HMM Parameter Estimation: the Baum-Welch
algorithm
PA154 Jazykové modelování (6.1) Pavel Rychlý
pary@fi.muni.cz
March 23, 2020
Source: Introduction to Natural Language Processing (600.465) Jan Hajic, CS Dept., Johns Hopkins Univ. www.cs.jhu.edu/~hajic
HMM: The Tasks
■ HMM(the general case):
► five-tuple (S, So, V, P5, Py), where:
► S = {si, S2,..., sr} is the set of states, So is the initial state,
► Y = {yi, 3/2, • • •,yy} 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
► (Task 3) Estimating the parameters (transition/output distributions)
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A variant of Expectation-Maximization
■ ldea(~EM, for another variant see LM smoothing (lect. 3)):
Start with (possibly random) estimates of Ps and Py.
► Compute (fractional) "counts" of state transitions/emissions taken, from Ps and Py, given data Y
► Adjust the estimates of Ps and Py from these "counts" (using MLE, i.e. relative frequency as the estimate).
■ Remarks:
► many more parameters than the simple four-way smoothing
► no proofs here; see Jelinek Chapter 9
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Setting
■ HMM (without Ps, Py)(S, S0, Y), and data T = {y; e V}/=1>>>|T|
► will use T ^ | 7"|
■ HMM structure is given: (S, So)
■ Ps- Typically, one wants to allow "fully connected" graph
► (i.e. no transitions forbidden ^ no transitions set to hard 0)
► why? —)► we better leave it on the learning phase, based on the data!
► sometimes possible to remove some transitions ahead of time
■ Py : should be restricted (if not, we will not get anywhere!)
► restricted ^ hard 0 probabilities of p(y|s,s/)
► "Dictionary": states     words, "m:n" mapping on S x Y (in general)
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Initialization
■ For computing the initial expected "counts"
■ Important part
► EM guaranteed to find a local maximum only (albeit a good one in most cases)
■ Py initialization more important
► fortunately, often easy to determine
► together with dictionary     vocabulary mapping, get counts, then MLE
■ Ps initialization less important
► e.g. uniform distribution for each p(.|s)
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Data structures
Will need storage for:
► The predetermined structure of the HMM (unless fully connected —>► need not to keep it!)
► The parameters to be estimated (Ps, Py)
► The expected counts (same size as (Ps, Py))
► The training data T = {y,- <G Y}-,=i...t
► The trellis (if f.c):
t T    Size: T X S (Precisely, |T|X|S|)
Each trellis state:
tno [float] numbers ^ ^ ~ ....... XV S
(forward/backward)       k£ k2> %3> k3 £
Q Q Q Q ®£and1henS<>me)
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The Algorithm Part I
□ Initialize P5, Py
Q Compute "forward" probabilities:
► follow the procedure for trellis (summing), compute a(s, /) everywhere
► use the current values of Ps, Py(p(s!\s), p(y|s, s')) :
') = Es^s' a(s> / - 1) x p(s'|s) x p(y/|s, s')
► NB: do not throw away the previous stage!
& Compute " backward" probabilities
► start at all nodes of the last stage, proceed backwards, /3(s, /)
► i.e., probability of the "tail" of data from stage i to the end of data
P(s',i) = Es'<-s/3(s>/ +x) x p(sIs') x P(yi+i\s'is)
► also, keep the /3(s, /) at all trellis states
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The Algorithm Part II
Collect counts:
► for each output/transition pair compute
c(y,s,s') = S1=0 k.ly=55+ia(s,i) p(s'|s) p(yj+1|s,s') p(s\i+l)
> ZJ~—í
prefix prob.    . .   ,      .. . tail prob
one pass through data, / this transition prob
only stop at (output) y x output prob
c(s, s') = J2yeY c(y, s, s') (assuming all observed y-, in Y) cls) = Es'esc(s>s') Reestimate: pf(sf\s) — c{s,s')/c{s)   p'{y\s,s') — c(y,s,s,)/c(s,s/)
Repeat 2-5 until desired convergence limit is reached
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Baum-Welch: Tips & Tricks
■ Normalization badly needed
► long training data —>* extremely small probabilities
■ Normalize a,/3 using the same norm.factor: as follows:
► compute a(s, /) as usual (Step 2 of the algorithm), computing the sum A/(/) at the given stage / as you go.
► at the end of each stage, recompute all alphasffor each state s):
a*(s,i) = a(s,i)/N(i)
► use the same A/(/) for f3s at the end of each backward (Step 3) stage:
P*(s,i) = /3(s,i)/N(i)
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Example
► S - short article, L - long article, C,V - word starting w/consonant, vowel
thus, only "the" is ambiguous (a, an, the - not members of C,V)
■ Task: pronunciation of "the"
■ Solution: build HMM, fully connected, 4 states:
► S - short article, L - long article, C,V - word starting w/consona vowel
► thus, only "the" is ambiguous (a, an, the - not members of C,V
■ Output form states only (p(w|s, s;) = p(w|s/))
•   DataY: an    egg    and    a   piece of   the big      ....     die end Trellis: ©    © O ....... O
© ©
O O G,
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Example: Initialization
■ Output probabilities:
► Pinit(w\c) = c(c, w)/c(c); where c(S, the) = c(/_, the) = c(the)/2 (other than that, everything is deterministic)
■ Transition probabilities:
► Pinit(c'\c) = l/4(uniform)
■ Don't forget:
► about the space needed
► initialize a(X, 0) = 1 (X : the never-occuring front buffer st.)
► initialize /3(s, 7") = 1 for all s (except for s = X)
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Fill in alpha, beta
■ Left to right, alpha:
o(s', /) = X^s-s>s' a(s'' — 1) x p(s'ls) x p(wi\s')< where s' is the output from states
■ Remember normalization (N(i)).
■ Similary, beta (on the way back from the end).
an    egg    and    a   piece of   the big      ....     the end
O
©.i Q,
ß(V, 6) = ß(L,7)p<L|V)p (the L)+ «jKQ ß(S,7)p(S|V)p(the,S)
a(V, 8) = a(U7)p(C|L)p(bÍ8, C)+ a<S,7)p(C|S)pCbi&C)
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Counts & Reestimation
■ One pass through data
■ At each position /, go through all pairs (s;,s;+i)
■ Increment appropriate counters by frac. counts (Step 4):
► inc(y/+i, sh s/+i) = a(sh /)p(s/+i|s/)p(y/+i|s/+i)b(s/+i5/+i)
► c(y, s/, s,-+i)+ = inc (for y at pos i+1)
► c(s/, s,-+i)+ = inc (always)
► c(s,-)+ = inc (always)
of   the big
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HMM: Final Remarks
■ Parameter "tying"
► keep certain parameters same     just one "counter" for all of them)
► any combination in principle possible
► ex.: smoothing (just one set of lambdas)
■ Real Numbers Output
► Y of infinite size (/?, Rn)
► parametric (typically: few) distribution needed (e.g., "Gaussian")
■ "Empty" transitions: do not generate output
► ~ vertical areas in trellis; do not use in "counting"
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