A note on parametric Bayesian inference via gradient flows

In this note, we summarize several recent developments for efficient sampling methods for parameters based on Bayesian inference. To reformulate those sampling methods, we use different formulations for gradient flows on the manifold in the parameter space, including strong form, weak form and De Giorgi type duality form. The gradient flow formulations will cover some applications in deep learning, ensemble Kalman filter for data assimilation,