Pattern Recognition and Machine Learning

13.2. Hidden Markov Models 621 Figure 13.12 Illustration of the forward recursion (13.36) for evaluation of theαvariables. In th ...

622 13. SEQUENTIAL DATA Figure 13.13 Illustration of the backward recursion (13.38) for evaluation of theβvariables. In this fra ...

13.2. Hidden Markov Models 623 Thus we can evaluate the likelihood function by computing this sum, for any conve- nient choice o ...

624 13. SEQUENTIAL DATA This completes the E step, and we use the results to find a revised set of parameters θnewusing the M-st ...

13.2. Hidden Markov Models 625 Figure 13.14 A fragment of the fac- tor graph representation for the hidden Markov model. χψn g 1 ...

626 13. SEQUENTIAL DATA Figure 13.15 A simplified form of fac- tor graph to describe the hidden Markov model. h fn z 1 zn− 1 zn ...

13.2. Hidden Markov Models 627 we obtain the beta recursion given by (13.38). Again, we can verify that the beta variables thems ...

628 13. SEQUENTIAL DATA From the product rule, we then have p(x 1 ,...,xn)= ∏n m=1 cm (13.57) and so α(zn)=p(zn|x 1 ,...,xn)p(x ...

13.2. Hidden Markov Models 629 Finally, we note that there is an alternative formulation of the forward-backward algorithm (Jord ...

630 13. SEQUENTIAL DATA Figure 13.16 A fragment of the HMM lattice showing two possible paths. The Viterbi algorithm efficiently ...

13.2. Hidden Markov Models 631 Intuitively, we can understand the Viterbi algorithm as follows. Naively, we could consider expli ...

632 13. SEQUENTIAL DATA Figure 13.17 Section of an autoregressive hidden Markov model, in which the distribution of the observat ...

13.2. Hidden Markov Models 633 Figure 13.18 Example of an input-output hidden Markov model. In this case, both the emission prob ...

634 13. SEQUENTIAL DATA Figure 13.19 A factorial hidden Markov model com- prising two Markov chains of latent vari- ables. For c ...

13.3. Linear Dynamical Systems 635 and so we can transform the model into an equivalent standard HMM having a single chain of la ...

636 13. SEQUENTIAL DATA make a greater contribution than less recent ones. Although this sort of intuitive argument seems plausi ...

13.3. Linear Dynamical Systems 637 model for that particular observation. However, the latent variables{zn}are no longer treated ...

638 13. SEQUENTIAL DATA 13.3.1 Inference in LDS We now turn to the problem of finding the marginal distributions for the latent ...

13.3. Linear Dynamical Systems 639 where we have defined Pn− 1 =AVn− 1 AT+Γ. (13.88) We can now combine this result with the fir ...

640 13. SEQUENTIAL DATA zn− 1 zn zn Figure 13.21 The linear dynamical system can be viewed as a sequence of steps in which incre ...