Bayesian Comparison of Latent Variable Models: Conditional Versus Marginal Likelihoods.

Area of Science:

• Statistics
• Psychometrics
• Computational Statistics

Background:

• Bayesian methods for latent variable models typically sample latent variables.
• Model comparison often overlooks the distinction between conditional and marginal likelihoods.
• This oversight can significantly impact research findings in psychometric modeling.

Purpose of the Study:

• To clarify the distinction between conditional and marginal likelihoods in Bayesian model comparison.
• To illustrate the impact of this distinction on model selection criteria like DICs and WAICs.
• To provide recommendations for applying these criteria to models with latent variables.

Main Methods:

• Focused on comparing conditional and marginal Deviance Information Criteria (DICs) and Watanabe-Akaike Information Criteria (WAICs).
• Examined the implications of treating latent variables as parameters versus integrating them out.
• Connected criteria to cross-validation strategies: marginal WAIC to leave-one-cluster-out and conditional WAIC to leave-one-unit-out.

Main Results:

• The choice between conditional and marginal approaches depends on the prediction goal: current clusters or new clusters.
• Marginal WAIC aligns with leave-one-cluster-out cross-validation.
• Conditional WAIC aligns with leave-one-unit-out cross-validation.

Conclusions:

• Researchers must carefully consider the conditional versus marginal distinction when comparing Bayesian models with latent variables.
• The choice of criterion (conditional or marginal) should align with the specific predictive goals of the study.
• Proper application of DICs and WAICs ensures valid model selection in psychometric and related fields.