1. Ch 13. Sequential Data (1/2) Pattern Recognition and Machine Learning, C. M. Bishop, 2006.
Summarized by Kim Jin-young
Biointelligence Laboratory, Seoul National University
많이 부족하지만 열심히 하겠습니다.^^

2. (C) 2007, SNU Biointelligence Lab, http://bi.snu.ac.kr/
Contents
13.1 Markov Models
13.2 Hidden Markov Models
Maximum likelihood for the HMM
The forward-backward algorithm
The sum-product algorithm for the HMM
Scaling factors
The Viterbi Algorithm
Extensions of the HMM
(C) 2007, SNU Biointelligence Lab, 

3. (C) 2007, SNU Biointelligence Lab, http://bi.snu.ac.kr/
Sequential Data
Data dependency exists according to a sequence
Weather data, DNA, characters in sentence
i.i.d. assumption doesn’t hold
Sequential Distribution
Stationary vs. Nonstationary
Markov Model
No latent variable
State Space Models
Hidden Markov Model (discrete latent variables)
Linear Dynamical Systems
(C) 2007, SNU Biointelligence Lab, 

4. (C) 2007, SNU Biointelligence Lab, http://bi.snu.ac.kr/
Markov Models
Markov Chain
State Space Model
(free of Markov assumption of any order with reasonable no. of extra parameters)
(C) 2007, SNU Biointelligence Lab, 

5. Hidden Markov Model (overview)
Introduction of discrete latent vars. (based on prior knowledge)
Examples
Coin toss
Urn and ball
Conditional Random Field
MRF globally conditioned by observation sequence X
CRF relaxes independence assumption by HMM
(C) 2007, SNU Biointelligence Lab, 

6. Hidden Markov Model (example)
Lattice Representation
Left-to-right HMM
<Handwriting Recognition>
(C) 2007, SNU Biointelligence Lab, 

7. (C) 2007, SNU Biointelligence Lab, http://bi.snu.ac.kr/
Hidden Markov Model
Given the following,
Joint prob. dist. for HMM is:
Whose elements are:
(observation,latent var,model parameters)
K : 상태의 수 / N : 총 시간
Zn-1j,nk : 시각 n-1에서 j상태였다가 시각 n에서 k상태로 transition
(initial latent node)
K : 상태의 수 / N : 총 시간
Zn-1jnk : 시각 n-1에서 j상태였다가 시각 n에서 k상태로 transition
(cond. dist. among
latent vars)
(emission prob.)
(C) 2007, SNU Biointelligence Lab, 

8. EM Revisited (slide by Ho-sik Seok)
General EM
Maximizing the log likelihood function
Given a joint distribution p(X, Z|Θ) over observed variables X and latent variables Z, governed by parameters Θ
Choose an initial setting for the parameters Θold
E step Evaluate p(Z|X,Θold )
M step Evaluate Θnew given by Θnew = argmaxΘQ(Θ ,Θold) Q(Θ ,Θold) = ΣZ p(Z|X, Θold)ln p(X, Z| Θ)
It the covariance criterion is not satisfied, then let Θold  Θnew
(C) 2007, SNU Biointelligence Lab, 

9. Estimation of HMM Parameter (using M.L.)
The Likelihood Function
Using EM Algorithm
E-Step
(marginalization over latent vars Z)
M-Step
g(Znk)는 n시점에 k상태에 위치할 확률을, x(Zn-1j,Znk)는 n-1시점에 k상태에서 n시점에 j상태로 transition할 확률을 나타냄
(C) 2007, SNU Biointelligence Lab, 

10. Forward-backward Algorithm
(probability of observation)
Probability for a single latent variable
Defining alpha & beta Recursively
Used for evaluating the prob. Of observation
g(Znk)는 n시점에 k상태에 위치할 확률을, x(Zn-1j,Znk)는 n-1시점에 k상태에서 n시점에 j상태로 transition할 확률을 나타냄
(?)
(C) 2007, SNU Biointelligence Lab, 

11. Sum-product Algorithm
(probability of observation)
Factor graph representation
Same result as before
g(Znk)는 n시점에 k상태에 위치할 확률을, x(Zn-1j,Znk)는 n-1시점에 k상태에서 n시점에 j상태로 transition할 확률을 나타냄
(C) 2007, SNU Biointelligence Lab, 

12. (C) 2007, SNU Biointelligence Lab, http://bi.snu.ac.kr/
The Viterbi Algorithm
(most likely state sequence)
From max-sum algorithm
Joint dist. by the most probable path
g(Znk)는 n시점에 k상태에 위치할 확률을, x(Zn-1j,Znk)는 n-1시점에 k상태에서 n시점에 j상태로 transition할 확률을 나타냄
(C) 2007, SNU Biointelligence Lab, 

13. (C) 2007, SNU Biointelligence Lab, http://bi.snu.ac.kr/
References
HMM
A Tutorial On Hidden Markov Models And Selected Applications In Speech Recognition (Rabiner)
CRF Introduction
(C) 2007, SNU Biointelligence Lab,