Summarized by Kim Jin-young

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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 http://bi.snu.ac.kr/ 많이 부족하지만 열심히 하겠습니다.^^

(C) 2007, SNU Biointelligence Lab, http://bi.snu.ac.kr/ Contents 13.1 Markov Models 13.2 Hidden Markov Models 13.2.1 Maximum likelihood for the HMM 13.2.2 The forward-backward algorithm 13.2.3 The sum-product algorithm for the HMM 13.2.4 Scaling factors 13.2.5 The Viterbi Algorithm 13.2.6 Extensions of the HMM (C) 2007, SNU Biointelligence Lab, http://bi.snu.ac.kr/

(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, http://bi.snu.ac.kr/

(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, http://bi.snu.ac.kr/

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, http://bi.snu.ac.kr/

Hidden Markov Model (example) Lattice Representation Left-to-right HMM <Handwriting Recognition> (C) 2007, SNU Biointelligence Lab, http://bi.snu.ac.kr/

(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, http://bi.snu.ac.kr/

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, http://bi.snu.ac.kr/

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, http://bi.snu.ac.kr/

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, http://bi.snu.ac.kr/

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, http://bi.snu.ac.kr/

(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, http://bi.snu.ac.kr/

(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 http://www.inference.phy.cam.ac.uk/hmw26/papers/crf_intro.pdf (C) 2007, SNU Biointelligence Lab, http://bi.snu.ac.kr/