DeepAI AI Chat
Log In Sign Up

On Bayesian posterior mean estimators in imaging sciences and Hamilton-Jacobi Partial Differential Equations

by   Jérôme Darbon, et al.

Variational and Bayesian methods are two approaches that have been widely used to solve image reconstruction problems. In this paper, we propose original connections between Hamilton–Jacobi (HJ) partial differential equations and a broad class of Bayesian methods and posterior mean estimators with Gaussian data fidelity term and log-concave prior. Whereas solutions to certain first-order HJ PDEs with initial data describe maximum a posteriori estimators in a Bayesian setting, here we show that solutions to some viscous HJ PDEs with initial data describe a broad class of posterior mean estimators. These connections allow us to establish several representation formulas and optimal bounds involving the posterior mean estimate. In particular, we use these connections to HJ PDEs to show that some Bayesian posterior mean estimators can be expressed as proximal mappings of twice continuously differentiable functions, and furthermore we derive a representation formula for these functions.


Optimally weighted loss functions for solving PDEs with Neural Networks

Recent works have shown that deep neural networks can be employed to sol...

Partial differential equations on hypergraphs and networks of surfaces: derivation and hybrid discretizations

We introduce a general, analytical framework to express and to approxima...

Leveraging Multi-time Hamilton-Jacobi PDEs for Certain Scientific Machine Learning Problems

Hamilton-Jacobi partial differential equations (HJ PDEs) have deep conne...

Robust estimation in controlled branching processes: Bayesian estimators via disparities

This paper is concerned with Bayesian inferential methods for data from ...

PDE Evolutions for M-Smoothers in One, Two, and Three Dimensions

Local M-smoothers are interesting and important signal and image process...

On the cost of Bayesian posterior mean strategy for log-concave models

In this paper, we investigate the problem of computing Bayesian estimato...