HMM: The Two Tasks
HMM Algorithms: Trellis and Viterbi
PA154 Jazykové modelování (5.2)
Pavel Rychlý
pary@fi.muni.cz
March 16, 2020
HMM (the general case):
► five-tuple (S, So, Y, Ps, Py), where:
► S = {si, 52,. ■ ■, sj} is the set of states, So is the initial,
► Y = {yi,y2, ■ ■ ■ ,yv} is the output alphabet,
► Ps(sj\sj) is the set of prob. distributions of transitions,
► Py(yk\si, Sj) is the set of output (emission) probability distributions.
Given an HMM & an output sequence Y — {yi,y2: - - - ,yk} (Task 1) compute the probability of Y;
(Task 2) compute the most likely sequence of states which has generated Y.
Source: Introduction to Natural Language Processing (600.465) Jan Hajic, CS Dept., Johns Hopkins Univ. www.c5.jhu.edLi/~li.ijn
PA154 Jazykové modelováni (5.2)
HMM Algorithms: Trellis and Viterbi
Trellis - Deterministic Output
Creating the Trellis: The Start
HMM:
Trellis:
time/position t
2 3 4..
p(toe)=x.6 X .88 X 1 + x.4 X .1 X =.568
- trellis state: (HMM state, position) Y: t
- each state: holds one number (prob)
- probability or Y: Za in the last state
(X,0) = 1 a(A, 1) = .6 «(D, 2) = .568 a(8,3) = .568 a(C, 1) = .4
Start in the start state (x),
► its q(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) ot(x,0)
l
... and forget about the 0-th stage
position/stage
yi: -
HMM Algorithms: Trellis and Viterbi
HMM Algorithms; Trellis and Viterbi
Trellis: The Next Step
Trellis: The Last Step
position/stage
Suppose we are in stage /, i=l Creating the next stage:
► create all trellis state in the next stage which generate y,-+1, but only those reachable from any of the stage-/ states
► set their a(state, i + 1) to: Ps(state\ prev.state) xa(prev.state, i) (add up all such numbers on arcs going to a common trellis state)
► .. .and forget about stage ;
Continue until "output" exhausted
- |V| = 3: until stage 3
Add together all the a(state,\Y\)
That's the P(Y).
Observation (pleasant): ► memory usage max: 2|S * multiplicationsmax: |5|2|Y
last position/stage
i
P(Y)=.568
PA154 Jazykové modeloN
HMM Algorithms: Trellis and Viterbi
HMM Algorithms: Trellis and Viterbi
Trellis: The General Case (still, bigrams)
General Trellis: The Next Step
Start as usual:
► start state (x), set its a(x,0) to 1.
,.06 e,.12
p(toe) = .48 X .616 X .6 + .2 X 1 X .176 +
.2 X 1 X .12 2* .237
a = 1
We are in stage /: position/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) =
— incoming arcs
Py(y,-_i_i|sŕaŕe,prev.state) x
a(prev.statej)
O..06
HMM Algorithms: Trellis and Viterbi
. and forget about stage / as usual
HMM Algorithms: Trellis and Viterbi
Trellis: The Complete Example
Stage:
2------í 3
= .024 + .177408 = .201408
D A       P(Y) = P(tue)= 236608
HMM Algorithms: Trellis and Viterbi
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:
p(toe) = .6 x .88 x .07 S .037 Multiple paths not possible —► trellis not really needed
HMM Algorithms: Trellis and Viterbi
Trigrams with Classes
More interesting:
► n-gram class LM: p(w;\w;-2, W/-l) = p(w;|q)p(q|q-2, c/-l)
—► states are pairs of classes (c;_i,c,-), 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 > p (too) = .6 > p(toy) = -6 > p(ííy) = .6 X
< -6 -00665
< -3 -00665
< -1 Sŕ -00111 [ eŕ .0072
PA154 Jazykové modelováni (5.2)
HMM Algorithms: Trellis and Viterbi
Class Trigrams: the Trellis
■ Trellis generation (Y = "toy"):
p(t|C) = 1 p(o|V) = .3 p(e|V) = .6 p(y|V) = .1
o,e,y o,e,y
again, trellis useful not really neede
a = .1584 Si .00111
a = .6 x .88 x .3
PA154 Jazykové modelováni (5.2)
V ■    1- n
HMM Algorithms: Trellis and Viterbi
Overlapping Classes
Imagine that classes may overlap
► e.g. v is sometimes vowel sometimes consonant, belongs to V as well as C:
b,e,y,r p(t|C) = .3
p(r|C) = .7 p(o|V) = .1 p(e|V) = .3 p(y|v) = .4 p(r|V) = .2
o,e,y,r     o,e,y,r t,r
p(try) = ?
Overlapping Classes: Trellis Example
p(t|C) = .3 p(r|C) = .7 p(o|V) = .1 p(e|V) = .3 p(y|v) = .4 p(r|V) = .2
o,e,y,r o,e,y,r
HMM Algorithms: Trellis and Viterbi
HMM Algorithms: Trellis and Viterbi
Trellis: Remarks
The Viterbi Algorithm
So far, we went left to right (computing a) Same result: going right to left (computing ft)
► 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 numbers & addition problems with many transitions
■ 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(s0,s1,s2,.. -,sk,y1,y2,.. .,yk) = = argmaxsPni=1..k p(yi^, s;_i)p(s;|s;_i)
HMM Algorithms: Trellis and Viterbi
HMM Algorithms: Trellis and Viterbi
The Crucial Observation
Viterbi Example
Imagine the trellis build as before (but do not compute the as yet; assume they are o.k.); stage /:
NB: remember previous state from which we got the n (.3, .32|= .32
this is certainly the "backwards'
(D,2). .. but forward (M. Property: Limited History)
v classification (C or V?, sequence?):
o,e,y,r
o,e,y,r o,e,y,r
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 modelováni (5.2)
HMM Algorithms: Trellis and Viterbi
PA154 Jazykové modelováni (5.2)
HMM Algorithms: Trellis and Viterbi
Viterbi Computation
n-best State Sequences
p(t|Q
P(r|C) p(o|V) p(e|V)
p(y|v)
p(r|V)
o,e,y,r o,e,y,r
0-8 X 1 X -7
HMM Algorithms: Trellis and Viterbi
Tracking Back the n-best paths
Backtracking-style algorithm:
► Start at the end, in the best of the n states (s^est)
► Put the other n-1 best nodes/back pointer pairs on stack, except those leading from s^est 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.
HMM Algorithms: Trellis and Viterbi
Keep track of n best "back pointers": Ex.: n= 2: Two "winners": VCV (best) CCV {2nd best)
Ctc = -03528 > ='.01411 v>c = -056 X . = -01792 = a,
Pruning
HMM Algorithms: Trellis and Viterbi
Sometimes, too many trellis states in a stage:
[A )     a = .002
criteria: (a) a < threshold
(b) Ztt < threshold
(c) # of states > threshold
(get rid of smallest a)
HMM Algorithms: Trellis and Viterbi