Bayesian Filtering on Graphs

Graph filters are ubiquitous for processing data over graphs. However, most filters obtained from data are point-estimates and may be sensitive to changes in topology or data distributions. Thus, modeling uncertainty in filters is critical to quantify confidence in analyses or improve downstream tasks in low-data regimes. We introduce a Bayesian framework for graph filter design, termed Bayesian graph filters. Given input-output realizations on a graph, we obtain the posterior filter and prior filter precision hyper-parameters via a constrained EM algorithm. The posterior filter leads to uncertainty in its frequency response, which has implications for stability. We study the stability via the integral Lipschitz (IL) property and derive a lower bound for the probability of Bayesian filters being IL. Results show that Bayesian filters can be more stable across the spectrum and under perturbations, provide uncertainty estimates and can outperform point filters on multiple tasks.