HMM Algorithms: Trellis and Viterbi
PA154 Jazykové modelovaní (5.2)
Pavel Rychlý
pary@fi.muni.cz
March 20, 2019
Source: Introduction to Natural Language Processing (600.465) Jan Hajic, CS Dept., Johns Hopkins Univ. www.cs.jhu.edu/~hajic
H M M: The Two Tasks
■ HMM (the general case):
► five-tuple (S, So, Y, Ps, Py), where:
► S = {si, S2,..., St} is the set of states, So is the initial,
► Y = {yi,3/2,.. •, yv} is the output alphabet,
► Ps(sj\si) is the set of prob. distributions of transitions,
► Py{yk\s;1 Sj) is the set of output (emission) probability distributions.
■ Given an HMM & an output sequence Y = {yi,y2, • • • , y/c}
(Task 1) compute the probability of Y;
(Task 2) compute the most likely sequence of states which has generated Y.
PA154 Jazykové modelování (5.2)
HMM Algorithms: Trellis and Viterbi
2/22
Trellis - Deterministic Output
HMM
Trellis:
time/position t
1 2 3 4.
p(toe)=x.6 x .88 x 1+ x.4 x .1 x 1 = .568
- trellis state: (HMM state, position)
- each state: holds one number (prob):a
- probability or Y: la in the last state
Y:      t        o e
a(x, 0) = 1 ot(A, 1) = .6 a(D, 2) = .568 a(ß, 3) a(C, 1) = .4
.568
PA154 Jazykové modelování (5.2)
HMM Algorithms: Trellis and Viterbi
3/22
Creating the Trellis: The Start
■ Start in the start state (x),
► its a(x,0) to 1.
■ Create the first stage:
► get the first "output" symbol yi
► create the first stage (column)
► but only those trellis states
which generate yi
► set their a(state,l) to the Ps(state\x) a(x, 0)
■ . .. and forget about the 0-th stage
PA154 Jazykové modelování (5.2)
HMM Algorithms: Trellis and Viterbi
Trellis: The Next Step
position/stage
■ Suppose we are in stage /, i=i
■ Creating the next stage:
► create all trellis state in the next stage which generate y,+i, but only those reachable from any of the stage-/ states
► set their a(state, i + 1) to: Ps(state\ prev.state) xa(prev.state, /) (add up all such numbers on arcs going to a common trellis state)
► ... and forget about stage /
PA154 Jazykové modelování (5.2)
HMM Algorithms: Trellis and Viterbi
Trellis: The Last Step
Continue until "output" exhausted - Y\ = 3: until stage 3
Add together all the a(state,\Y\)
That's the P(Y).
Observation (pleasant):
► memory usage max: 2|S
► multiplicationsmax:  S 2 Y
last position/stage
a .568
a — .568
P(Y)=.568
PA154 Jazykové modelování (5.2)
HMM Algorithms: Trellis and Viterbi
6/22
Trellis: The General Case (still, bigrams)
Start as usual:
► start state (x), set its a(x,0) to 1
O..06
e,.12
p(toe) = .48 x .616 x .6 + .2 x 1 x .176 +
.2 x 1 x .12 ^ .237
a = 1
PA154 Jazykové modelování (5.2)
HMM Algorithms: Trellis and Viterbi
7/22
General Trellis: The Next Step
We are in stage /:
► Generate the next stage i+1 as before (except now arcs generate output, thus use only those arcs marked by the output symbol y,+i)
► For each generated state compute a(state,i + 1) =
position/stage 0
incoming arcs
a(prev.state,i)
o,.06
Py(yi+i\state,prev.state) x
e,.12
a = 1
a = .48
a = .2
and forget about stage / as usual
PA154 Jazykové modelování (5.2)
HMM Algorithms: Trellis and Viterbi
8/22
Trellis: The Complete Example
The Case of Trigrams
■ Like before, but:
► states correspond to bigrams,
► output function always emits the second output symbol of the pair (state) to which the arc goes:
Multiple paths not possible —>> trellis not really needed
PA154 Jazykové modelování (5.2)
HMM Algorithms: Trellis and Viterbi
10/22
Trigrams with Classes
More interesting:
► n-gram class LM: p(i/i/;|i/i/;_2, = p(w/|c/)p(c/|c/_2, Q_i)
—>» states are pairs of classes (c,-_i, q), and emit "words":
(letters in our example)
p(t | C) = 1 usual, p(o|V) = .3 non-p(e|V) = .6 overlapping P(y|V) = 1 classes
o,e,y
o,e,y
p(toe) = .6 X 1 X .88 X .3 X .07 X .6 ^ .00665 p(teo) = .6 X 1 X .88 X .6 X .07 X .3 ^ .00665 p(toy) = .6 X 1 X .88 X .3 X .07 X .1 ^ .00111 p(tty) = .6 X 1 X .12 x 1 X 1 X .1 ^ .0072
PA154 Jazykové modelování (5.2)
HMM Algorithms: Trellis and Viterbi
11/22
Class Trigrams: the Trellis
■ Trellis generation (Y = "toy"):
o,e,y      o,e,y t
a = .6 x .88 x .3
PA154 Jazykové modelování (5.2)
Y-   t n
HMM Algorithms: Trellis and Viterbi
^^^^ 12/22
Overlapping Classes
■ Imagine that classes may overlap
► e.g. V is sometimes vowel sometimes as C:
o,e,y,r     o,e,y,r t,r
consonant, belongs to V as well
p(t|C) =	.3
P(r|C) =	.7
p(o|V) =	.1
p(e|V) =	.3
p(y|V) =	.4
p(r|V) =	.2
p(try) = ?
PA154 Jazykové modelování (5.2)
HMM Algorithms: Trellis and Viterbi
13/22
Overlapping Classes: Trellis Example
o,e,y,r      o,e,y,r t,r
a = .18 X .88 X .2 = .03168
PA154 Jazykové modelování (5.2)
HMM Algorithms: Trellis and Viterbi
Trellis: Remarks
■ So far, we went left to right (computing a)
■ Same result: going right to left (computing /3)
► supposed we know where to start (finite data)
■ In fact, we might start in the middle going left and right
■ Important for parameter estimation
(Forward-Backward Algortihm alias Baum-Welch)
■ Implementation issues:
► scaling/normalizing probabilities, to avoid too small numbe & addition problems with many transitions
PA154 Jazykové modelování (5.2)
HMM Algorithms: Trellis and Viterbi
The Viterbi Algorithm
■ Solving the task of fmding the most likely sequence of states which generated the observed data
■ i.e., finding
Sbest = argmaxsP(S\Y) which is equal to (Y is constant and thus P(Y) is fixed):
Sbest = argmaxsP{S,Y) = = argmaxsP(so,s1,s2,.. .,sklylly2,... ,y/c) = = argmaxsPUi=lmmk p(yi|s/, s/_i)p(s/|s/_i)
PA154 Jazykové modelování (5.2)
HMM Algorithms: Trellis and Viterbi
16/22
"he Crucial Observation
■ Imagine the trellis build as before (but do not compute the as yet; assume they are o.k.); stage /:
stage
a = .6
?... max!
this is certainly the "backwards" maximum to (D,2)... but
it cannot change even whenever we go forward (M. Property: Limited History)
PA154 Jazykové modelování (5.2)
HMM Algorithms: Trellis and Viterbi
17/22
Viterbi Example
■ V classification (C or V?, sequence?):
p(t|C) = .3 p(r|C) = .7 p(o|V) = .1 p(e|V) = .3
p(y|V) = .4 p(r|V) = .2
argmaxxyz p(rry|XYZ) = ?
Possible state seq.:
(x, V)(V, C)(C, V)[VCV],(x,C)(C, C)(C, V)[CCV],(x, C)(C, V)(V, V)[CVV]
PA154 Jazykové modelovaní (5.2)
HMM Algorithms: Trellis and Viterbi
18/22
Viterbi Computation
Y:
p(t|C)
P(r|C) p(o|V) p(e|V)
p(y|v)
p(r|V)
,3 7 .1 .3 .4 .2
a in trellis state:
o
best prob from start to here
o,e,y,r o,e,y,r
18 X .12 x .7 : .03528
a = .07392 x .07 X .4 = .002070
X .88 X .2 ľ07392
a =
.0.8 X 1 X .7 = .056
= .0141 avc = .(
= .or
a =
.4 X .2 .08
PA154 Jazykové modelování (5.2)
HMM Algorithms: Trellis and Viterbi
19/22
n-best State Sequences
Y:
a. = 1
Keep track of n best "back pointers":
Ex.: n= 2: Two "winners": VCV (best) CCV (2nd best)
a = .18 X .12 x .7 = .03528
a = .07392 X .07 X .4 =.002070
.88 x .2
PA154 Jazykové modelování (5.2)
a = .0.8 X 1 X .7 = .056
a = .4 X .2 = .08
HMM Algorithms: Trellis and Viterbi
aC,C = -03528 X 1 >
='.01411 olvc = .056 X .8 X = .01792 = amax
20/22
Tracking Back the n-best paths
■ Backtracking-style algorithm:
► Start at the end, in the best of the n states (sbest)
► Put the other n-1 best nodes/back pointer pairs on stack, except those leading from Sbest to the same best-back state.
■ Follow the back "beam" towards the start of the data, spitting out nodes on the way (backwards of course) using always only the best back pointer.
■ At every beam split, push the diverging node/back pointer pairs onto the stack (node/beam width is sufficient!).
■ When you reach the start of data, close the path, and pop the topmost node/back pointer(width) pair from the stack.
■ Repeat until the stack is empty; expand the result tree if necessary.
PA154 Jazykové modelování (5.2)
HMM Algorithms: Trellis and Viterbi
21/22
runing
Sometimes, too many trellis states in a stage:
a .002
a
a
a
a
.043 = .001 = .231 = .002
criteria: (a) a < threshold
(b) Y.-K < threshold
(c) # of states > threshold
(get rid of smallest a)
a .000003
a .000435
a .0066
PA154 Jazykové modelování (5.2)
HMM Algorithms: Trellis and Viterbi
22/22