#### Objective Bayesian Analysis of Spatially Correlated Data

James Berger, Victor De Oliveira and Bruno Sanso

Spatially varying phenomena are often modeled using Gaussian random
fields, specified by their mean function and covariance function.
The spatial correlation structure of these models is commonly specified
to be of a certain form (e.g., spherical, power exponential, rational
quadratic, or Matern) with a small
number of unknown parameters. We consider objective Bayesian analysis
of such spatial models, when the mean function of the Gaussian random
field is specified as in a linear model. It is thus necessary to
determine an objective (or default) prior distribution for the unknown
mean and covariance parameters of the random field.
We first show that common choices of default prior distributions,
such as the constant prior and the independent Jeffreys prior, typically
result in improper posterior distributions for this model. Next,
the reference prior for the model is developed, and is shown to
yield a proper posterior distribution. A further attractive property
of the reference prior is that it can be used directly for computation
of Bayes factors or posterior probabilities of hypotheses to compare
different correlation functions, even though the reference prior is
improper. An illustration is given using a spatial data set of
topographic elevations.

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