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.

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.

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

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.

MCMC applications in Finance: Odds and Ends, and others

Some applications of Bayesian modeling and MCMC in genomics

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.