The word "Bayes" suggests that we are updating a distribution using data, to get a posterior distribution.
The fact that the Bayesian information criterion (BIC) is used to select a model from a set of models, suggests that it is called BIC because we are selecting the model with the highest posterior, or something like that.
However, it's not clear to me from the definition of BIC that that's the case. Is it? What is the reason that the Bayesian information criterion is called "Bayesian"?