Abstract: We present a class of information metric optimization methods for high-dimensional Bayesian sampling problems. First, two examples of information geometries in probability spaces, such as the Fisher-Rao and the Wasserstein-2 spaces, are studied. Then, focusing on the Wasserstein-2 metric, we introduce accelerated gradient and Newton flows to design fast and efficient sampling algorithms. We also present practical sampling approximations for the proposed dynamics in high-dimensional sample spaces. They contain optimal transport natural gradient methods, projected Wasserstein gradient methods, and convex neural network approximation of the Wasserstein gradient. Finally, numerical experiments, including PDE-constrained Bayesian inferences and parameter estimations in COVID-19 modeling, demonstrate the effectiveness of the proposed optimization-oriented sampling algorithms.