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4. Markov Chains - Statistics

4. Markov Chains A discrete time process{Xn, n= 0,1,2, ..}with discretestate spaceXn {0,1,2, ..}is aMarkov chainif it has theMarkov property:P[Xn+1=j|Xn=i, Xn 1=in 1, .. , X0=i0]=P[Xn+1=j|Xn=i] In words, the past is conditionally independentof the future given the present state of theprocess or given the present state, the pastcontains no additional information on the futureevolution of the system. The Markov property is common in probabilitymodels because, by assumption, one supposesthat the important variables for the system beingmodeled are all included in the state space. We considerhomogeneousMarkov Chains forwhichP[Xn+1=j|Xn=i] =P[X1=j|X0=i].1 Example:physical systems. If the state spacecontains the masses, velocities and accelerations ofparticles subject to Newton s laws of mechanics,the system in Markovian (but not random!)

Example: physical systems.If the state space contains the masses, velocities and accelerations of particles subject to Newton’s laws of mechanics, the system in Markovian (but not random!)

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