Which expression corresponds to the 'Trigram calculation' for P(w1,w2,w3)?

Which expression corresponds to the 'Trigram calculation' for P(w1,w2,w3)?

• A. P(w1,w2,w3)=P(w1)⋅P(w2∣w1)⋅P(w3∣w2)
• B. P(w1,w2,w3)=P(w1)⋅P(w2)⋅P(w3)
• C. P(w1,w2,w3)=P(w1)⋅P(w2∣w1)⋅P(w3)
• D. P(w1,w2,w3)=P(w1)⋅P(w2∣w1)⋅P(w3∣w2,w1)

Answer: (D)

Trigram calculation uses the chain rule to factor a three-word sequence into successive conditional probabilities, where each word depends on the two words before it. The first word is unconditional: P(w1). The second word depends on the first: P(w2|w1). The third word, in a trigram model, depends on both preceding words: P(w3|w1, w2). Since the two preceding words form the conditioning context, you can write this as P(w3|w2, w1). Multiplying these together gives the joint probability: P(w1, w2, w3) = P(w1) · P(w2|w1) · P(w3|w1, w2). This is the form that uses both prior words to predict the next, which is what the trigram model specifies. The other forms either drop a dependence or assume independence, corresponding to unigram or bigram models rather than trigram.

Trigram calculation uses the chain rule to factor a three-word sequence into successive conditional probabilities, where each word depends on the two words before it. The first word is unconditional: P(w1). The second word depends on the first: P(w2|w1). The third word, in a trigram model, depends on both preceding words: P(w3|w1, w2). Since the two preceding words form the conditioning context, you can write this as P(w3|w2, w1). Multiplying these together gives the joint probability: P(w1, w2, w3) = P(w1) · P(w2|w1) · P(w3|w1, w2). This is the form that uses both prior words to predict the next, which is what the trigram model specifies. The other forms either drop a dependence or assume independence, corresponding to unigram or bigram models rather than trigram.