HMM

#Computers
A model that tries to predict the hidden variables of a markov model

Topics

Markov Chains

$\displaystyle e_{k}(b)=P(Y_{t}=b|X_{t}=k)$

Emission probability, or probability we observe $\displaystyle b$ given hidden variable state $\displaystyle k$

$\displaystyle P(x_{1:T})=\pi_{x_{1}}\prod_{t = 1}^{T-1}q_{x_{t+1}x_{t}}$

$\displaystyle v_{t}(k)=\text{max}{x{1}:t-1}P(y_{1:t},x_{1:t-1},x_{t}=k)$

Probability of most probable path (MPP) that ends in state $\displaystyle k$

$\displaystyle v_{1}(k)=P(y_{1},x_{1}=k)=e_{k}(y_{1})\pi_{k}$

$\displaystyle v_{t}(k)=\text{max}{l}v{t-1}(l)q_{kl}e_{k}(y_{t})$

$\displaystyle \text{MPP}{t}(k)=\text{argmax}{l}v_{t-1}(l)q_{kl}e_{k}(y_{t})$