  We introduce a framework for Newton's flows in probability space with
information metrics, named information Newton's flows. Here two information
metrics are considered, including both the Fisher-Rao metric and the
Wasserstein-2 metric. A known fact is that overdamped Langevin dynamics
correspond to Wasserstein gradient flows of Kullback-Leibler (KL) divergence.
Extending this fact to Wasserstein Newton's flows, we derive Newton's Langevin
dynamics. We provide examples of Newton's Langevin dynamics in both
one-dimensional space and Gaussian families. For the numerical implementation,
we design sampling efficient variational methods in affine models and
reproducing kernel Hilbert space (RKHS) to approximate Wasserstein Newton's
directions. We also establish convergence results of the proposed information
Newton's method with approximated directions. Several numerical examples from
Bayesian sampling problems are shown to demonstrate the effectiveness of the
proposed method.
