Designing expressive Graph Neural Networks (GNNs) is an important topic in graph machine learning fields. Despite the existence of numerous approaches proposed to enhance GNNs based on Weisfeiler-Lehman (WL) tests, what GNNs \emph{can and cannot} learn still lacks a deeper understanding. This paper adopts a fundamentally different approach to examine the expressive power of GNNs from a probabilistic perspective. By establishing connections between GNNs' predictions and the central inference problems of probabilistic graphical models (PGMs), we can analyze previous GNN variants with a novel hierarchical framework and gain new insights into their node-level and link-level behaviors. Additionally, we introduce novel methods that can provably enhance GNNs' ability to capture complex dependencies and make complex predictions. Experiments on both synthetic and real-world datasets demonstrate the effectiveness of our approaches.