Predictive Bayesian model comparison often relies on leave-one-out (LOO) cross-validation criteria such as the expected log predictive density (ELPD). However, model rankings can be overly sensitive to outliers and tail mismatch because ELPD is based on the log score. We propose a score-matched generalized ELPD framework that replaces the log score by a Bregman scoring rule to update model parameters through a generalized posterior and to evaluate LOO predictive utility. Candidate posterior predictive distributions are ranked by out-of-sample utility under the chosen scoring rule, yielding a direct proper-score generalization of standard ELPD. We focus especially on the $β$-divergence family, where $β$ controls the sensitivity of predictive comparison to low-density observations. Under model misspecification, the procedure asymptotically selects the model whose predictive distribution is closest to the data-generating process under the chosen Bregman divergence. A simulation study and applications to microbial and forensic data show that the generalized ELPD can change the selected model through reduced sensitivity to low-density observations.

翻译：基于预测的贝叶斯模型比较通常依赖于留一法交叉验证准则，例如期望对数预测密度。然而，由于ELPD基于对数评分，模型排序可能对异常值和尾部不匹配过度敏感。我们提出一种评分匹配的广义ELPD框架，通过布雷格曼评分规则替代对数评分，借助广义后验分布更新模型参数并评估留一法预测效用。候选后验预测分布依据所选评分规则下的样本外效用进行排序，从而实现对标准ELPD的直接适当评分泛化。我们特别关注β-散度族，其中β控制预测比较对低密度观测的敏感度。在模型设定错误的情况下，该过程渐进地选择在所选布雷格曼散度下预测分布最接近数据生成过程的模型。仿真实验及微生物与法医学数据应用表明，广义ELPD通过降低对低密度观测的敏感度，能够改变所选模型。