HMM Formalism K K S,K,∏,A,B S:S.SN are the values for the hidden states K: kI.kM are the values for the observations
6 HMM Formalism • {S, K, P, A, B} • S : {s1…sN } are the values for the hidden states • K : {k1…kM } are the values for the observations S S S K K K S K S K
HMM Formalism B K K S,K,∏,A,B ∏={π} are the initial state probabilities A=(ai are the state transition probabilities B=bik are the observation state probabilities
7 HMM Formalism • {S, K, P, A, B} • P = {pi} are the initial state probabilities • A = {aij} are the state transition probabilities • B = {bik} are the observation state probabilities A B A A A B B S S S K K K S K S K
Inference in an hmm Probability Estimation: Compute the probability of a given observation sequence Decoding: Given an observation sequence compute the most likely hidden state sequence Parameter Estimation: Given an observation sequence find a model that most closely fits the observation
8 Inference in an HMM • Probability Estimation: Compute the probability of a given observation sequence • Decoding: Given an observation sequence, compute the most likely hidden state sequence • Parameter Estimation: Given an observation sequence, find a model that most closely fits the observation
Probability estimation Given an observation sequence and a model compute the probability of the observation sequence O=(01….On),=(A,B,) Compute P(Olu
9 Compute ( | ) ( ... ), ( , , ) 1 P O O = o oT = A B P o1 ot-1 ot ot+1 oT Given an observation sequence and a model, compute the probability of the observation sequence Probability Estimation
Probability estimation P(O|x,p)=ba0b2…ba
10 Probability Estimation T oT P O X bx o bx o bx ( | , ) ... 1 1 2 2 = o1 ot-1 ot ot+1 oT x1 xt-1 xt xt+1 xT