Hidden Markov Model (HMM)

observation sequence: U={ut | t∈}={u1,u2,...,uT}

hidden state sequence: S={st | t∈}={s1,s2,...,sT}

number of observations: T number of states: M

parameter sets: λ=(π,A,B)

π = (π k), π k = P (s 1 = k), 1 \leq l \leq M

A = (a k l), a k l = P (s t + 1 = l | s t = k), 1 \leq l, k \leq M

B = (b k (u)), b k (u) = P (u t = u | s t = k), 1 \leq k \leq M

P (U, S | λ) = P (S | λ) P (U | S, λ) = π s 1 b s 1 (u 1) a s 1, s 2 b s 2 (u 2) . . . a s T - 1, s T b s T (u T)

P (U | λ) = \sum S P (U, S | λ) = \sum S π s 1 b s 1 (u 1) a s 1, s 2 b s 2 (u 2) . . . a s T - 1, s T b s T (u T)

αk(t)≡P(u1:t, st=k | λ) with initial αk(1)=πkbk(u1) and recursions αk(t)=(∑Ml=1αl(t−1)alk)bk(ut)
βk(t)≡P(ut+1:T | st=k,λ) with initial βk(T)=1 and recursions βk(t)=(∑Ml=1aklbl(ut+1)βl(t+1))
Lk(t)≡P(st=k, | U,λ) , conditional probability of being in state k at time t with given observations U .

$L k (t) \equiv P (s t = k, | U, λ) = P ( U , s t = k | λ ) P ( U | λ ) = P ( u 1 : t , s t = k | λ ) P ( u t + 1 : T | s t = k , λ ) P ( U | λ ) = α k ( t ) β k ( t ) P ( U | λ )$
Hk,l(t)≡P(st=k,st+1=l | U,λ) , conditional probability of being in state k at time t and in state l at time t+1 with given observations U .

$H k, l (t) \equiv P (s t = k, s t + 1 = l | U, λ) = P ( U , s t = k , s t + 1 = l | λ ) P ( U | λ ) = α k ( t ) a k l b l ( u t + 1 ) β l ( t + 1 ) P ( U | λ )$

Compute P(U | λ) using Forward-Backward algorithm

P (U | λ)) = P (u 1 : t, u t + 1 : T | λ) = \sum k = 1 M P (u 1 : t, s t = k, u t + 1 : T | λ)

= \sum k = 1 M P (u 1 : t, s t = k | λ) P (u t + 1 : T | u 1 : t, s t = k, λ) = \sum k = 1 M α k (t) β k (t) = \sum k = 1 M α k (T)

Compute Ŝ =argmaxSP(S | U) using Viterbi algorithm

W (S) \equiv - log P (S, U | λ) = - [log π s 1 b s 1 (u 1) + \sum t = 2 T log a s t - 1, s t b s t (u t)]

then problem II can be written as

S ̂ = arg max S P (S | U, λ) = arg max S P ( S , U | λ ) P ( U | λ ) = arg max S P (S, U | λ) = arg min S W (S)

with further define

V t (t) = min s 1, s 2, . . ., s t - 1 [- log P (s 1 : t - 1, s t = k, U | λ)]

= max s 1, s 2, . . ., s t - 1 [log π s 1 b s 1 (u 1) + \sum t' = 2 t - 1 log a s t' - 1, s t' b s t' (u t') + log a s t - 1, k b k (u t)]

Initialize V1(k)=−logπkbk(u1) ∀ 1≤k≤M
Compute Vt(k) with recursions
$V t (k) = max l (V t - 1 (l) - log a l k b k (u t)) \forall 1 \leq k \leq M, 2 \leq t \leq T$
Get the minimum value W(S)=min1≤k≤MVT(k)
Trace back to find the optimal state path {sT,sT−1,...,s1}

Compute λ̂ =argmaxλP(u | λ) using Baum-Welch algorithm

Baum-Welch algorithm is the EM estimation of HMM parameters

E-Step: under current set of parameters λold , compute αk(t) , βk(t) , Lk(t) , Hk,l(t)

M-Step: update parameters
+ For Discrete HMM:

π ̂ k = L k ( 1 ) \sum M k = 1 L k ( 1 ) a ̂ k l = \sum T - 1 t = 1 H k , l ( t ) \sum T - 1 t = 1 L k ( t ) b ̂ k (j) = \sum T t = 1 L k ( t ) 1 u t = j \sum T t = 1 L k ( t )

For Gaussian HMM:
$π ̂ k = L k ( 1 ) \sum M k = 1 L k ( 1 ) a ̂ k l = \sum T - 1 t = 1 H k , l ( t ) \sum T - 1 t = 1 L k ( t ) μ ̂ k = \sum T t = 1 L k ( t ) u t \sum T t = 1 L k ( t ) Σ ̂ k = \sum T t = 1 L k ( t ) ( u t - μ k ) ( u t - μ k ) T \sum T t = 1 L k ( t )$