LM Smoothing (The EM Algorithm)
PA154 Jazykové modelování (3) Pavel Rychlý
pary@fi.muni.cz
March 9, 2017
Source: Introduction to Natural Language Processing (600.465) Jan Hajic, CS Dept., Johns Hopkins Univ. www.cs.jhu.edu/~hajic
"he Zero Problem
■ "Raw" n-gram language model estimate:
- necessarily, some zeros
► Imany: trigram model    2.16 x 1014 parameters, data ~109 words -which are true 0?
► optimal situation: even the least grequent trigram would be seen several times, in order to distinguish it's probability vs. other trigrams
► optimal situation cannot happen, unfortunately (open question: how many data would we need?)
—> we don't know
- we must eliminate zeros
■ Two kinds of zeros: p(w|h) = 0, or even p(h) = 0!
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Why do we need Nonzero Probs?
■ To avoid infinite Cross Entropy:
- happens when an event is found in test data which has not been seen in training data
H(p) = oo: prevents comparing data with > 0 "errors"
■ To make the system more robust
- low count estimates:
► they typically happen for "detailed" but relatively rare appearances
- high count estimates: reliable but less "detailed"
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Eliminating the Zero Probabilites: Smoothing
Get new p'(w) (same ft): almost p(w) but no zeros Discount w for (some) p(w) > 0: new p'(w) < p(w)
Ewediscounted(P(W) ~ P'(w)) = D
Distribute D to all w; p(w) = 0: new p'(w) > p(w) - possibly also to other w with low p(w)
For some w (possibly): p'(w) = p(w)
Make sure E^eqPW = 1 There are many ways of smoothing
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Smoothing by Adding 1
Simplest but not really usable:
Predicting words w from a vocabulary V, training data T:
p\w\h)
c(h, w) +1 c(h) + | V
► for non-conditional distributions: pf{w) = ^+\v\ Problem if | V\ > c(h) (as is often the case; even >> c(h)\) Example:
Training data: <s> what is it what is small?   |T| = 8
V = {what, is, it, small, ?,<s> ,flying, birds, are, a, bird,.}, |V| = 12 p(it) = .125, p(what) = .25, p(.)=0   p(what is it?) = .252 x .1252 9* .001
p(it is flying.) = .125x.25 x 02 = 0 p'(it) = .1, p'(what) = .15, p'(what is it?) = .15M2 ^ .0002
P'(-) = -05
p'(it is flying.) = .1 x .15 x .052 ^ .00004
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Adding less than 1
Equally simple:
Predicting word w from a vocabulary V, training data T:
„ ...    c(h7w) + X
P= «h) + x\v\- A<1
► for non-conditional distributions: pf{w) = Example:
Training data: <s> what is it what is small?   |T| = 8
V = {what, is, it, small, ?,<s> ,flying, birds, are, a, bird,.}, |V| = 12 p(it) = .125, p(what) = .25, p(.)=0   p(what is it?) = .252 x .1252 ^ .001
p(it is flying.) = .125x.25 x 02 = 0
Use A = .1
p'(it) ^ .12, p'(what) ^ .23, p'(what is it?) = .232 x .122 ^ .0007
p'(.) ^ -01
p'(it is flying.) = .12x.23 x .012 ^ .000003
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Good-Turing
Suitable for estimation from large data
- similar idea: discount/boost the relative frequency estimate:
Pr(w)
(c(w) + 1) x A/(c(w) + 1)
|7| x N(c(w))
where A/(c) is the count of words with count c (count-of-counts) specifically, for c(w) = 0 (unseen words), pr(w) = |
- good for small counts (< 5-10, where N(c) is high)
- normalization! (so that we have J2WP'(W) = 1)
PA154 Jazykove modelovani (3)
LM Smoothing
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Good-Turing: An Example
Remember: pr{w)
(c(i/i/)+1)xA/(c(i/i/)+1) \T\xN(c{w))
Training data:
<s> what is it what is small?   |T| = 8
V = {what, is, it, small, ?,<s> ,flying, birds, are, a, bird,.}, |V| = 12 p(it) = .125, p(what) = .25, p(.)=0   p(what is it?) = .252 x .1252 ^ .001
p(it is flying.) = .125x.25 x 02 = 0
■ Raw estimation (A/(0) = 6, A/(1) = 4, A/(2) = 2, N(i) = 0, for / > 2):
pr(it) = (1+1)xN(1+1)/(8xN(1)) = 2x2/(8x4) = .125 pr(what) = (2+1)xN(2+1)/(8xN(2)) = 3x0/(8x2) = 0:
keep orig. p(what) pr(.) = (0+1)xN(0+1)/(8xN(0)) = 1x4/(8x6) ^ .083
■ Normalize (divide by 1.5 = J2we\v\ Pr(w)) ancl compute:
p'(it) = .08, p'(what) ^ .17, p'(.) = .06
p'(what is it?) = .172 x .082 ^ .0002
p'(it is flying.) = .082 x .17 x .062 ^ .00004
LM Smoothing
8/17
Smoothing by Combination: Linear Interpolation
Combine what?
► distribution of various level of detail vs. reliability
n-gram models:
► use (n-1)gram, (n-2)gram, uniform
—> reliability <— detail
Simplest possible combination: - sum of probabilities, normalize:
► p(0|0) = .8, p(1|0) = .2, p(0|1) = 1, p(1|1) = 0, p(0) = .4, p(1) = .6
► p'(0|0) = .6, p'(1|0) = .4, p'(1|0) = .7, p'(1|1) = .3
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Typical n-gram LM Smoothing
■ Weight in less detailed distributions using A = (A0, A-i, A2, A3):
A2P2(W/-|        + AiPi(iv,) + A0/| 1/
■ Normalize:
A/ > 0, E,=0 A/ = 1 is sufficient (A0 = 1 - E,=i A/)(n = 3)
■ Estimation using MLE:
- fix the p3, p2, Pi and |V| parameters as estimated from the training data
-then find such {A/} which minimizes the cross entropy (maximazes probablity of data): ^(pK^/I^))
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Held-out Data
■ What data to use?
- try training data T: but we will always get A3 = 1
► why? let P/t be an i-gram distribution estimated using r.f. from T)
► minimizing HT(p'\) over a vector A, p'A = A3p37 + A2p27 + MPm + A0/| V\
- remember HT(p\) = H(p37-) + D(p3r||p'a); Psv-fixed H(p37) fixed, best) -which p\ minimizes HT(p\)? Obviously, a p\ for which D(p37-||p'a) = 0
- ...and that's p37 (because D(p||p) = 0, as we know)
- ...and certainly p'A = p3rifA3 = 1 (maybe in some other cases, too).
- (P'a = 1 x p37- + 0 x p27 + 1 x Pit- + 0/|V|)
- thus: do not use the training data for estimation of A!
► must hold out part of the training data (heldout data, H)
► ...call remaining data the (true/raw) training data, T
► the test data S (e.g., for comparison purposes): still different data!
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The Formulas
-1       I AVI
Repeat: minimizing — J2i=\ loQ2(Px(^i\hi)) over A
H
P'x(wi\hi) = = A3P3(lV/	= P'x(Wi	W,_2, Wj_^) = 7-i) + A2A>(w/	+ \<\P\(Wj) + A0|^|
			
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The (Smoothing) EM Algorithm
□ Start with some A, such that A > 0 for all j e 0..3 H Compute "Expected Counts" for eachAy.
B Compute new set of Ay, using "Next A" formula.
□ Start over at step 2, unless a termination condition is met.
■ Termination condition: convergence of A.
- Simply set an s, and finish if |Ay - Xj^extl < £ for each j (step 3).
■ Guaranteed to converge: follows from Jensen's inequality, plus a technical proof.
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Remark on Linear Interpolation Smoothing
■ "Bucketed Smoothing":
- use several vectors of A instead of one, based on (the frequency of) history: A(h)
► e.g. for h = (micrograms,per) we will have
A(h) = (.999, .0009, .00009, .00001) (because "cubic" is the only word to follow...)
- actually: not a separate set for each history, but rather a set for "similar" histories ("bucket"): ---^^
A(b(h)), where b: V2    N (in the case of trigrams) b classifies histories according to their reliability (-frequency)
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Bucketed Smoothing: The Algorithm
■ First, determine the bucketing function b (use heldout!):
- decide in advance you want e.g. 1000 buckets
- compute the total frequency of histories in 1 bucket {fmax{b))
- gradually fill your buckets from the most frequent bigrams so that the sum of frequencies does not exceed fmax{b) (you might end up with slightly more than 1000 buckets)
■ Divide your heldout data according to buckets
■ Apply the previous algorithm to each bucket and its data
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Simple Example
■ Raw distribution (unigram only; smooth with uniform):
p(a) = .25, p(b) = .5, p(a) = 1/64 for a e {c.r}, = 0 for the rest: s, t, u, v, w, x, y, z
■ Heldout data: baby; use one set of A
(A-i: unigram, A0: uniform)
■ Start with A0 = A1 = .5:
c(Ai) = .5x.5/.27 + .5x.25/.14 + .5x.5/.27 + .5x0/.02 = 2.27 c(A0) = .5x.04/.27 + .5x.04/.14 + .5x.04/.27 + .5x.04/.02 = 1.28 Normalize M,next = .68, \0,next = -32
Repeat from step 2 (recomputep'a first for efficient computation, then
Finish when new lambdas almost equal to the old ones (say, < 0.01 difference).
pfx{b) = .5 x .5+ .5/26 pl(a) = .5 x .25+ .5/26 Px(y) = -5x0 + .5/26:
.27 .14 .02
c(A,), ...).
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Some More Technical Hints
■ Set V = {all words from training data}.
► You may also consider V = T u H, but it does not make the coding in any way simpler (in fact, harder).
► But: you must never use the test data for your vocabulary
■ Prepend two "words" in front of all data:
► avoids beginning-of-data problems
► call these index -1 and 0: then the formulas hold exactly
■ When cn(w,h) = 0:
► Assing 0 probability to pn(w|h) where cn_i (h) > 0, but a uniform probablity (1/|V|) to those pn(w|h) where cn_i (h) = 0 (this must be done both when working on the heldout data during EM, as well as when computing cross-entropy on the test data!)
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