Language Modeling is the task of predicting what word comes next. More formally: given a sequence of words \(\boldsymbol{x}^{(1)}, \boldsymbol{x}^{(2)}, \ldots, \boldsymbol{x}^{(t)}\), compute the probability distribution of the next word \(\boldsymbol{x}^{(t+1)}\) :
\[
P\left(\boldsymbol{x}^{(t+1)} | \boldsymbol{x}^{(t)}, \ldots, \boldsymbol{x}^{(1)}\right)
\]
You can also think of a Language Model as a system that assigns probability to a piece of text. For example, if we have some text \(\boldsymbol{x}^{(1)}, \ldots, \boldsymbol{x}^{(T)}\), then the
probability of this text (according to the Language Model) is:
\[
\begin{aligned} P\left(\boldsymbol{x}^{(1)}, \ldots, \boldsymbol{x}^{(T)}\right) &=P\left(\boldsymbol{x}^{(1)}\right) \times P\left(\boldsymbol{x}^{(2)} | \boldsymbol{x}^{(1)}\right) \times \cdots \times P\left(\boldsymbol{x}^{(T)} | \boldsymbol{x}^{(T-1)}, \ldots, \boldsymbol{x}^{(1)}\right) \\ &=\prod_{t=1}^{T} P\left(\boldsymbol{x}^{(t)} | \boldsymbol{x}^{(t-1)}, \ldots, \boldsymbol{x}^{(1)}\right) \end{aligned}
\]
Question: How to learn a Language Model?
Answer: learn a n-gram Language Model!
Definition: A n-gram is a chunk of n consecutive words.
Idea: Collect statistics about how frequent different n-grams
are, and use these to predict next word.
First we make a simplifying assumption: \(\boldsymbol{x}^{(t+1)}\) depends only on the preceding n-1 words. The conditional prob is:
\[P\left(\boldsymbol{x}^{(t+1)} | \boldsymbol{x}^{(t)}, \ldots, \boldsymbol{x}^{(1)}\right)=P\left(\boldsymbol{x}^{(t+1)} | \boldsymbol{x}^{(t)}, \ldots, \boldsymbol{x}^{(t-n+2)}\right)=\frac{{P\left(\boldsymbol{x}^{(t+1)}, \boldsymbol{x}^{(t)}, \ldots, \boldsymbol{x}^{(t-n+2)}\right)}}{{P\left(\boldsymbol{x}^{(t)}, \ldots, \boldsymbol{x}^{(t-n+2)}\right)}}
\]
Question: How do we get these n-gram and (n-1)-gram probabilities?
Answer: By counting them in some large corpus of text!
原文：https://www.cnblogs.com/curtisxiao/p/10656859.html