David Simpson, Queensland University of Technology, Australia

Krylov Subspace Methods for Sampling from Large Gaussian Markov Random Fields

Abstract

Many applications in spatial statistics, geostatistics and image
analysis require efficient techniques for sampling from large Gaussian
Markov random fields (GMRFs). One method for sampling from a GMRF is
to form y = A^{-1}b + x, where x is a sample from the zero mean GMRF
with symmetric positive definite (SPD) precision matrix A. Typically,
the first term can be approximated using a conjugate gradient method,
while the second term is usually computed using a sparse Cholesky
decomposition [1]. In this talk, I will discuss an alternate approach
to sampling from zero mean GMRFs based on Krylov subspace
approximations to the matrix-vector product x = A^{-1/2}z, where z is
a vector of i.i.d. standard normal variables. The only assumption this
method makes about A is that it is possible to calculate its product
with a vector efficiently. Furthermore, as it does not require a
Cholesky decomposition, this method can be used to tackle a variety of
problems in computational statistics, including problems in which the
precision matrix depends non-linearly on unknown
hyperparameters. Issues relating to the implementation of this method
will also be discussed, including the restarting and preconditioning
of Lanczos approximations to functions of SPD matrices.

References
[1] H. Rue, Fast sampling of Gaussian Markov random fields, J. R. Statist.
Soc. B, 63 (2001), pp. 325-338.