Many Bayesian model selection problems, such as variable selection or cluster analysis, start by setting prior model probabilities on a structured model space. Based on a chosen loss function between models, model selection is often performed with a Bayes estimator that minimizes the posterior expected loss. The prior model probabilities and the choice of loss both highly affect the model selection results, especially for data with small sample sizes, and their proper calibration and careful reflection of no prior model preference are crucial in objective Bayesian analysis. We propose risk equilibrium priors as an objective choice for prior model probabilities that only depend on the model space and the choice of loss. Under the risk equilibrium priors, the Bayes action becomes indifferent before observing data, and the family of the risk equilibrium priors includes existing popular objective priors in Bayesian variable selection problems. We generalize the result to the elicitation of objective priors for Bayesian cluster analysis with Binder's loss. We also propose risk penalization priors, where the Bayes action chooses the simplest model before seeing data. The concept of risk equilibrium and penalization priors allows us to interpret prior properties in light of the effect of loss functions, and also provides new insight into the sensitivity of Bayes estimators under the same prior but different loss. We illustrate the proposed concepts with variable selection simulation studies and cluster analysis on a galaxy dataset.

翻译：许多贝叶斯模型选择问题（如变量选择或聚类分析）首先在结构化的模型空间上设定先验模型概率。基于选定的模型间损失函数，模型选择通常通过最小化后验期望损失的贝叶斯估计器进行。先验模型概率与损失函数的选择均会对模型选择结果产生显著影响，尤其在样本量较小的数据中；因此，在客观贝叶斯分析中，如何合理校准先验并谨慎反映无先验模型偏好至关重要。本文提出风险均衡先验作为先验模型概率的客观选择，该先验仅依赖于模型空间与损失函数。在风险均衡先验下，贝叶斯行动在观测数据前呈现无差异状态，且风险均衡先验族涵盖了贝叶斯变量选择问题中现有的主流客观先验。我们将该结论推广至绑定损失函数下的贝叶斯聚类分析中的客观先验构建，同时提出风险惩罚先验——在该先验下，贝叶斯行动在数据观测前选择最简模型。风险均衡与惩罚先验的概念不仅允许我们从损失函数效应的角度解释先验性质，还为同一先验在不同损失函数下贝叶斯估计器的敏感性提供了新见解。我们通过变量选择模拟实验及星系数据集的聚类分析对所提概念进行了验证。