4 articles.
One bimodal target, one Gaussian, and two directions of the same divergence. Minimising KL(P||Q) puts the Gaussian across both modes with almost no mass where the target actually lives; minimising KL(Q||P) puts it on one mode at a value of 0.6931 nats, which is ln 2 to four decimals and not a coincidence. Each fit is judged catastrophic by the other objective, 2.0976 against 15.2799.
Entropy is not a summary of a distribution but a floor that the best code meets to the last decimal, the surcharge for using the wrong distribution is exactly the loss every classifier already minimises, and mutual information puts a hard ceiling on everything downstream of a sensor. Three results, each unusually sharp.
How next-word prediction turns unlabelled text into supervision, why cross entropy is just negative average log probability, what perplexity really measures, and why a model that completes text fluently still cannot follow an instruction.
Measuring uncertainty in bits: Shannon entropy and why the logarithm is base 2, information gain worked on a split, and how cross-entropy and KL divergence relate to entropy and to the loss functions used to train classifiers.