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I know what Markov Chains are (the probabilities for the next state of the chain is dependent only on the current state) What I would prefer to know is why are

I know what Markov Chains are (the probabilities for the next state of the chain is dependent only on the current state) What I would prefer to know is why are they more useful than a more nuanced model with non-fixed probabilities, or memory of previous states? - given that these are not really harder to simulate. It seems to me that Markov Chains are very often used as an inappropriate simplification. Is there some mathematical advantage to using them?

coherentpony 12y ago
What you are referring to are non-Markov processes. That is, process with a memory. These can be useful too, but the reason Markovian processes are preferred is because of the independence (lack of memory). Independence is a sufficient condition for many theorems that use Markov processes to prove things about average behaviour.
- learnstats2 12y ago
  Yes, I understand. But why does a lack of memory help? What useful theorems are there?
dangerlibrary 12y ago
The formal definition of a single Markov Chain is one with a fixed transition matrix. In practice, many software implementations of Markov models update/modify the transition matrix as new information becomes available. Formally, you're no longer working with the same Markov chain, but that doesn't mean that the model isn't useful.
- learnstats2 12y ago
  Right; I don't think this should be described as a Markov Chain variant, since it breaks the fundamental property of a Markov Chain; Any old simulation works in this way.
thomaspaine 12y ago
Markov chains have a lot of nice features you can derive from them, like convergence to a stationary distribution. It's hard to say that it's an inappropriate simplification if it happens to be pragmatic (like most simplifications). For example PageRank can be modeled using markov chains. You can surely develop more complicated models but markov chains provide a good starting point.
IndianAstronaut 12y ago
Markov chains easily allow you to attach cost functions to them. Not particularly data mining oriented, but good for running simulations and cost-benefit.

coherentpony 12y ago

What you are referring to are non-Markov processes. That is, process with a memory. These can be useful too, but the reason Markovian processes are preferred is because of the independence (lack of memory). Independence is a sufficient condition for many theorems that use Markov processes to prove things about average behaviour.

learnstats2 12y ago
Yes, I understand. But why does a lack of memory help? What useful theorems are there?

dangerlibrary 12y ago

The formal definition of a single Markov Chain is one with a fixed transition matrix. In practice, many software implementations of Markov models update/modify the transition matrix as new information becomes available. Formally, you're no longer working with the same Markov chain, but that doesn't mean that the model isn't useful.

learnstats2 12y ago
Right; I don't think this should be described as a Markov Chain variant, since it breaks the fundamental property of a Markov Chain; Any old simulation works in this way.

thomaspaine 12y ago

Markov chains have a lot of nice features you can derive from them, like convergence to a stationary distribution. It's hard to say that it's an inappropriate simplification if it happens to be pragmatic (like most simplifications). For example PageRank can be modeled using markov chains. You can surely develop more complicated models but markov chains provide a good starting point.

IndianAstronaut 12y ago

Markov chains easily allow you to attach cost functions to them. Not particularly data mining oriented, but good for running simulations and cost-benefit.