1. Machine Learning 4 Machine Learning 4 Hidden Markov Models

2. The Problem to Be Solved

3. More Specifically Given a sequence of acoustic observations Most probable sequence of words Corresponding to speaker’s intent

4. Two Items Sequence The signal is observable, the output is not.

5. Framed a Different Way: The Ice Cream Task A climatologist in 2799 wants to reconstruct the weather in Baltimore during 2012 Baltimore is now under water Jacob Eisner, who lived in Baltimore in the early 21 st century kept a diary. His diary, through much historical drama, became the property of the Missouri Historical Society, a short walk from Washington University where the climatologist works. This diary, besides containing lots of dreary stuff about emotional states, contains a record of how many ice cream cones Jason ate each day that summer. What was the sequence of hot and cold days during the eventful summer of 2012?

6. Note two items: Sequence: ice cream comes Observation: sequence of ice cream cones Hidden: sequence of hot and cold days We presume: There is a probabilistic relationship between the sequence of ice cones and the sequence of hot and cold days

7. Dr. Eisner 2012 (not eating ice cream)

8. Dr. Markov circa 1900 (not eating ice cream either)

9. Model of Newspaper Vending Machine as FSA

10. Markov Chains Each a ij is an index into a table Gives transition probabilities

11. Weather Model from Luger (p. 375) S1 = sunny, s2 = cloudy, s3 = foggy, s4 = rainy

12. Invented Gender/Handedness data Male (M)Female (F)Total Left (L)5813 Right (R)347 Total81220

13. As a Hidden Markov Model

14. P(LLL)

15. (!) There must be a better way P(LLL) = (.625 * 625 *.625 *.4 *.4 *.4) + (.625 *.625 *.667 *.4 *.4 *.6) + (.625 *.667 *.625 *.4 *.6 *.4) (.625 *.667 *.667 *.4 *.6 *.6) + (.667 *.625 *.625 *.6 *.4 *.4) + (.667 *.625 *.625 *.6 *.4 *.6) + (.667 *.667 *.625 *.6 *.6 *.4) + (.667 *.667 *.667 *.6 *.6 *.6) =.015625 +.0250125 +.0250125 +.02669334 +.0250125 +.03751875 +.04004001 +.064096048 =.259010648

16. The Ice Cream HMM

17. A: priors matrix StartHotCold Start 0.0.8.2 Hot 0.0.7.3 Cold 0.0.4.6 B: likelihoods matrix 1 Cone2 Cones3 Cones Hot.2.4 Cold.5.4.1 Ice Cream Task Rows labeled by prior state/conditioning event

18. A: priors matrix startfemalemale start 0.0.6.4 female 0.0.6.4 male 0.0.6.4 B: likelihoods matrix leftright Female.67.33 Male.625.375 Gender Task Rows labeled by prior state/conditioning event

19. Forward Algorithm

20. Forward Trellis b j (o t ).0464