HMM

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

Topics

• Markov Chains

$e_k(b)=P(Y_t=b|X_t=k)$

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

$P(x_{1:T})={\pi }_{x_1}\prod _{t=1}^{T-1}{q}_{x_{t+1}{x}_t}$

$v_t(k)={\text{max}}_{x_1:t-1}P(y_{1:t},x_{1:t-1},x_t=k)$

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

$v_1(k)=P(y_1,x_1=k)=e_k(y_1){\pi }_k$

$v_t(k)={\text{max}}_l{v}_{t-1}(l)q_{kl}{e}_k(y_t)$

${\text{MPP}}_t(k)={\text{argmax}}_l{v}_{t-1}(l)q_{kl}{e}_k(y_t)$