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.

We propose to use deep learning to estimate parameters in statistical models when standard likelihood estimation methods are computationally infeasible. We show how to estimate parameters from max-stable processes, where inference is exceptionally challenging even with small datasets but simulation is straightforward. We use data from model simulations as input and train deep neural networks to learn statistical parameters. Our neural-network-based method provides a competitive alternative to current approaches, as demonstrated by considerable accuracy and computational time improvements. It serves as a proof of concept for deep learning in statistical parameter estimation and can be extended to other estimation problems.

We examine several aspects of explicability of a classification system built from neural networks. The first aspect is the pairwise explicability, which is the ability to provide the most accurate prediction when the range of possibilities is narrowed to just two. Next we consider explicability in development, which means ability to make incremental improvement in prediction accuracy based on observed deficiency of the system. Inherent stochasticity of neural network based classifiers can be interpreted using likelihood randomness explicability. Finally, sureness explicability indicates confidence of the classifying system to make any prediction at all. These concepts are examined in the framework of pairwise coupling, which is a non-trainable metamodel that originated during development of support vector machines. Several methodologies are evaluated, of which the key one is shown to be the choice of the pairwise coupling method. We compare two methods: the established Wu-Lin-Weng method with the recently proposed Bayes covariant method. Our experiments indicate that the Wu-Lin-Weng method gives more weight to a single pairwise classifier, whereas the latter tries to balance information from the whole matrix of pairwise likelihoods. This translates into higher accuracy, and better sureness predictions for the Bayes covariant method. Pairwise coupling methodology has its costs, especially in terms of the number of parameters (but not necessarily in terms of training costs). However, when additional explicability aspects beyond accuracy are desired in an application, the pairwise coupling models are a promising alternative to the established methodology.