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# Constituent Structure

Distribution of Words in Sentences: N-grams, Phrase Structure Syntax and Parsing

## unigram

Probability of each token chosen randomly (and independently of other tokens)
$unigram(t) = {Count(times\ t\ appearing)\over{Count(total\ word\ appearings)}}$

### Markov Assumption

Probability of each token chosen randomly (and independently of other tokens)

## bigram

Probability of a token given the previous token
$bigram(t, t_{previous}) = {{Count({t_{previous}}\rightarrow{t})}\over{Count(t_{previous})}}$

### Example

Count(the) = 69_971
Count(the -> same) = 628
bigram(same, the) = count(the -> same) / count(the) = 628 / 69_971 = 0.0898

1. 1.
Include probability that a word occurs at the beginning of a sentence, i.e. bigram(the, START)
2. 2.
Include probability that a token occurs at the end of a sentence, e.g. bigram(END, .)
3. 3.
Include non-zero probability for case when an unknown word follows a known one.

### Backoff Model

If a bigram has a zero count, "backoff" (use) the unigram of the word.
That is to replace bigram(current_word, previous_word) with unigram(current_word).

### Markov Assumption

Probability of a word depends only on the previous word.

## Trigrams, 4-grams, N-grams

### Trigram Probability

$trigram(t, t_{-1}, t_{-2})={Count({t_{-2}}\rightarrow{t_{-1}}\rightarrow{t})\over{Count({t_{-2}}\rightarrow{t_{-1}})}}$
Example: count(the -> same -> as) / count(the -> same)

### 4-gram Probability

$fourgram(t, t_{-1}, t_{-2}, t_{-3})={Count({t_{-3}}\rightarrow{t_{-2}}\rightarrow{t_{-1}}\rightarrow{t})\over{Count({t_{-3}}\rightarrow{t_{-2}}\rightarrow{t_{-1}})}}$
Example: count(the -> same -> as -> an) / count(the -> same -> as)

### N-gram Probability

$ngram(t, t_{-1}, ..., t_{-n+1})={Count({t_{-n+1}}\rightarrow{...}\rightarrow{t_{-1}}\rightarrow{t})\over{Count({t_{-n+1}}\rightarrow{...}\rightarrow{t_{-1}})}}$

### Markov Assumptions

Trigram Model: probability of a word depends only on the previous two words.
N-gram Model: probability of a word depends only on the previous N-1 words.
Probability of a sentence = Product of probabilities of each word.

## Noun Phrases and Noun Groups

Both can have left modifiers. Only noun phrases can have right modifiers.
• A noun group consists of: left modifiers of the head noun and the head noun
• We will assume that all punctuation and coordinate conjunctions are outside of a noun group