A discussion of ‘prior-based Bayesian information criterion’

Wewould like to thank the authors (Bayarri et al., 2018) for their interesting and provoking paper, and we wish to discuss some issues related to sample size in general and the number of covariates in the context of linear regression model when using the Bayesian information criteria (BIC) for model selection. Schwarz (1978) was the first to develop tools for estimating the dimension of parameters among distributions in exponential family and consequently, introduce the BIC to serve as an approximation to the Bayesian posterior probability of a given model. The BIC has been used in a broad context and has been widely adapted for model selection despite that there are situations where the BIC might not be appropriate. Returning to its root as in this discussion paper is essential when the model and the data structure markedly deviate from the original context. The original BIC criterion targets models arose from distributions belonging to an exponential family which permits a neat and simple analytical form after Laplace approximation. The neatness of this form is a blessing, but unfortunately, can be a curse as well. When the data is deprived of the independent and identically distributed (iid) structure, a blind application of BIC will not survive a close scrutiny. As discussed in Bayarri et al. (2018), the sample size in BIC becomes problematic. The prior-based BIC (PBIC) proposed in Bayarri et al. (2018) is essential to overcome these issues. This paper timely draws our attention to many unsettling issues related to the use of BIC in non-standard situations.