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520.666 Information Extraction from Speech and Text Homework # 4: Alternative strategies for smoothing a bigram language model
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Information Extraction from Speech and Text
Homework # 4
Due March 10, 2023.
In class, we discussed linear interpolation for smoothing a bigram language model, namely
P(wjv) = f(wjv) + (1 )f(w);
wheref(j) andf() denoted the appropriate relative frequency estimates, and
was chosen so as to maximize the probability of some held-out data.
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This homework considers alternative strategies for smoothing a bigram language model
by directly modifying the counts observed in the training data. In particular, let C(v;w)
denote the count of a bigram hv;wiin the training text , and letC(v;w) be the modi ed
count. For some constant >0, consider the three cases
(i)C(v;w) =C(v;w) +,
(ii)C(v;w) =C(v;w) +C(w), and
(iii)C(v;w) =C(v;w) +C(v)f(w).
In each case, the smoothed bigram probability is calculated as
P(wjv) =C(v;w)
P
w02VC(v;w0):
LetN(v;w) denote the count of a bigram hv;wiin the held-out textH.
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Derive an expression for the that maximizes the log-probability
P(H) =Xv2VXw2VN(v;w) logP(wjv)
of the held-out text in each of the three cases (i), (ii) and (iii) above.
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Show that if N(v;w) =C(v;w) for all bigramshv;wi, then the optimal value is = 0
in each case. Why is this an expected result?
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Show, in each case, that Pmay be written as the linear interpolation of a bigram and
a lower order language model, though not necessarily f(w).
P(wjv) =f2(wjv) + (1 )f1(w);
i.e., identify f1,f2and
, and discuss the merits/drawbacks of each smoothing strategy.
After finishing the homework, carefully review all sections of Chapter 4 again.
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