  This paper proposes a new approach for Bayesian and maximum likelihood
parameter estimation for stationary Gaussian processes observed on a large
lattice with missing values. We propose an MCMC approach for Bayesian
inference, and a Monte Carlo EM algorithm for maximum likelihood inference. Our
approach uses data augmentation and circulant embedding of the covariance
matrix, and provides exact inference for the parameters and the missing data.
Using simulated data and an application to satellite sea surface temperatures
in the Pacific Ocean, we show that our method provides accurate inference on
lattices of sizes up to 512 x 512, and outperforms two popular methods:
composite likelihood and spectral approximations.
