7 t2 pxt qt pqt qt1 48 t hence the factorization

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Unformatted text preview: qt )p(qt |qt−1 ) (4.7) t=2 p(xt |qt )p(qt |qt−1 ) = (4.8) t Hence, the factorization properties of an HMM are immediate consequences of the meaning of the HMM graph (perhaps just seeing the graph is less work). Prof. Jeff Bilmes EE596A/Winter 2013/DGMs – Lecture 4 - Jan 23rd, 2013 page 4-17 (of 239) HMMs HMMs as GMs Other HMM queries What HMMs can do MPE Summ HMM Probabilistic Queries Queries associated with HMMs The quantities we typically wish to compute for an HMM include: Compute p(qt |x1:t ), or the filtering problem. Compute p(qt |x1:s ), with t > s, or the prediction problem. Compute p(qt |x1:u ), with t < u, or the smoothing problem. Above three named from linear systems literature in EE (e.g., Kalman filters). Note: above includes p(qt |x1:T ) for t ∈ {1, 2, . . . , T }. Also needed query is p(qt , qt+1 |xr:s ) (often r = 1 and s = T ). In all above cases, we need to sum out hidden variables from joint distributions. E.g., p(qt |x1:T ) = p(qt , x1:T )/p(x1:T ), so also need p(x1:T ). I.e., we compute both the numerator and denominator in each of the above queries. Next few slides show how this relates to clique potentials in the HMM graph. Prof. Jeff Bilmes EE596A/Winter 2013/DGMs – Lecture 4 - Jan 23rd, 2013 page 4-18 (of 239) HMMs HMMs as GMs Other HMM queries What HMMs can do MPE Summ HMM - parameter names, homogeneous case Recall parameter names, time-homogeneous case. 1 P (Qt = j |Qt−1 = i) = aij or [A]ij is a first-order time-homogeneous transition matrix. Prof. Jeff Bilmes EE596A/Winter 2013/DGMs – Lecture 4 - Jan 23rd, 2013 page 4-19 (of 239) HMMs HMMs as GMs Other HMM queries What HMMs can do MPE Summ HMM - parameter names, homogeneous case Recall parameter names, time-homogeneous case. 1 P (Qt = j |Qt−1 = i) = aij or [A]ij is a first-order time-homogeneous transition matrix. 2 P (Q1 = i) = πi is the initial state distribution. Prof. Jeff Bilmes EE596A/Winter 2013/DGMs – Lecture 4 - Jan 23rd, 2013 page 4-19 (of 239) HMMs HMMs as GMs Other HMM queries What HMMs can do MPE Summ HMM - parameter names, homogeneous case Recall parameter names, time-homogeneous case. 1 P (Qt = j |Qt−1 = i) = aij or [A]ij is a first-order time-homogeneous transition matrix. 2 P (Q1 = i) = πi is the initial state distribution. 3 P (Xt = x|Qt = i) = bi (x) is the observation distribution for the current state being in configuration i. Prof. Jeff Bilmes EE596A/Winter 2013/DGMs – Lecture 4 - Jan 23rd, 2013 page 4-19 (of 239) HMMs HMMs as GMs Other HMM queries What HMMs can do MPE Summ HMM - parameter names, homogeneous case Recall parameter names, time-homogeneous case. 1 P (Qt = j |Qt−1 = i) = aij or [A]ij is a first-order time-homogeneous transition matrix. 2 P (Q1 = i) = πi is the initial state distribution. 3 P (Xt = x|Qt = i) = bi (x) is the observation distribution for the current state being in configuration i. Notice that there are a fixed number of parameters regardless of the length T . In other words, parameters are shared across all time. T...
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