Question

Use the Chain Rule to write the formula for the probability of sentence "w1 w2 w3...

Use the Chain Rule to write the formula for the probability of sentence "w1 w2 w3 w4 w5":

P(w1, w2, w3, w4, w5) = P(w1) * ...

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Answer #1

We have the formula for conditional probability as:

P(A,B) = p(A|B) p(B)

We can extend this for three variables,

P(A,B,C) = P(A| B,C) P(B,C) = P(A|B,C) P(B|C) P(C)

and in general to n variables:

P(A1, A2, ..., An) = P(A1| A2, ..., An) P(A2| A3, ..., An) P(An-1|An) P(An).

Thus using the above-mentioned formula, we can write:

P(w1, w2, w3, w4, w5) = P (w5, w4, w3, w2, w1)

= P( w5|w4, w3, w2, w1) * P(w4|w3, w2, w1) * P(w3|w2, w1) * P( w2| w1) * P(w1)

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