Natural Language Processing · M2 · formulas

Trigram estimates: two-word history, not two denominators

N-grams & probability

Formulas explained through full worked examples

A trigram model predicts the next word from the two preceding words. The numerator is the count of a triple. The denominator is the count of its two-word history with an available next token. Do not multiply two independent bigram probabilities and call that a trigram probability.

P(w_i\mid w_{i-2},w_{i-1})=\frac{C(w_{i-2},w_{i-1},w_i)}{C(w_{i-2},w_{i-1})}

Question. A corpus contains 12 occurrences of “prime minister”. Eight are followed by “of”, three by “said”, and one by “visited”. Estimate the three trigram probabilities.

Given

History h = (prime, minister). Count of this history = 12. Triple counts for of, said, visited are 8, 3, 1.

To find

The conditional distribution of the next word.

Formula and why it applies

P(w\mid prime,minister)=C(prime,minister,w)/12

All three candidates share one two-word history, so they share denominator 12.

Solution

Step 1. Calculate of

P(of\mid prime,minister)=8/12=2/3

Step 2. Calculate said

P(said\mid prime,minister)=3/12=1/4

Step 3. Calculate visited

P(visited\mid prime,minister)=1/12

Step 4. Confirm normalization

8/12+3/12+1/12=12/12=1

Final answer. The probabilities are 2/3, 1/4 and 1/12.

Question. Train a trigram model on “I like tea”, “I like coffee” and “you like tea”. Use two start markers and one end marker. Score “I like tea”.

Given

Three sentences. Padded first sentence is <s> <s> I like tea </s>; pad the other two in the same way.

To find

Four trigram prediction probabilities and the sentence probability.

Formula and why it applies

P=\prod_i P(w_i\mid w_{i-2},w_{i-1})

Each prediction uses a two-token context; the two start markers supply the first history.

Solution

Step 1. Count the first-word event

The history (<s>,<s>) occurs three times, and two are followed by I.

P(I\mid\langle s\rangle,\langle s\rangle)=2/3

Step 2. Count the second-word event

(<s>,I) occurs twice; both are followed by like.

P(like\mid\langle s\rangle,I)=2/2=1

Step 3. Count the third-word event

(I,like) occurs twice; one is followed by tea.

P(tea\mid I,like)=1/2

Step 4. Count the end event

(like,tea) occurs twice; both are followed by end.

P(\langle/s\rangle\mid like,tea)=2/2=1

Step 5. Multiply all four

P=\frac23\times1\times\frac12\times1=\frac13

Final answer. Trigram P(I like tea) = 1/3. It differs from the bigram estimate because the context and denominator changed.

Question. A trigram model supplies P(we|<s>,<s>)=0.4, P(study|<s>,we)=0.5, P(NLP|we,study)=0.8 and P(</s>|study,NLP)=0.25. Score “we study NLP”.

Given

Four trigram conditional factors. The end marker is included.

To find

The complete sentence probability.

Formula and why it applies

P=P(we\mid s,s)P(study\mid s,we)P(NLP\mid we,study)P(end\mid study,NLP)

Use each supplied two-word history exactly; do not swap its word order.

Solution

Step 1. Insert the values

P=0.4\times0.5\times0.8\times0.25

Step 2. Multiply the first pair

0.4\times0.5=0.2

Step 3. Include NLP

0.2\times0.8=0.16

Step 4. Include the end

0.16\times0.25=0.04

Final answer. P = 0.04.

Yaad rakho: Trigram: teen-word count upar, pehle do-word count neeche.

Why larger n is not automatically better. More context can capture local order more precisely, but each longer history occurs less often. Parameter count, storage and sparsity grow. A model trained on Shakespeare may assign poor probabilities to financial news because training and test distributions differ. Keep a held-out test set; do not evaluate only on memorized training sentences.