A Bayesian Perspective on Uncertainty Quantification for Estimated Graph Signals

TL;DR

This paper introduces a Bayesian framework for quantifying uncertainty in graph signal estimation, interpreting traditional methods as posterior Gaussian distributions shaped by graph structure and priors.

Contribution

It offers a novel probabilistic perspective that unifies various graph signal estimation methods under a Bayesian approach, enabling uncertainty quantification.

Findings

Conventional estimation methods are reinterpreted as posterior Gaussian means.

The framework naturally incorporates assumptions like smoothness and bandlimitedness.

It provides a way to characterize the uncertainty around estimated signals.

Abstract

We present a Bayesian perspective on quantifying the uncertainty of graph signals estimated or reconstructed from imperfect observations. We show that many conventional methods of graph signal estimation, reconstruction and imputation, can be reinterpreted as finding the mean of a posterior Gaussian distribution, with a covariance matrix shaped by the graph structure. In this perspective, assumptions of signal smoothness as well as bandlimitedness are naturally expressible as the choice of certain prior distributions; noisy, noise-free or partial observations are expressible in terms of certain likelihood models. In addition to providing a point estimate, as most standard estimation strategies do, our probabilistic framework enables us to characterize the shape of the estimated signal distribution around the estimate in terms of the posterior covariance matrix.