  In spatial statistics, fast and accurate parameter estimation, coupled with a
reliable means of uncertainty quantification, can be challenging when fitting a
spatial process to real-world data because the likelihood function might be
slow to evaluate or wholly intractable. In this work, we propose using
convolutional neural networks to learn the likelihood function of a spatial
process. Through a specifically designed classification task, our neural
network implicitly learns the likelihood function, even in situations where the
exact likelihood is not explicitly available. Once trained on the
classification task, our neural network is calibrated using Platt scaling which
improves the accuracy of the neural likelihood surfaces. To demonstrate our
approach, we compare neural likelihood surfaces and the resulting maximum
likelihood estimates and approximate confidence regions with the equivalent for
exact or approximate likelihood for two different spatial processes: a Gaussian
process and a Brown-Resnick process which have computationally intensive and
intractable likelihoods, respectively. We conclude that our method provides
fast and accurate parameter estimation with a reliable method of uncertainty
quantification in situations where standard methods are either undesirably slow
or inaccurate. The method is applicable to any spatial process on a grid from
which fast simulations are available.
