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Math Help - Correlation in Markov Chains

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    Correlation in Markov Chains

    I have a n dimensional Markov Chain

    There are n number of nodes and the values in a state basically indicate the condition of each of the nodes. Each node can have 2 values either 0 or 1.
    e.g. 0001 (Indicates that there are 4 nodes and the fourth node is in the OFF state (0=On, 1=OFF)

    For reference, I have attached a 2D Markov chain. every state is connected to every other state.

    Considering the attached diagram, I have to assign the transition probability values, but there is correlation among some of the nodes which lie in the same group. e.g. if nodes 3 and 4 lie in the same group, and the fourth node goes to off state then the probability of the third node going to OFF state as well is high, since they lie in the same group. i.e. (Probability of 0001->0011 is high since the nodes 3 and 4 lie in the same group.) How can i represent this in the Markov Chain.

    One way probably to do this using conditional probability. I have no clear idea on how to introduce this correlation.

    Could some one help me with this?

    Thanks,
    KOustubh
    Attached Thumbnails Attached Thumbnails Correlation in Markov Chains-math.png  
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