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Markov.md vault backup: 2023-12-22 17:58:38 2023-12-22 17:58:38 +00:00
README.md vault backup: 2023-06-07 15:00:18 2023-06-07 15:00:18 +01:00

tags
ai
maths

Hidden Markov Models - JWMI Github Rabiner - A Tutorial on Hidden Markov Models and Selected Applications in Speech Recognition

  • Stochastic sequences of discrete states
    • Transitions have probabilities
  • Desired output not always produced the same
    • Same pronunciation

P(X|M)=\left(\prod_{t=1}^Ta_{x_{t-1}x_t}\right)\eta_{x_T}
a_{x_0x_1}=\pi_{x_1}

1st Order

  • Depends only on previous state
    • Markov assumption
P(x_t=j|x_{t-1}=i,x_{t-2}=h,...)\approx P(x_t=j|x_{t-1}=i)
  • Described by state-transition probabilities
a_{ij}=P(x_t=j|x_{t-1}=i), 1\leq i,j\leq N
  • \alpha
    • State transition
  • For N states
    • N  by N matrix of state transition probabilities

Weather

A=\left\{a_{ij}\right\}=\begin{bmatrix} 0.4 & 0.3 & 0.3\\ 0.2 & 0.6 & 0.2 \\ 0.1 & 0.1 & 0.8 \end{bmatrix}

rain, cloud, sun across columns and down rows

A=\{\pi_j,a_{ij},\eta_i\}=\{P(x_t=j|x_{t-1}=i)\}

Start/End

  • Null states
    • Entry/exit states
    • Don't generate observations

\pi_j=P(x_1=j) \space 1 \leq j \leq N
  • Sub j because probability of kicking off into that state
\eta_i=P(x_T=i) \space 1 \leq i \leq N
  • Sub i because probability of finishing from that state

State Duration

  • Probability of staying in state decays exponentially
p(X|x_1=i,M)=(a_{ii})^{\tau-1}(1-a_{ii})

  • Given, a_{33}=0.8
  • $\times0.8$ repeatedly
    • Stay in state