The phrase “inference to the best explanation” was coined by Gilbert Harman in his attempt to explain abductive inference but it conveys the wrong impression to anyone who takes it as a substitute for the whole course of inquiry rather than just its starting point. Peirce himself was always very clear about this.
Bayes’ theorem is a deductive identity that adds no information to the situation, nor is that its job. It does not add rows or columns to the contingency matrix or make the observations that populate its cells. Those are jobs for the independent capacities of abductive and inductive reasoning.