The availability of data from multiple heterogeneous environments has motivated methods that remain reliable under distributional shifts. When the joint distribution of response and predictors varies across environments, the response may still depend on a subset of predictors through an invariant mechanism. Existing methods typically assess candidate invariant sets through pooled stability criteria, treating environmental variation as nuisance. In this paper, we propose a Bayesian framework that explicitly separates a shared response mechanism from environment-specific or response-dependent associations, exploiting heterogeneity as evidence for structure learning. A competitive spike-and-slab prior is designed to force each predictor to compete between invariant and non-invariant spurious effects. Under a tractable working model, we establish invariant model selection consistency and posterior contraction for invariant coefficients. We further study the presence of irrelevant predictors, characterize posterior concentration on an equivalent invariant class, and introduce a post-selection refinement that consistently recovers the minimal invariant model. Simulations and a real application illustrate the robustness and finite-sample efficiency of the proposed method.

翻译：来自多个异质环境的数据推动了在分布偏移下保持可靠性的方法发展。当响应变量与预测变量的联合分布随环境变化时，响应变量可能仍通过不变机制依赖于预测变量的某个子集。现有方法通常通过联合稳定性准则评估候选不变集，将环境变化视为干扰因素。本文提出一个贝叶斯框架，明确分离共享响应机制与环境特定或响应依赖的关联，利用异质性作为结构学习的证据。设计了一种竞争性尖峰-板先验，强制每个预测变量在不变效应与非不变伪效应之间竞争。在可处理的工作模型下，我们建立了不变模型选择一致性和不变系数的后验收缩性质。进一步研究了无关预测变量的存在性，刻画了等价不变类上的后验集中性，并引入了一种后选择精炼方法，能够一致地恢复最小不变模型。模拟实验和实际应用验证了所提方法的稳健性与有限样本有效性。