If I build a Markov chain based on a statistical analysis of word sequences in Hamlet, and then use it to produce a new sentence that isn't found in the text of that work, I have not created a derivative work of Hamlet under any applicable sense of that term.
Uh, that is exactly what a derivative work is. You literally specify that Hamlet is an input to your work. I believe you're conflating derivative with transformative. You're certainly creating a transformative derivation of Hamlet, but you are by definition creating a derivative work by training a Markov chain on the text of Hamlet.
The obvious follow up here is whether an LLM is creating transformative derivations or not. A lot of folks argue that yes, an LLM spitting out statistically sampled code that matches existing code is not transformative and is (or might be) infringing the terms of the license it was released under. Others argue that there's not an exact copy of the original source in the LLM's weights so by definition it must be a transformative work. I think it's a pretty obvious "somewhere in the middle" that is gonna make a bunch of lawyers a whole lot of money.
Personally, I don't care one way or the other. I'm one of the folks that thinks software shouldn't be copyright-able in the first place.