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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 417 (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 418 (of 239) HMMs HMMs as GMs Other HMM queries What HMMs can do MPE Summ HMM  parameter names, homogeneous case Recall parameter names, timehomogeneous case.
1 P (Qt = j Qt−1 = i) = aij or [A]ij is a ﬁrstorder
timehomogeneous transition matrix. Prof. Jeﬀ Bilmes EE596A/Winter 2013/DGMs – Lecture 4  Jan 23rd, 2013 page 419 (of 239) HMMs HMMs as GMs Other HMM queries What HMMs can do MPE Summ HMM  parameter names, homogeneous case Recall parameter names, timehomogeneous case.
1 P (Qt = j Qt−1 = i) = aij or [A]ij is a ﬁrstorder
timehomogeneous 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 419 (of 239) HMMs HMMs as GMs Other HMM queries What HMMs can do MPE Summ HMM  parameter names, homogeneous case Recall parameter names, timehomogeneous case.
1 P (Qt = j Qt−1 = i) = aij or [A]ij is a ﬁrstorder
timehomogeneous transition matrix.
2 P (Q1 = i) = πi is the initial state distribution.
3 P (Xt = xQt = 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 419 (of 239) HMMs HMMs as GMs Other HMM queries What HMMs can do MPE Summ HMM  parameter names, homogeneous case Recall parameter names, timehomogeneous case.
1 P (Qt = j Qt−1 = i) = aij or [A]ij is a ﬁrstorder
timehomogeneous transition matrix.
2 P (Q1 = i) = πi is the initial state distribution.
3 P (Xt = xQt = 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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