CMPSCI 240: Reasoning Under Uncertainty
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1 CMPSCI 240: Reasoning Under Uncertainty Lecture 24 Prof. Hanna Wallach April 24, 2012
2 Reminders Check the course website: ~wallach/courses/s12/cmpsci240/ Eighth homework is due on Friday
3 Grades 1. Add discussion section scores, divide by 14, multiply by Add homework scores, divide by 250, multiply by Divide midterm score by 100, multiply by Add 1 3 to obtain your score (max. possible is 70) 5. If your score is 40 or less, you re in danger of getting a D
4 Recap
5 Last Time: Matrices and Vectors for Markov Chains Transition probability matrix: A = p 11 p 12 p p 1S p 21 p 21 p p 2S p S1 p S2 p S3... p SS
6 Last Time: Matrices and Vectors for Markov Chains Transition probability matrix: A = p 11 p 12 p p 1S p 21 p 21 p p 2S p S1 p S2 p S3... p SS State probability vector: v (t) with v (t) i = P(X t =i)
7 Last Time: Matrices and Vectors for Markov Chains Transition probability matrix: A = p 11 p 12 p p 1S p 21 p 21 p p 2S p S1 p S2 p S3... p SS State probability vector: v (t) with v (t) i = P(X t =i) n-step transition probabilities: v (n) = v (0) A n
8 Last Time: Steady State If va = v then v is a steady state distribution
9 Last Time: Steady State If va = v then v is a steady state distribution Most Markov chains have a unique steady state distribution that is approached by successive time steps (applications of transition matrix A) from any starting distribution
10 Last Time: Steady State If va = v then v is a steady state distribution Most Markov chains have a unique steady state distribution that is approached by successive time steps (applications of transition matrix A) from any starting distribution va = v defines a set of n + 1 simultaneous equations
11 Examples of Steady State e.g., draw the transition probability graph and find the steady state distribution for a Markov chain with 4 states and A =
12 Irreducible Markov Chains The steady state distribution is v = 0, 0, 0.375, 0.625
13 Irreducible Markov Chains The steady state distribution is v = 0, 0, 0.375, There is no path from states 3 or 4 to states 1 or 2
14 Irreducible Markov Chains The steady state distribution is v = 0, 0, 0.375, There is no path from states 3 or 4 to states 1 or 2 A Markov chain is irreducible if there exists a path in the transition graph from every state to every other state
15 Irreducible Markov Chains The steady state distribution is v = 0, 0, 0.375, There is no path from states 3 or 4 to states 1 or 2 A Markov chain is irreducible if there exists a path in the transition graph from every state to every other state If a Markov chain is not irreducible, then it is reducible
16 Communicating Classes State i communicates with state j, i.e., i j, if there is some way of reaching state j from state i and vice versa
17 Communicating Classes State i communicates with state j, i.e., i j, if there is some way of reaching state j from state i and vice versa A set of states C is a communicating class if i j for every i, j C and no i C communicates with some j / C
18 Communicating Classes State i communicates with state j, i.e., i j, if there is some way of reaching state j from state i and vice versa A set of states C is a communicating class if i j for every i, j C and no i C communicates with some j / C C is closed if the probability of leaving it is zero
19 Communicating Classes State i communicates with state j, i.e., i j, if there is some way of reaching state j from state i and vice versa A set of states C is a communicating class if i j for every i, j C and no i C communicates with some j / C C is closed if the probability of leaving it is zero A Markov chain is irreducible if its state space forms a single communicating class, i.e., i, j C for all i, j S
20 Examples of Communicating Classes e.g., how many communicating classes are there if A =
21 Examples of Communicating Classes e.g., how many communicating classes are there if A = e.g., what does this say about the reducibility of the chain?
22 Examples of Steady State e.g., draw the transition probability graph and find the steady state distribution for a Markov chain with 4 states and A =
23 Examples of Steady State e.g., draw the transition probability graph and find the steady state distribution for a Markov chain with 4 states and A = e.g., is this Markov chain irreducible?
24 Periodic Markov Chains v (0) = 0.250, 0.250, 0.250, 0.250
25 Periodic Markov Chains v (0) = 0.250, 0.250, 0.250, v (1) = 0.375, 0.312, 0.125, 0.188
26 Periodic Markov Chains v (0) = 0.250, 0.250, 0.250, v (1) = 0.375, 0.312, 0.125, v (2) = 0.375, 0.281, 0.125, 0.219
27 Periodic Markov Chains v (0) = 0.250, 0.250, 0.250, v (1) = 0.375, 0.312, 0.125, v (2) = 0.375, 0.281, 0.125, v (3) = 0.375, 0.281, 0.125, 0.219
28 Periodic Markov Chains v (0) = 0.250, 0.250, 0.250, v (1) = 0.375, 0.312, 0.125, v (2) = 0.375, 0.281, 0.125, v (3) = 0.375, 0.281, 0.125, v (4) = 0.375, 0.281, 0.125, 0.219
29 Periodic Markov Chains v (0) = 0.250, 0.250, 0.250, v (1) = 0.375, 0.312, 0.125, v (2) = 0.375, 0.281, 0.125, v (3) = 0.375, 0.281, 0.125, v (4) = 0.375, 0.281, 0.125,
30 Periodic Markov Chains v (0) = 1.000, 0.000, 0.000, 0.000
31 Periodic Markov Chains v (0) = 1.000, 0.000, 0.000, v (1) = 0.000, 0.500, 0.000, 0.500
32 Periodic Markov Chains v (0) = 1.000, 0.000, 0.000, v (1) = 0.000, 0.500, 0.000, v (2) = 0.750, 0.000, 0.250, 0.000
33 Periodic Markov Chains v (0) = 1.000, 0.000, 0.000, v (1) = 0.000, 0.500, 0.000, v (2) = 0.750, 0.000, 0.250, v (3) = 0.000, 0.562, 0.000, 0.438
34 Periodic Markov Chains v (0) = 1.000, 0.000, 0.000, v (1) = 0.000, 0.500, 0.000, v (2) = 0.750, 0.000, 0.250, v (3) = 0.000, 0.562, 0.000, v (4) = 0.750, 0.000, 0.250, 0.000
35 Periodic Markov Chains v (0) = 1.000, 0.000, 0.000, v (1) = 0.000, 0.500, 0.000, v (2) = 0.750, 0.000, 0.250, v (3) = 0.000, 0.562, 0.000, v (4) = 0.750, 0.000, 0.250, v (5) = 0.000, 0.562, 0.000, 0.438
36 Periodic Markov Chains v (0) = 1.000, 0.000, 0.000, v (1) = 0.000, 0.500, 0.000, v (2) = 0.750, 0.000, 0.250, v (3) = 0.000, 0.562, 0.000, v (4) = 0.750, 0.000, 0.250, v (5) = 0.000, 0.562, 0.000, v (6) = 0.750, 0.000, 0.250, 0.000
37 Periodic Markov Chains v (0) = 1.000, 0.000, 0.000, v (1) = 0.000, 0.500, 0.000, v (2) = 0.750, 0.000, 0.250, v (3) = 0.000, 0.562, 0.000, v (4) = 0.750, 0.000, 0.250, v (5) = 0.000, 0.562, 0.000, v (6) = 0.750, 0.000, 0.250, v (7) = 0.000, 0.562, 0.000, 0.438
38 Periodic Markov Chains v (0) = 1.000, 0.000, 0.000, v (1) = 0.000, 0.500, 0.000, v (2) = 0.750, 0.000, 0.250, v (3) = 0.000, 0.562, 0.000, v (4) = 0.750, 0.000, 0.250, v (5) = 0.000, 0.562, 0.000, v (6) = 0.750, 0.000, 0.250, v (7) = 0.000, 0.562, 0.000, v (8) = 0.750, 0.000, 0.250,
39 Periodic Markov Chains Chain is always in 1 or 3 on even t and 2 or 4 on odd t
40 Periodic Markov Chains Chain is always in 1 or 3 on even t and 2 or 4 on odd t X = {1, 3} Y = {2, 4} and vice versa
41 Periodic Markov Chains Chain is always in 1 or 3 on even t and 2 or 4 on odd t X = {1, 3} Y = {2, 4} and vice versa If the chain is in a state in X at time t, then at time t + 2 it must return to a state in X ; the same is true for Y
42 Periodic Markov Chains A state i is periodic if the probability of returning to i is zero except at regular intervals (e.g., every 2 time steps)
43 Periodic Markov Chains A state i is periodic if the probability of returning to i is zero except at regular intervals (e.g., every 2 time steps) If all states are periodic, then the chain is periodic
44 Periodic Markov Chains A state i is periodic if the probability of returning to i is zero except at regular intervals (e.g., every 2 time steps) If all states are periodic, then the chain is periodic An irreducible Markov chain is periodic if there is some k > 1 such that A k is the transition matrix of a reducible chain
45 Steady State Periodic: chain moves from group to group in a periodic way
46 Steady State Periodic: chain moves from group to group in a periodic way Reducible: some states can t be reached from others
47 Steady State Periodic: chain moves from group to group in a periodic way Reducible: some states can t be reached from others Steady state: if a Markov chain is aperiodic and irreducible then there must be some v such that for some t P(X t =i) v i ɛ for any starting distribution and any positive, real ɛ
48 For Next Time Check the course website: ~wallach/courses/s12/cmpsci240/ Eighth homework is due on Friday
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