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## INLA-MRA: A Bayesian inference method for large spatiotemporal datasets

### Luc VillandrĂ© (HEC MontrĂ©al, University of Toronto)

Automated data collection creates major computational hurdles for several common geostatistical Bayesian models, that involve a latent Gaussian field whose covariance matrix has number of rows equal to sample size. To evaluate the associated density, the Cholesky decomposition of the matrix is necessary. If the matrix is dense, the decomposition is computed with an algorithm whose computational complexity is cubic in the number of rows. As a result, evaluation becomes intractable once sample size exceeds 100,000. Moreover, the computational burden of each likelihood calculation increases quickly with the number of observations. It follows that conventional numerical algorithms for posterior estimation, that require thousands of such evaluations, are ill-suited for massive datasets. The method we propose, INLA-MRA, overcomes those issues by using sparse inverse covariance matrices, and by estimating marginal posteriors with an importance sampling strategy inspired by INLA. We apply INLA-MRA to land surface temperature data collected in May 2012 in the state of Maharashtra, India, by satellites equipped with the MODIS imaging sensor. Thick cloud cover often results in a large number of missing values, which we aim to infer with our method. We find that INLA-MRA can produce sensible parameter and hyperparameter marginal posterior estimates, as well as realistic prediction surfaces characterised by low mean squared prediction error.