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11. Markov Chains (MCs) 2

11. Markov Chains (MCs) 2. Courtesy of J. Bard, L. Page, and J. Heyl. 11.2.1 n-step transition probabilities (review). Transition prob. matrix. n-step transition prob. from state i to j is n-step transition matrix (for all states) is then For instance, two step transition matrix is.

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11. Markov Chains (MCs) 2

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  1. 11. Markov Chains (MCs) 2 Courtesy of J. Bard, L. Page, and J. Heyl

  2. 11.2.1 n-step transition probabilities (review)

  3. Transition prob. matrix • n-step transition prob. from state i to j is • n-step transition matrix (for all states) is then • For instance, two step transition matrix is

  4. Chapman-Kolmogorov equations • Prob. of going from state i at t=0, passing though statek at t=m, and ending at state j at t=m+n is • In matrix notation,

  5. 11.2.2 state probabilities

  6. State probability (pmf of an RV!) • Let p(n) = {pj(n)}, for all jE, be the row vector of state probs. at time n (i.e., state prob. vector) • Thus, p(n) is given by • From the initial state • In matrix notation

  7. How an MC changes (Ex 11.10, 11.11) 0.9 0.1 0.8 • A two-state system Silence (state 0) Speech (state 1) 0.2 Suppose p(0)=(1,0) Suppose p(0)=(0,1) p(1) = p(0)P = (0.9, 0.1) Then p(1) = p(0)P= (0,1)P = (0.2, 0.8) p(2) = (1,0)P2 = (0.83, 0.17) p(2)= (0.2,0.8)P= (0,1)P2 = (0.34, 0.66) p(4)= (1,0)P4 = (0.747, 0.253) p(4)= (0,1)P4 = (0.507, 0.493) p(8)= (1,0)P8 = (0.686, 0.314) p(8)= (0,1)P8 = (0.629, 0.371) p(16)= (1,0)P16 = (0.668, 0.332) p(16)= (0,1)P16 = (0.665, 0.335) p(32)= (1,0)P32 = (0.667, 0.333) p(32)= (0,1)P32 = (0.667, 0.333) p(64)= (1,0)P64 = (0.667, 0.333) p(64)= (0,1)P64 = (0.667, 0.333)

  8. Independence of initial condition

  9. The lesson to take away • No matter what assumptions you make about the initial probability distribution, • after a large number of steps, • the state probability distribution is approximately (2/3, 1/3) See p.666, 667

  10. 11.2.3 steady state probabilities

  11. State probabilities (pmf) converge • As n, then transition prob. matrix Pn approaches a matrix whose rows are equal to the same pmf. • In matrix notation, • where 1 is a column vector of all 1’s, and =(0, 1, … ) • The convergence of Pn implies the convergence of the state pmf’s

  12. Steady state probability • System reaches “equilibrium” or “steady state”, • i.e., n, pj(n)  j, pi(n-1)  i • In matrix notation, • here  is stationary state pmf of the Markov chain • To solve this,

  13. 0.90.1 0.20.8 Speech activity system  = P • From the steady state probabilities (1, 2) = (1, 2) 1 = 0.91 + 0.12 2 = 0.21 + 0.82 1 + 2 = 1 1 = 2/3= 0.667 2 = 1/3= 0.333

  14. Question 11-1: Alice, Bob and Carol are playing Frisbee. Alice always throws to Carol. Bob always throws to Alice. Carol throws to Bob 2/3 of the time and to Alice 1/3 of the time. In the long run, what percentage of the time do each of the players have the Frisbee?

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