Global water-level systems have undergone large changes in recent years. Moderate Resolution Imaging Spectroradiometer (MODIS) land cover data can be used to monitor and understand these changes. However, such data are extremely high-dimensional making them difficult to process while accounting for all sources of uncertainty. Bayesian hierarchical models are commonly implemented using Markov Chain Monte Carlo (MCMC) methods which can be computationally expensive for large datasets. Exact Posterior Regression (EPR) provides an efficient alternative for inference in latent Gaussian hierarchical models under exponential families without requiring MCMC or approximate Bayesian strategies. EPR has not been developed for multinomial data, and we extend this class of models to the multinomial setting. Additionally, we explore signal-to-noise covariance properties of discrepancy models, which we expect to be present in our data. We extend EPR to multinomial likelihoods via a stick-breaking construction to analyze water-level changes. Through simulation studies, we compare the proposed multinomial EPR approach with alternative models implemented using Gibbs sampling and the Integrated Nested Laplace Approximation (INLA). We evaluate each approach based on regression parameter estimation, category probability estimation, and computational efficiency. Overall, EPR and INLA achieve substantial reductions in computation time while EPR shows improved metrics in estimation and prediction.

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