A Modified Bayesian Criterion for Model Selection in Mixed and Hierarchical Frameworks

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.