Scalable marginalization of latent variables for correlated data
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Marginalization of latent variables or nuisance parameters is a fundamental aspect of Bayesian inference and uncertainty quantification. In this work, we focus on scalable marginalization of latent variables in modeling correlated data, such as spatio-temporal or functional observations. We first introduce Gaussian processes (GPs) for modeling correlated data and highlight the computational challenge, where the computational complexity increases cubically fast along with the number of observations. We then review the connection between the state space model and GPs with Matérn covariance for temporal inputs. The Kalman filter and Rauch-Tung-Striebel smoother were introduced as a scalable marginalization technique of computing the likelihood and making predictions of GPs without approximation. We then introduce recent efforts on extending the scalable marginalization idea to linear model of coregionalization for multivariate correlated output and spatio-temporal observations. In the final part of this work, we introduce a novel marginalization technique to estimate interaction kernels and to forecast particle trajectories. The achievement lies in sparse representation of covariance function, then applying conjugate gradient for solving the computational challenges and improving predictive accuracy. The computational advances achieved in this work outline a wide range of applications in molecular dynamic simulation, cellular migration and agent based models.
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