MIT16_410F10_lec21

MIT16_410F10_lec21 - 16.410/413 Principles of Autonomy and...

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Unformatted text preview: 16.410/413 Principles of Autonomy and Decision Making Lecture 21: Intro to Hidden Markov Models the Baum-Welch algorithm Emilio Frazzoli Aeronautics and Astronautics Massachusetts Institute of Technology November 24, 2010 E. Frazzoli (MIT) Lecture 21: HMMs November 24, 2010 1 / 23 Assignments Readings Lecture notes [AIMA] Ch. 15.1-3, 20.3. Paper on Stellar: L. Rabiner, A tutorial on Hidden Markov Models... E. Frazzoli (MIT) Lecture 21: HMMs November 24, 2010 2 / 23 Outline 1 Decoding and Viterbis algorithm 2 Learning and the Baum-Welch algorithm E. Frazzoli (MIT) Lecture 21: HMMs November 24, 2010 3 / 23 Decoding Filtering and smoothing produce distributions of states at each time step. Maximum likelihood estimation chooses the state with the highest probability at the best estimate at each time step. However, these are pointwise best estimate: the sequence of maximum likelihood estimates is not necessarily a good (or feasible) trajectory for the HMM! How do we find the most likely state history , or state trajectory ? (As opposed to the sequence of point-wise most likely states?) E. Frazzoli (MIT) Lecture 21: HMMs November 24, 2010 4 / 23 Example: filtering/smoothing vs. decoding 1/4 Three states: X = { x 1 , x 2 , x 3 } . Three possible observations: Z = { 2 , 3 } . Initial distribution: = (1 , , 0). Transition probabilities: T = 0 0 . 5 0 . 5 0 0 . 9 0 . 1 1 Observation probabilities: M = . 5 0 . 5 . 9 0 . 1 . 1 0 . 9 x 1 x 2 x 3 0.5 0.5 0.9 0.1 1 Observation sequence: Z = (2 , 3 , 3 , 2 , 2 , 2 , 3 , 2 , 3) . E. Frazzoli (MIT) Lecture 21: HMMs November 24, 2010 5 / 23 Example: filtering/smoothing vs. decoding 2/4 Using filtering: t x 1 x 2 x 3 1 1.0000 2 0.1000 0.9000 3 0.0109 0.9891 4 0.0817 0.9183 5 0.4165 0.5835 6 0.8437 0.1563 7 0.2595 0.7405 8 0.7328 0.2672 9 0.1771 0.8229 The sequence of point-wise most likely states is: (1 , 3 , 3 , 3 , 3 , 2 , 3 , 2 , 3) . The above sequence is not feasible for the HMM model! E. Frazzoli (MIT) Lecture 21: HMMs November 24, 2010 6 / 23 Example: filtering vs. smoothing vs. decoding 3/4 Using smoothing: t x 1 x 2 x 3 1 1.0000 2 0.6297 0.3703 3 0.6255 0.3745 4 0.6251 0.3749 5 0.6218 0.3782 6 0.5948 0.4052 7 0.3761 0.6239 8 0.3543 0.6457 9 0.1771 0.8229 The sequence of point-wise most likely states is: (1 , 2 , 2 , 2 , 2 , 2 , 3 , 3 , 3) . E. Frazzoli (MIT) Lecture 21: HMMs November 24, 2010 7 / 23 Viterbis algorithm As before, let us use the Markov property of the HMM. Define k ( s ) = max X 1:( k- 1) Pr X 1: k = ( X 1:( k- 1) , s ) , Z 1: k | (i.e., k ( s ) is the joint probability of the most likely path that ends at state s at time k , generating observations Z 1: k .) Clearly, k +1 ( s ) = max q ( k ( q ) T q , s ) M s , z k +1 This can be iterated to find the probability of the most likely path...
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This note was uploaded on 12/26/2011 for the course SCIENCE 16.410 taught by Professor Prof.brianwilliams during the Fall '10 term at MIT.

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MIT16_410F10_lec21 - 16.410/413 Principles of Autonomy and...

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