Some fixes in LM part regarding ngram history length + MLE ngram #13
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lecture_note.tex
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| on the $n-1$ preceding symbols only, meaning | ||
| \begin{align*} | ||
| p(w_k | w_{<k}) \approx p(w_k | w_{k-n}, w_{k-n+1}, \ldots, w_{k-1}). | ||
| % p(w_k | w_{<k}) \approx p(w_k | w_{k-n}, w_{k-n+1}, \ldots, w_{k-1}). % history length should be n-1 |
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lecture_note.tex
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| This results in | ||
| \begin{align*} | ||
| p(S) \approx \prod_{t=1}^T p(w_t | w_{t-n}, \ldots, w_{t-1}). | ||
| p(S) \approx \prod_{t=1}^T p(w_t | w_{t-n+1}, \ldots, w_{t-1}). % history should have n-1 length |
lecture_note.tex
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| The biggest issue of having an $n$-gram that never occurs in the training corpus | ||
| is that any sentence containing the $n$-gram will be given a zero probability | ||
| is that any sentence containing such $n$-gram will be given a zero probability |
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So given the definition of n-gram the text is 100% correct but in the formulas there are always histories of length n, which is probably a typo. I have also added small explanation about why relative freq. ngram estimator is optimal from the MLE perspective.