On the Proper Forms of BIC for Model Order Selection

The Bayesian Information Criterion (BIC) is often presented in a form that is only valid in large samples and under a certain condition on the rate at which the Fisher Information Matrix (FIM) increases with the sample length. This form has been improperly used previously in situations in which the conditions mentioned above do not hold. In this correspondence, we describe the proper forms of BIC in several practically relevant cases that do not satisfy the above assumptions. In particular, we present a new form of BIC for high signal-to-noise ratio (SNR) cases. The conclusion of this study is that BIC remains one of the most successful existing rules for model order selection, if properly used.