Monday, 6 April 2020

Bigram probability estimate of a word sequence

Bigram probability estimate of a word sequence, Probability estimation for a sentence using Bigram language model


Bigram Model - Probability Calculation - Example Problem


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Solved Example:

Let us solve a small example to better understand the Bigram model. For this we need a corpus and the test data. Let us assume that the following is a small corpus;

Training corpus:
<s> I am from Vellore </s>
<s> I am a teacher </s>
<s> students are good and are from various cities</s>
<s> students from Vellore do engineering</s>

Test data:
<s> students are from Vellore </s>

Let us find the Bigram probability of the given test sentence. I explained the solution in two methods, just for the sake of understanding. the second method is the formal way of calculating the bigram probability of a sequence of words.

Method 1 
As per the Bigram model, the test sentence can be expanded as follows to estimate the bigram probability;
P(<s> students are from Vellore </s>)
                               = P(students | <s>) * P(are | students) * P(from | are) 
                                 * P(Vellore | from) *    P(</s> | Vellore)

To estimate bigram probabilities, we can use the following equation;
[Hint – count of sentence start (<s>) = 4, count of string <s> students = 1]
[Hint – count of word students = 2, count of string students are = 1]
[Hint – count of word are = 2, count of string are from = 1]
[Hint – count of word from = 3, count of string from Vellore = 2]
[Hint – count of word Vellore = 2, count of string Vellore </s> = 1]
P(<s> students are from Vellore </s>)
                        = P(students | <s>) * P(are | students) * P(from | are) 
                             * P(Vellore | from) * P(</s> | Vellore)
                        = 1/4 * 1/2 * 1/2 * 2/3 * 1/2 = 0.0208

Method 2
Formal way of estimating the bigram probability of a word sequence:
The bigram probabilities of the test sentence can be calculated by constructing Unigram and bigram probability count matrices and bigram probability matrix as follows;
Unigram count matrix
<s>
students
are
from
Vellore
4
2
2
3
2

Bigram count matrix


wn


students
are
from
Vellore
</s>


wn-1
<s>
1
0
0
0
0
students
0
1
1
0
0
are
0
0
1
0
0
from
0
0
0
2
0
Vellore
0
0
0
0
1

Bigram probability matrix (normalized by unigram counts)


wn


students
are
from
Vellore
</s>


wn-1
<s>
1/4
0/4
0/4
0/4
0/4
students
0/2
1/2
1/2
0/2
0/2
are
0/2
0/2
1/2
0/2
0/2
from
0/3
0/3
0/3
2/3
0/3
Vellore
0/2
0/2
0/2
0/2
1/2

P(<s> students are from Vellore </s>)
                          = P(students | <s>) * P(are | students) * P(from | are) 
                              * P(Vellore | from) * P(</s> | Vellore)
                          = 1/4 * 1/2 * 1/2 * 2/3 * 1/2 = 0.0208

The probability of the test sentence as per the bigram model is 0.0208.

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