  Existing methods for graph out-of-distribution (OOD) generalization primarily
rely on empirical studies on synthetic datasets. Such approaches tend to
overemphasize the causal relationships between invariant sub-graphs and labels,
thereby neglecting the non-negligible role of environment in real-world
scenarios. In contrast to previous studies that impose rigid independence
assumptions on environments and invariant sub-graphs, this paper presents the
theorems of environment-label dependency and mutable rationale invariance,
where the former characterizes the usefulness of environments in determining
graph labels while the latter refers to the mutable importance of graph
rationales. Based on analytic investigations, a novel variational inference
based method named ``Probability Dependency on Environments and Rationales for
OOD Graphs on Real-world Data'' (DEROG) is introduced. To alleviate the adverse
effect of unknown prior knowledge on environments and rationales, DEROG
utilizes generalized Bayesian inference. Further, DEROG employs an EM-based
algorithm for optimization. Finally, extensive experiments on real-world
datasets under different distribution shifts are conducted to show the
superiority of DEROG. Our code is publicly available at
https://anonymous.4open.science/r/DEROG-536B.
