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Maximum a Posterior

Maximum a Posterior

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Maximum a Posterior

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  1. Maximum a Posterior Presented by 陳燦輝

  2. Maximum a Posterior • Introduction • MAP for Discrete HMM • Prior Dirichlet • MAP for Semi-Continuous HMM • Prior Dirichlet + normal-Wishart • Segmental MAP Estimates • Conclusion • Appendix-Matrix Calculus

  3. Introduction • HMM parameter estimators have been derived purely from the training observation sequences without any prior information included. • There may be many cases in which the prior information about the parameters is available, ex : previous experience

  4. Introduction (cont)

  5. Discrete HMM Definition :

  6. Discrete HMM Q-function :

  7. Discrete HMM Q-function :

  8. Discrete HMM Q-function : 同理

  9. Discrete HMM R-function :

  10. Discrete HMM Q

  11. Discrete HMM Initial probability

  12. Discrete HMM Transition probability

  13. Discrete HMM observation probability

  14. Discrete HMM

  15. Discrete HMM • How to choose the initial estimate for ? • One reasonable choice of the initial estimate is the mode of the prior density.

  16. Discrete HMM • What’s the mode ? • So applying Lagrange Multiplier we can easily derive above modes. • Example :

  17. Discrete HMM • Another reasonable choice of the initial estimate is the mean of the prior density. • Both are some kind of summarization of the available information about the parameters before any data are observed.

  18. SCHMM

  19. SCHMM independent

  20. Model 1 Model 2 Model M SCHMM

  21. SCHMM Q-function :

  22. SCHMM Q-function :

  23. SCHMM

  24. SCHMM Initial probability • Differentiating w.r.t and equate it to zero.

  25. SCHMM Transition probability • Differentiating w.r.t and equate it to zero.

  26. SCHMM Mixture weight • Differentiating w.r.t and equate it to zero.

  27. SCHMM • Differentiating w.r.t and equate it to zero. • Differentiating w.r.t and equate it to zero.

  28. SCHMM • Full Covariance matrix case :

  29. SCHMM • Full Covariance matrix case :

  30. SCHMM • Full Covariance matrix case :

  31. SCHMM • Full Covariance matrix case : (1) (2) (3)

  32. SCHMM • Full Covariance matrix case :

  33. SCHMM Full Covariance • The initial estimate can be chosen as the mode of the prior PDF • And also can be chosen as the mean of the prior PDF

  34. SCHMM • Diagonal Covariance matrix case : • Then and

  35. SCHMM • Diagonal Covariance matrix case :

  36. SCHMM Diagonal Covariance • Diagonal Covariance matrix case :

  37. SCHMM Diagonal Covariance • Diagonal Covariance matrix case :

  38. SCHMM • Diagonal Covariance matrix case :

  39. SCHMM • Diagonal Covariance matrix case : (1) (2) (3)

  40. SCHMM Diagonal Covariance • Diagonal Covariance matrix case :

  41. SCHMM Diagonal Covariance • The initial estimate can be chosen as the mode of the prior PDF • And also can be chosen as the mean of the prior PDF

  42. Segmental MAP Estimates

  43. Segmental MAP Estimates DHMM

  44. Segmental MAP Estimates SCHMM

  45. Conclusion • The important issue of prior density is discussed. • Some application : • Model adaptation, HMM training, IR(?)

  46. Appendix-Matrix Calculus(1) • Notation:

  47. Appendix-Matrix Calculus(2) • Properties 1: • proof • Properties 1— Extension: • proof

  48. Appendix-Matrix Calculus(3) • Properties 2: • proof

  49. Appendix-Matrix Calculus(4) • Properties 3: • proof

  50. Appendix-Matrix Calculus(5) • Properties 4: • proof