  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 Mat{\'e}rn covariance for temporal
inputs. The Kalman filter and Rauch-Tung-Striebel smoother were introduced as a
scalable marginalization technique for computing the likelihood and making
predictions of GPs without approximation. We then introduce recent efforts on
extending the scalable marginalization idea to the 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 forecast particle
trajectories. The achievement lies in the 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.
