Language model in natural language processing, Bigram Trigram and Ngram language models
Language Model in Natural Language Processing
A statistical language model is a probability distribution over sequences of strings/words, and assigns a probability to every string in the language.
Language models are based on a
probabilistic description of language phenomena. Probabilities are used to pick
the most frequent of several alternatives. Hence, probabilistic langue models used
in various NLP applications including spelling correction, speech recognition,
syntactic analysis, text summarization, word prediction, etc. to predict the most probable word sequence (or sentence). You would probably understand on "how probability contributes on some of these applications?" from the example below;
Example:
NLP
Application

Probability
comparison between correct and wrong sequences of words


Spell correction

➔

P(I love flowers) > P(I lvoe flowers)

Speech recognition

➔

P(I saw a man) > P(eyes awe a man)

Sentence formation

➔

P(I am doing law) > P(am law doing I)

Word prediction

➔

P(Please turn the
lights on) > P(Please turn the
lights right)

How to compute the probability of a sentence?
A sentence (W) is a sequence of words (w_{1},
w_{2}, …, w_{n}) and the probability of the same can be
calculated as follows;
P(W) = P(w_{1}, w_{2}, …, w_{n})
Also, the probability of an upcoming word
can be calculated of a given word sequence;
P(w_{n}  w_{1},
w_{2}, …, w_{n1})
The model that calculates either P(W) or P(w_{1}, w_{2}, …, w_{n})
is called the language model.
How to calculate P(w_{1},
w_{2}, …, w_{n})?
P(w_{1},
w_{2}, …, w_{n}) is a joint probability. Let
us calculate the joint probability P(A, B) for two events A and B.
The joint probability can be calculated
using the conditional probability as follows;
Conditional probability,
which can be rewritten as,
P(A,
B) = P(A  B) * P(B)
By Bayes’ theorem to manipulate conditional
probability, we can further expand the joint probability equation as follows;
P(A,
B) = P(A) * P(B  A)
Chain rule of probability:
The above derived joint probability can be
extended to more variables as follows;
P(A,
B, C, D) = P(A) * P(B  A) * P(C  A, B) * P(D  A, B, C)
This is referred as chain rule. This can be generalized
and used to calculate the joint probability of our word sequence P(w_{1}, w_{2}, …, w_{n})
as follows;
The chain rule can be used to find the
probability of given sequence of words. For example, probability of the
sentence (word sequence) “the prime minister of our country” can be calculated
as follows;
Now, if you like to compute the probability
of the word sequence W, we need to calculate each component in it. For instance,
for the example given in Equation 2, we need to measure the component P(ourthe prime minister of). One way
to measure this is relative frequency
count which is as follows;
This can be read as, "out of the number of times we saw
‘the prime minister of’ in a corpus, how
many times was it followed by the word ‘our’".
Due to sparseness of data and the length of
the given test data (sentence), one or more of the components in the above
equation may 0 which leads the result to 0. How
do we handle this problem?
Markov assumption:
The probability of a word depends only on
the previous word or previous couple of words. According to this assumption, we
do not need to look too far in the history to predict the future. Instead, we
can approximate the history by just the last few words. As per the Markov assumption,
one of the components in the Equation 2 of our example can be modified as
follows;
P(our  the prime minister of) ≈ P(our  of)
OR
P(our  the prime minister of) ≈ P(our  minister of)
Here, the first one is bigram model, and the second one is trigram model.
As per the bigram model, we are making the
following approximation;
P(w_{n}
 w_{1}, w_{2}, …, w_{n1}) ≈ P(w_{n}  w_{n1})
As per the trigram model, we are making the
following approximation;
P(w_{n}  w_{1}, w_{2}, …, w_{n1})
≈ P(w_{n}  w_{n1}, w_{n1})
These models can be generalized as Ngram
model which looks N1 words from the history. The Ngram approximation to the conditional
probability of the next words in a sequence is;
P(w_{n}
 w_{1}, w_{2}, …, w_{n1}) ≈ P(w_{n}  w_{nk},
…, w_{n1})
Here, k represents the particular model. For
instance, k=1 refers to Unigram model, k=2 refers to Bigram model, k=3 refers
to Trigram model, and so on.

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