Diffusion Processes with Hidden States from ... - FU Berlin, FB MI

3.7 Baum-Welch-AlgorithmAlgorithm 3.1 (EM Algorithm).• Expectation-Step (E-Step): This step evaluates the expectation value Q based on the givenparameter estimate λ k .• Maximization-Step (M-Step): This step determines the refined parameter set λ k+1 by maximizingthe expectation:λ k+1 = argmaxQ(λ,λ k ). (3.31)λ⇒ The maximization guarantees, that L (λ k+1 ) ≥ L (λ k ).A very good introduction to the field of EM-algorithms is given in [5].3.7 Baum-Welch-AlgorithmIn this section, we derive the EM algorithm for finding the maximum-likelihood estimate of theparameters of a hidden Markov model given a set of observed feature vectors. This algorithm iscalled Baum-Welch algorithm. It is a special instance of the EM algorithm based on HMMs (seein [21]).The probability of seeing the partial sequence o 1 ,...,o t and ending up in state i at time t is givenby:α i (t) = P(O 1 = o 1 ,...,O t = o t ,Q t = i|λ).The variable α i (t) is called the forward variable. We can efficiently define α i (t) recursively as:• α i (1) = πb i (o 1 )• α j (t + 1) = ∑ N i=1 (α i(t)a i, j )b j (o t+1 )• P(O|λ) = ∑ N i=1 α i(T ).This is the probability for the observation of the sequence o 1 ,...,o T under given parameterλ. It is given by summation over all α i at fixed time T .The backward procedure is similar. The probability to find a partial sequence o t+1 ,...,o T , giventhat we started at state i at time t isβ i (t) = P(O t+1 = o t+1 ,...,O T = o T |Q t = i,λ).Accordingly the variable β i (t) is called the backward variable. Analogously we can define β i (t):• β i (T ) = 125