Simply looking at your mathematical claims and comparing them to the definitions of:
Discriminant information (discrete case):
I(P||Q) = - \sum_{x \in X} P(x)log(\frac{Q(x)}{P(x)})
The fraction is a log transformation of a ratio of probabilities at a particular x, while the information is a probability weighted sum of those log transformed probability ratios.
The Bayes Factor as defined in the James Berger et. al. paper I’ve linked to in a number of threads (including above):
The Bayes factor is virtually identical to the KL information divergence metric, in that it is the integrated likelihood, with a uniform prior being a special case where the Bayesian posterior and Frequentist “confidence” distribution (set of all intervals with \alpha = [0 ... 1] coincide (in large samples).
Brockett, P. L. (1991). Information theoretic approach to actuarial science: A unification and extension of relevant theory and applications. Transactions of the Society of Actuaries, 43, 73-135. PDF