I won’t pretend to know the technical details (as the other replies do) but I want to make a point for the “pedagogical” effect here, which I agree with the author. The way I interpret the article, it’s not supposed to be a deep, theoretical treatise on the subject; more of an introductory, “intuitive” take on it. This works for those who need to either learn the concept to begin with, or refresh their memories if they don’t work with it every day. I think it’s a given that any intuitive take on a mathematical concept will always oversimplify things, with the underlying assumption that, if you actually need to know more, you’re going to have to dive deeper somewhere else. The most important thing I think is to help the reader build a rough conceptual understanding of the concept such that they can reason about it, instead of simply memorizing the terms.
I see, where you're coming from, but in that particular case, the "intuitive" explanation (walk on a graph) is far less intuitive for me than the proper explanation, that a Markov process is memoryless. That said, I used MCMC in the past to do physics simulation, where the Markov property also applies to the underlying physical process. So maybe it's just me.
No single explanation works for most. That's why you need multiple ways to explain things and how the standard education system fails at disseminating information.
>I won’t pretend to know the technical details (as the other replies do) but I want to make a point for the “pedagogical” effect here, which I agree with the author. The way I interpret the article, it’s not supposed to be a deep, theoretical treatise on the subject; more of an introductory, “intuitive” take on it.
There are ways to teach things at a conceptual level without teaching them incorrectly though, especially when writing for an adult audience.
Yeah, the hard part of MCMC is going from finite state and time spaces to continuous. But the general concepts and intuition for finite state/time get a LOT more complicated, even though they are ultimately fairly intuitive generalizations of the finite case. Maybe this is a hot take, but I did do my PhD on MCMC and to be honest I think most of the details for the more general case are superfluous for a lot of people who just want to use MCMC, since it boils down to "it basically just works algorithmically the same way as the easier to understand case of finite state/time but beware the actual details of proving that get very involved." The conclusion of the theory for MCMC is that it ends up being pretty darn lenient in terms of caveats and edge cases, so in practice as a user there aren't that many gotchas to not being really well versed in the deepest parts of the theory.
A counterpoint is that finding texts and references that do handle the more advanced cases in depth is quite difficult, bordering on impossible if you want something approachable. I recreated the general proofs for at least the continuous state case as a chapter in my dissertation and stringing together an end-to-end proof with the proper references required quite a lot of digging and discussions with my (co-)advisor.
Yep, and that's shooting in the foot the "people" (I stay polite) trying to hide this intuitive "pedagogical" perspective for some agenda of their own...