[lambdaphage]

The Markov property does not require dependence on the previous state-- it requires that the distribution of the next state given the history of the chain be the distribution of the next state given the current state. The lottery trivially satisfies this property since tomorrow's winner doesn't depend on today's winner at all.

[ltjohnson]

This is correct. Sadly, the first paragraph of the article contains some glaring errors.

"For Markov chains to be effective the current state has to be dependent on the previous state in some way;"

This is trivially untrue. A sequence of independently and identically distributed (iid) random variables is a Markov chain. An iid sequence is clearly effective at many things (e.g. Monte Carlo integration).

"Not every process has the Markov Property, such as the Lottery, this weeks winning numbers have no dependence to the previous weeks winning numbers." As lambdaphage pointed out, the Lottery does have the Markov property.

[kernel_sanders]

I'm interested in the above article and the above two comments, however, I don't understand the Lottery example. Can you clarify how it does have the Markov property?

I'm not seeing how the distribution of possible winning numbers relates at all to the current state. I'm trying to phrase this in the language of the above two comments. Help me out if I've got it all wrong. =)

[tlarkworthy]

The Markov property is that you can model the system as being dependent on the immediately previous state + noise.

The lottery ignores the previous state and is defined purely by the noise, so it is a (trivial) markov process.

More complex systems depend on the entire history (e.g. to model a poker player you have to consider all of their actions up to the current). Newtonian systems are markov, if you know the state of the system you can run it forward in time deterministically. Even if your knowledge of the state of the Newtonian system is not fully known, you can still run the distribution of states forward in time precisely.

current_state = previous_state + process_noise

typically expressed in matrix math but the idea is as simple as that.