Neural Likelihood Surfaces for Spatial Processes with Computationally Intensive or Intractable Likelihoods

In spatial statistics, fast and accurate parameter estimation coupled with a reliable means of uncertainty quantification can be a challenging task when fitting a spatial process to real-world data because the likelihood function might be slow to evaluate or intractable. In this work, we propose using convolutional neural networks (CNNs) 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 maximum likelihood estimates and approximate confidence regions constructed from the neural likelihood surface with the equivalent for exact or approximate likelihood for two different spatial processes: a Gaussian Process, which has a computationally intensive likelihood function for large datasets, and a Brown-Resnick Process, which has an intractable likelihood function. We also compare the neural likelihood surfaces to the exact and approximate likelihood surfaces for the Gaussian Process and Brown-Resnick Process, 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.