In this work, we propose a modified Bayesian Information Criterion (BIC) specifically designed for mixture models and hierarchical structures. This criterion incorporates the determinant of the Hessian matrix of the log-likelihood function, thereby refining the classical Bayes Factor by accounting for the curvature of the likelihood surface. Such geometric information introduces a more nuanced penalization for model complexity. The proposed approach improves model selection, particularly under small-sample conditions or in the presence of noise variables. Through theoretical derivations and extensive simulation studies-including both linear and linear mixed models-we show that our criterion consistently outperforms traditional methods such as BIC, Akaike Information Criterion (AIC), and related variants. The results suggest that integrating curvature-based information from the likelihood landscape leads to more robust and accurate model discrimination in complex data environments.

翻译：本研究提出了一种专为混合模型与分层结构设计的改进贝叶斯信息准则（BIC）。该准则通过引入对数似然函数海森矩阵的行列式，利用似然曲面的曲率信息对经典贝叶斯因子进行修正。这种几何信息的融入为模型复杂度提供了更精细的惩罚机制。所提出的方法显著提升了模型选择性能，尤其在小样本条件或存在噪声变量的场景下表现突出。通过理论推导和广泛的模拟研究（涵盖线性模型与线性混合模型），我们证明该准则在各项指标上均优于传统方法，包括标准BIC、赤池信息准则（AIC）及其相关变体。研究结果表明，在复杂数据环境中，融合似然函数曲面的几何信息能够实现更鲁棒且更精确的模型判别。