Hidden Markov Model at a glance

Contents

  1. Markov Chain
  2. Hidden Markov Model
  3. 3 problems
  4. Sample with HMM

Markov Chain

Homogeneous Markov chain

Markov Chain

The second-order Markov chain

Markov Chain

rain cloud sun
rain 0.4 0.3 0.3
cloud 0.2 0.6 0.2
sun 0.1 0.1 0.8

HMM

HMM

HMM

Observed a data set

X = \{X_1,...,X_M\}
X={X1,...,XM}X = \{X_1,...,X_M\}

Latent vars

S = \{S_1,...,S_N\}
S={S1,...,SN}S = \{S_1,...,S_N\}

HMM params

\theta = \{\pi, A, B\}
θ={π,A,B}\theta = \{\pi, A, B\}
A = \{a_{ij}\} \quad a_{ij} = P(z_t = S_j|z_{t-1}=S_i) \quad 1\leq i,j \leq N
A={aij}aij=P(zt=Sjzt1=Si)1i,jNA = \{a_{ij}\} \quad a_{ij} = P(z_t = S_j|z_{t-1}=S_i) \quad 1\leq i,j \leq N
B = \{b_{jk}\} \quad b_{jk} = P(x_t = X_k|z_{t}=S_j) \quad 1\leq j \leq N,1 \leq k \leq M
B={bjk}bjk=P(xt=Xkzt=Sj)1jN,1kMB = \{b_{jk}\} \quad b_{jk} = P(x_t = X_k|z_{t}=S_j) \quad 1\leq j \leq N,1 \leq k \leq M

Three Problems

  • Evaluation Problem
  • Decode Problem
  • Learning Problem

Evaluation Problem

Solve with Forward-Backward algorithm

Have observed   

X=(x_1x_2...x_T)
X=(x1x2...xT)X=(x_1x_2...x_T)

and

\theta = \{\pi, A, B\}
θ={π,A,B}\theta = \{\pi, A, B\}

evaluate

P(X|\theta)
P(Xθ)P(X|\theta)

Forward-Backward

idea

P(X|\theta) = \sum_Z{P(X,Z|\theta)} = \sum_Z{P(X|Z,\theta)}P(Z|\theta)
P(Xθ)=ZP(X,Zθ)=ZP(XZ,θ)P(Zθ)P(X|\theta) = \sum_Z{P(X,Z|\theta)} = \sum_Z{P(X|Z,\theta)}P(Z|\theta)
P(X|\theta) = \sum_{z_1,z_2,...,z_T} \pi_{z_1}b_{z_1X_1}a_{z_1z_2}b_{z_2X_2}...
P(Xθ)=z1,z2,...,zTπz1bz1X1az1z2bz2X2...P(X|\theta) = \sum_{z_1,z_2,...,z_T} \pi_{z_1}b_{z_1X_1}a_{z_1z_2}b_{z_2X_2}...

with

\alpha_t(i) = P(x_1x_2...x_t, z_t = S_i|\theta)
αt(i)=P(x1x2...xt,zt=Siθ)\alpha_t(i) = P(x_1x_2...x_t, z_t = S_i|\theta)
=> \alpha_{t+1}(j)= b_{j x_{t+1}}\sum_{i=1}^N{\alpha_{t}(i)a_{ij}}
=>αt+1(j)=bjxt+1i=1Nαt(i)aij=> \alpha_{t+1}(j)= b_{j x_{t+1}}\sum_{i=1}^N{\alpha_{t}(i)a_{ij}}

Forward

Backward

\beta_t(i) = P(x_{t+1}x_{t+2}...x_T,z_t=S_i|\theta)
βt(i)=P(xt+1xt+2...xT,zt=Siθ)\beta_t(i) = P(x_{t+1}x_{t+2}...x_T,z_t=S_i|\theta)

Decode Problem

Have observed   

X=(x_1,...,x_T)
X=(x1,...,xT)X=(x_1,...,x_T)

and

\theta = \{\pi, A, B\}
θ={π,A,B}\theta = \{\pi, A, B\}

find

Z^*=(z_1z_2...z_T)
Z=(z1z2...zT)Z^*=(z_1z_2...z_T)

that

Z^*= argmax P(Z|X, \theta)
Z=argmaxP(ZX,θ)Z^*= argmax P(Z|X, \theta)

http://www.utdallas.edu/~prr105020/biol6385/2018/lecture/Viterbi_handout.pdf

Learning Problem

Given a HMM θ, and an observation history

X = ( x_1x_2 ... x_T )
X=(x1x2...xT)X = ( x_1x_2 ... x_T )

find new θ that explains the observations at least as well, or possibly better

P(X|\theta') \geq P(X|\theta)
P(Xθ)P(Xθ)P(X|\theta') \geq P(X|\theta)

Learning Problem

EM idea

Sample

Conclusion

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By Khanh Tran

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