  This article was motivated by the desire to improve Markov chain Monte Carlo
methods for spatial survival models in which the locations of individuals in
space are known. For a dataset comprising information on n individuals,
standard methods of MCMC-based inference involve computing the inverse of an n
by n matrix at each iteration. However with a judicious choice of auxiliary
variables on a regular grid with m prediction points it will be shown how to
fit an essentially equivalent model but with a substantially reduced
computational cost. For a fixed output grid, the computational cost of the new
method is reduced from O(n^3) to O(n); the cost of increasing the output grid
size being O(m\log m). Furthermore, the new method simultaneously solves the
problem of spatial prediction of functions of the latent field, which for
standard methods usually presents a further computational challenge. We apply
the new method to a spatial survival dataset previously analysed in Henderson
et. al 2002 and show how the new method can be applied to spatial and
spatiotemporal geostatistical datasets with the same computational benefits.
