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.
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.