Laboratory

13.50 - 14.30 Geneviève Lefebvre, Université du Quebec à Montréal

A Path Sampling Identity for Computing the Kullback-Leibler and J-Divergences with some applications

Using an identity that arises in the formulation of path sampling - a powerful method for estimating normalizing constants - I will present expressions for the Kullback-Leibler and J-divergences between two distributions from possibly different parametric families. These expressions naturally stem from path sampling when the geometric path is used to link the two extreme densities. I will then present two different contexts for which these expressions can prove themselves useful.

14.30 - 15.10 Eric Jacquier, HEC Montréal

MCMC applications in Finance: Odds and Ends, and others

15.10 - 15.50 Raphael Gottardo, IRCM, Montréal

Some applications of Bayesian modeling and MCMC in genomics

Many application areas within the field of genomics require sophisticated statistical techniques in order to deal with problems associated with large datasets, indirect measurements, complex underlying processes or any combination of these three. In a typical genomic experiment, the amount of replication is limited and the number of parameters (e.g. genes) is large, which makes the use of standard statistical methods difficult. Bayesian hierarchical models have been shown to be particularly well suited for these types of problems; they can be made quite flexible while borrowing strength from the data and using prior information when estimating the quantities of interest. Unfortunately, these models can be quite complex and cannot be fit without sophisticated MCMC algorithms. In this talk, I will review a few genomics problems along with some of the Bayesian models we have devised to solve these problems
15.50 - 16.10 Pause Café / Coffee Break
16.10 - 17.00 Alan Gelfand, Duke University

Dimension Reduction Approaches for Analyzing Large Spatial Datasets

Fitting hierarchical spatial models often involves expensive matrix decompositions whose computational complexity increases in cubic order with the number of spatial locations, rendering such models infeasible for large spatial data sets. This computational burden is exacerbated in multivariate settings with several spatially dependent response variables. It is also aggravated when data is collected at frequent time points and spatiotemporal process models are used.

Dimension reduction approaches provide one strategy for addressing this problem. Here, we argue for the use of predictive process models for spatial and spatiotemporal data. Every spatial (or spatiotemporal) process induces a predictive process model (in fact, arbitrarily many of them). The latter models project process realizations of the former to a lower-dimensional subspace; we achieve the flexibility to accommodate nonstationary, non-Gaussian, possibly multivariate, possibly spatiotemporal processes in the context of large datasets.

We discuss attractive theoretical properties of these predictive processes as well as a computationally feasible template encompassing these diverse settings. Finally, we illustrate with spatial modelling of forest biomass where interest lies in detecting how biomass changes across the landscape (as a continuous surface) and how homogeneous it is across the region. We employ point-referenced biomass data observed at 9, 500 locations obtained from the USDA Forest Service Forest Inventory and Analysis.