Size of the vocabulary in Laplace smoothing for a trigram language model












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Let's say we have a text document with $N$ unique words making up a vocabulary $V$, $|V| = N$. For a bigram language model with add-one smoothing, we define a conditional probability of any word $w_{i}$ given the preceeding word $w_{i-1}$ as: $$P(w_{i}|w_{i-1}) = frac{count(w_{i-1}w_{i}) + 1}{count(w_{i-1}) + |V|}$$ As far as I understand (or not) the conditional probability, and basing on a 3rd point of this Wikipedia article, $w_{i-1}$ might be assumed to be "constant" here, so by summing this expression for all possible $w_{i}$ we should obtain 1, and so it is, which is obvious.



However, I do not understand the answers given for this question saying that for n-gram model the size of the vocabulary should be the count of the unique (n-1)-grams occuring in a document, for example, given a 3-gram model (let $V_{2}$ be the dictionary of bigrams): $$P(w_{i}|w_{i-2}w_{i-1}) = frac{count(w_{i-2}w_{i-1}w_{i}) + 1}{count(w_{i-2}w_{i-1}) + |V_{2}|}$$ It just doesn't add up to 1 when we try to sum it for every possible $w_{i}$. Therefore - should the $|V|$ really be equal to the count of unique (n-1)-grams given an n-gram language model or should it be the count of unique unigrams?






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    5












    $begingroup$


    Let's say we have a text document with $N$ unique words making up a vocabulary $V$, $|V| = N$. For a bigram language model with add-one smoothing, we define a conditional probability of any word $w_{i}$ given the preceeding word $w_{i-1}$ as: $$P(w_{i}|w_{i-1}) = frac{count(w_{i-1}w_{i}) + 1}{count(w_{i-1}) + |V|}$$ As far as I understand (or not) the conditional probability, and basing on a 3rd point of this Wikipedia article, $w_{i-1}$ might be assumed to be "constant" here, so by summing this expression for all possible $w_{i}$ we should obtain 1, and so it is, which is obvious.



    However, I do not understand the answers given for this question saying that for n-gram model the size of the vocabulary should be the count of the unique (n-1)-grams occuring in a document, for example, given a 3-gram model (let $V_{2}$ be the dictionary of bigrams): $$P(w_{i}|w_{i-2}w_{i-1}) = frac{count(w_{i-2}w_{i-1}w_{i}) + 1}{count(w_{i-2}w_{i-1}) + |V_{2}|}$$ It just doesn't add up to 1 when we try to sum it for every possible $w_{i}$. Therefore - should the $|V|$ really be equal to the count of unique (n-1)-grams given an n-gram language model or should it be the count of unique unigrams?






    probability machine-learning







    share|cite|improve this question











    $endgroup$















      5












      5








      5





      $begingroup$


      Let's say we have a text document with $N$ unique words making up a vocabulary $V$, $|V| = N$. For a bigram language model with add-one smoothing, we define a conditional probability of any word $w_{i}$ given the preceeding word $w_{i-1}$ as: $$P(w_{i}|w_{i-1}) = frac{count(w_{i-1}w_{i}) + 1}{count(w_{i-1}) + |V|}$$ As far as I understand (or not) the conditional probability, and basing on a 3rd point of this Wikipedia article, $w_{i-1}$ might be assumed to be "constant" here, so by summing this expression for all possible $w_{i}$ we should obtain 1, and so it is, which is obvious.



      However, I do not understand the answers given for this question saying that for n-gram model the size of the vocabulary should be the count of the unique (n-1)-grams occuring in a document, for example, given a 3-gram model (let $V_{2}$ be the dictionary of bigrams): $$P(w_{i}|w_{i-2}w_{i-1}) = frac{count(w_{i-2}w_{i-1}w_{i}) + 1}{count(w_{i-2}w_{i-1}) + |V_{2}|}$$ It just doesn't add up to 1 when we try to sum it for every possible $w_{i}$. Therefore - should the $|V|$ really be equal to the count of unique (n-1)-grams given an n-gram language model or should it be the count of unique unigrams?






      probability machine-learning







      share|cite|improve this question











      $endgroup$




      Let's say we have a text document with $N$ unique words making up a vocabulary $V$, $|V| = N$. For a bigram language model with add-one smoothing, we define a conditional probability of any word $w_{i}$ given the preceeding word $w_{i-1}$ as: $$P(w_{i}|w_{i-1}) = frac{count(w_{i-1}w_{i}) + 1}{count(w_{i-1}) + |V|}$$ As far as I understand (or not) the conditional probability, and basing on a 3rd point of this Wikipedia article, $w_{i-1}$ might be assumed to be "constant" here, so by summing this expression for all possible $w_{i}$ we should obtain 1, and so it is, which is obvious.



      However, I do not understand the answers given for this question saying that for n-gram model the size of the vocabulary should be the count of the unique (n-1)-grams occuring in a document, for example, given a 3-gram model (let $V_{2}$ be the dictionary of bigrams): $$P(w_{i}|w_{i-2}w_{i-1}) = frac{count(w_{i-2}w_{i-1}w_{i}) + 1}{count(w_{i-2}w_{i-1}) + |V_{2}|}$$ It just doesn't add up to 1 when we try to sum it for every possible $w_{i}$. Therefore - should the $|V|$ really be equal to the count of unique (n-1)-grams given an n-gram language model or should it be the count of unique unigrams?






      probability machine-learning




      probability machine-learning






      share|cite|improve this question















      share|cite|improve this question













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      edited May 30 '16 at 23:21







      cafe_

















      asked May 30 '16 at 22:51









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          The slides here mention that V should be the size of the vocabulary which makes sense for me.






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            The slides here mention that V should be the size of the vocabulary which makes sense for me.






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                The slides here mention that V should be the size of the vocabulary which makes sense for me.






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                The slides here mention that V should be the size of the vocabulary which makes sense for me.







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                answered Feb 4 '18 at 5:09









                Avinash KumarAvinash Kumar

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