Hidden Markov Model at a glance
Contents
- Markov Chain
- Hidden Markov Model
- 3 problems
- 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}
Latent vars
S = \{S_1,...,S_N\}
S={S1,...,SN}
HMM params
\theta = \{\pi, A, B\}
θ={π,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=Sj∣zt−1=Si)1≤i,j≤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=Xk∣zt=Sj)1≤j≤N,1≤k≤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)
and
\theta = \{\pi, A, B\}
θ={π,A,B}
evaluate
P(X|\theta)
P(X∣θ)
Forward-Backward
idea
P(X|\theta) = \sum_Z{P(X,Z|\theta)} = \sum_Z{P(X|Z,\theta)}P(Z|\theta)
P(X∣θ)=∑ZP(X,Z∣θ)=∑ZP(X∣Z,θ)P(Z∣θ)
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(X∣θ)=∑z1,z2,...,zTπz1bz1X1az1z2bz2X2...
with
\alpha_t(i) = P(x_1x_2...x_t, z_t = S_i|\theta)
αt(i)=P(x1x2...xt,zt=Si∣θ)
=> \alpha_{t+1}(j)= b_{j x_{t+1}}\sum_{i=1}^N{\alpha_{t}(i)a_{ij}}
=>αt+1(j)=bjxt+1∑i=1Nαt(i)aij
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=Si∣θ)
Decode Problem
Have observed
X=(x_1,...,x_T)
X=(x1,...,xT)
and
\theta = \{\pi, A, B\}
θ={π,A,B}
find
Z^*=(z_1z_2...z_T)
Z∗=(z1z2...zT)
that
Z^*= argmax P(Z|X, \theta)
Z∗=argmaxP(Z∣X,θ)
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)
find new θ that explains the observations at least as well, or possibly better
P(X|\theta') \geq P(X|\theta)
P(X∣θ′)≥P(X∣θ)
Learning Problem
EM idea
Sample
Conclusion
Hidden Markov Model at a glance