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Instructor: Eyal Amir Grad TAs : Wen Pu, Yonatan Bisk Undergrad TAs : Sam Johnson, Nikhil Johri

CS 440 / ECE 448 Introduction to Artificial Intelligence Spring 2010 Lecture #23. Instructor: Eyal Amir Grad TAs : Wen Pu, Yonatan Bisk Undergrad TAs : Sam Johnson, Nikhil Johri. Today & Thursday. Time and uncertainty Inference: filtering, prediction, smoothing Hidden Markov Models (HMMs)

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Instructor: Eyal Amir Grad TAs : Wen Pu, Yonatan Bisk Undergrad TAs : Sam Johnson, Nikhil Johri

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  1. CS 440 / ECE 448Introduction to Artificial IntelligenceSpring 2010Lecture #23 Instructor: Eyal Amir Grad TAs: Wen Pu, Yonatan Bisk Undergrad TAs: Sam Johnson, Nikhil Johri

  2. Today & Thursday • Time and uncertainty • Inference: filtering, prediction, smoothing • Hidden Markov Models (HMMs) • Model • Exact Reasoning

  3. Time and Uncertainty • Standard Bayes net model: • Static situation • Fixed (finite) random variables • Graphical structure and conditional independence • In many systems, data arrives sequentially • Dynamic Bayes nets (DBNs) and HMMs model: • Processes that evolve over time

  4. Example (Robot Position) Pos2 Pos3 Pos1 Sensor 1 Sensor 3 Sensor2 Vel 1 Vel 2 Vel 3 Sensor1 Sensor3 Sensor 2

  5. Robot Position(With Observations) Pos2 Pos3 Pos1 Sens.A 1 Sens.A3 Sens.A2 Vel 1 Vel 2 Vel 3 Sens.B1 Sens.B3 Sens.B 2

  6. Inference Problem • State of the System at time t: • Probability distribution over states: • A lot of parameters

  7. Solution (Part 1) • Problem: • Solution: Markov Assumption • Assume is independent of given • State variables are expressive enough to summarize all relevant information about past • Therefore:

  8. Solution (Part 2) • Problem: • If all are different • Solution: • Assume all are the same • The process is time-invariant or stationary

  9. Inference in Robot Position DBN • Compute distribution over true position and velocity • Given a sequence of sensor values • Belief state: • Probability distribution over different states at each time step • Update belief state when a new set of sensor readings arrive

  10. Example • First order Markov assumption not exactly true in real world

  11. Example • Possible fixes: • Increase order of Markov process • Augment state, e.g., add Temp, Pressure Or battery to position and velocity

  12. Today • Time and uncertainty • Inference: filtering, prediction, smoothing • Hidden Markov Models (HMMs) • Model • Exact Reasoning • Dynamic Bayesian Networks • Model • Exact Reasoning

  13. Inference Tasks • Filtering: • Belief state: probability of state given the evidence • Prediction: • Like filtering without evidence • Smoothing: • Better estimate of past states • Most likelihood explanation: • Scenario that explains the evidence

  14. Filtering (forward algorithm) Xt+1 Xt Xt-1 Et-1 Et+1 Update: Et Predict: Recursive step

  15. Example

  16. Smoothing Forward backward

  17. SmoothingBackWard Step

  18. Most Likely Explanation • Finding most likely path Xt+1 Xt Xt-1 Et-1 Et+1 Et Most likely path to xt Plus one more update

  19. Most Likely Explanation • Finding most likely path Xt+1 Xt Xt-1 Et-1 Et+1 Et Called Viterbi

  20. Viterbi(Example)

  21. Viterbi(Example)

  22. Viterbi(Example)

  23. Viterbi(Example)

  24. Viterbi(Example)

  25. Today • Time and uncertainty • Inference: filtering, prediction, smoothing, MLE • Hidden Markov Models (HMMs) • Model • Exact Reasoning • Dynamic Bayesian Networks • Model • Exact Reasoning

  26. X1 X2 X3 Y1 Y3 Y2 Sparse transition matrix ) sparse graph Hidden Markov model (HMM) “True” state Phones/ words Noisy observations acoustic signal transitionmatrix Diagonal Matrix

  27. Forwards algorithm for HMMs Predict: Update:

  28. at|t-1 Xt+1 Xt-1 Xt bt+1 bt Yt-1 Yt+1 Yt Message passing view of forwards algorithm

  29. Forwards-backwards algorithm bt at|t-1 Xt-1 Xt Xt+1 bt Yt-1 Yt+1 Yt

  30. If Have Time… • Time and uncertainty • Inference: filtering, prediction, smoothing • Hidden Markov Models (HMMs) • Model • Exact Reasoning • Dynamic Bayesian Networks • Model • Exact Reasoning

  31. Dynamic Bayesian Network • DBN is like a 2time-BN • Using the first order Markov assumptions Time 0 Time 1 Standard BN Standard BN

  32. Dynamic Bayesian Network • Basic idea: • Copy state and evidence for each time step • Xt: set of unobservable (hidden) variables (e.g.: Pos, Vel) • Et: set of observable (evidence) variables (e.g.: Sens.A, Sens.B) • Notice: Time is discrete

  33. Example

  34. Inference in DBN Unroll: Inference in the above BN Not efficient (depends on the sequence length)

  35. T T(t+1) T(t+1) T 0.91 0.09 F 0.0 1.0 DBN Representation: DelC RHM R(t+1) R(t+1) T 1.0 0.0 F 0.0 1.0 RHMt RHMt+1 fRHM(RHMt,RHMt+1) Mt Mt+1 fT(Tt,Tt+1) Tt Tt+1 L CR RHC CR(t+1) CR(t+1) O T T 0.2 0.8 E T T 1.0 0.0 O F T 0.0 1.0 E F T 0.0 1.0 O T F 1.0 0.1 E T F 1.0 0.0 O F F 0.0 1.0 E F F 0.0 1.0 Lt Lt+1 CRt CRt+1 RHCt RHCt+1 fCR(Lt,CRt,RHCt,CRt+1)

  36. RHMt RHMt+1 Mt Mt+1 Tt Tt+1 s1 s2 ... s160 Lt Lt+1 s1 0.9 0.05 ... 0.0 s2 0.0 0.20 ... 0.1 . . . s160 0.1 0.0 ... 0.0 CRt CRt+1 RHCt RHCt+1 Benefits of DBN Representation Pr(Rmt+1,Mt+1,Tt+1,Lt+1,Ct+1,Rct+1 | Rmt,Mt,Tt,Lt,Ct,Rct) = fRm(Rmt,Rmt+1) * fM(Mt,Mt+1) * fT(Tt,Tt+1) * fL(Lt,Lt+1) * fCr(Lt,Crt,Rct,Crt+1) * fRc(Rct,Rct+1) • Only few parameters vs. • 25440 for matrix • Removes global exponential • dependence

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