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> A Markov chain is a sequence of random variables X1, X2, X3, . . . , Xt , . . . , such that the probability distribution of Xt+1 depends only on t and xt (Mar

• btaitelb 10y ago
  Markov chains are much simpler than Markov models. The lecture starts off with the simpler idea to motivate the more complex one that you're describing.
  • graycat 10y ago
    Sorry. The statement I quoted is just flatly wrong. Okay?
    • evanpw 10y ago
      It's more reasonable to parse "depends only on" as expressing conditional independence than unconditional, and that parsing has the added bonus of making the original statement correct.
      • graycat 10y ago
        No. Again, once again, over again, one more time, in a Markov process, including a discrete time Markov chain, even one with a finite state space, the past can, and in practice usually does, provide a lot of powerful information to predict the future. The future really can depend on the past. Typically the past and the future are not independent -- not even close. But, the Markov assumption (property) says that GIVEN (excuse my emphasis -- it is just crucial here) the present, the past does not contribute more information for predicting the future. That is the past and the future may not be independent at all, but the past and the future are conditionally independent given the present. To discuss the Markov property, desperately need the concept of conditional independence -- there is no substitute, no easier terminology or description. Alas, conditional independence is not so easy to discuss, even in the simplest cases. And in the grown up cases, need the Radon-Nikodym theorem of measure theory and the associated machinery of conditional expectation. Tellingly, the part I quoted never mentioned conditional independence or even conditioning. Sorry, again, the quote was wrong. I tried to correct the quote and, thus, keep readers trying to learn from being misled. My sympathies are with the readers trying to learn. Early in my career, I read lots of such elementary, easy to understand introductions to lots of topics in applied math. The biggie problem was, as here, the content was from not very good down to actually wrong. Later I got a first class, rock solid foundation in grad school. In particular my background in Markov processes was from a star student of E. Cinlar, a world class expert long at Princeton. And my Ph.D. dissertation was in stochastic optimal control where the Markov assumption is just crucial -- indeed, stochastic optimal control is also commonly called Markov decision processes. In addition I've applied Markov processes in some serious work in US national security. What I'm saying here is on the center of the target. What I quoted no one should try to learn from -- it's wrong.
      • mturmon 10y ago
        This is quite correct.
• mturmon 10y ago
  Out of a pretty big slide deck, explicitly intended for a beginner audience, this is a weird thing to pick on. The following is "Proposition 1.1" from the relevant section of a standard text (Bhattacharya and Waymire): "A stochastic process X0, X1, X2, ... has the Markov property if and only if for each n the conditional distribution of Xn+1 given X0...Xn is a function only of Xn." The only difference between the above Prop. 1.1 and the statement in TFA is that TFA omits to specify it means a conditional distribution of Xt+1 given X1...Xt (as opposed to an unconditional distribution). But, in literally the following line, TFA goes on to clarify what it means by stating the equation that the words correspond to, and the equation has the correct conditional distribution. I don't see reason for alarm here.