COMMENT: THE INFERENTIAL INFORMATION CRITERION FROM A BAYESIAN POINT OF VIEW

• Published in: Sociological methodology Vol. 48; no. 1; pp. 91 - 97
• Main Author: Vassend, Olav B.
• Format: Journal Article
• Language: English
• Published: Los Angeles, CA SAGE Publishing 01-08-2018; SAGE Publications; American Sociological Association
• Subjects: Bayesian analysis; Criteria; Probability; Symposium: Commentaries and Rejoinder
• ISSN: 0081-1750; 1467-9531
• DOI: 10.1177/0081175018794489

As Michael Schultz notes in his very interesting paper (this volume, pp. 52–87), standard model selection criteria, such as the Akaike information criterion (AIC; Akaike 1974), the Bayesian information criterion (BIC; Schwarz 1978), and the minimum description length principle (MDL; Rissanen 1978), are purely empirical criteria in the sense that the score a model receives does not depend on how well the model coheres with background theory. This is unsatisfying because we would like our models to be theoretically plausible, not just empirically successful. To remedy this situation, Schultz proposes an amended version of AIC that he calls the inferential information criterion (IIC). Mathematically, the IIC score of a model is a trade-off between the model’s AIC score and its conditional probability given a background theory, P(M|T). More precisely, IIC issues the following imperative:The IIC: Choose the model that has the minimal IIC score, where the IIC score of M is given by the following formula:IIC( )=−log ( | )+AIC( ). (1)From a statistical point of view, the IIC score is a rather odd hybrid, since it involves both the Bayesian-looking P(M|T) and the AIC score, AIC(M), which is typically considered a frequentist construct. The goal of this comment is to analyze IIC from a fully Bayesian point of view. As we will see, IIC has a Bayesian justification. Unfortunately, the Bayesian analysis will also show that there is reason to suspect that IIC will fail as an adequate model selection criterion in precisely the cases that Schultz is most interested in.