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The conditional extremes framework allows for event-based stochastic modelingof dependent extremes, and has recently been extended to spatial andspatio-temporal settings. After standardizing the marginal distributions andapplying an appropriate linear normalization, certain non-stationary Gaussianprocesses can be used as asymptotically-motivated models for the processconditioned on threshold exceedances at a fixed reference location and time. Inthis work, we adapt existing conditional extremes models to allow for thehandling of large spatial datasets. This involves specifying the model forspatial observations at $d$ locations in terms of a latent $m\ll d$ dimensionalGaussian model, whose structure is specified by a Gaussian Markov random field.We perform Bayesian inference for such models for datasets containing thousandsof observation locations using the integrated nested Laplace approximation, orINLA. We explain how constraints on the spatial and spatio-temporal Gaussianprocesses, arising from the conditioning mechanism, can be implemented throughthe latent variable approach without losing the computationally convenientMarkov property. We discuss tools for the comparison of models via theirposterior distributions, and illustrate the flexibility of the approach withgridded Red Sea surface temperature data at over $6,000$ observed locations.Posterior sampling is exploited to study the probability distribution ofcluster functionals of spatial and spatio-temporal extreme episodes.

High-dimensional modeling of spatial and spatio-temporal conditional extremes using INLA and Gaussian Markov random fields