# Language Model (LM)

A Language Model (LM) is a Statistical Model that estimates a probability function of a sequence of language units.

## References

### 2018a

• (Wikipedia, 2018) ⇒ https://en.wikipedia.org/wiki/language_model Retrieved:2018-4-8.
• A statistical language model is a probability distribution over sequences of words. Given such a sequence, say of length m, it assigns a probability $P(w_1,\ldots,w_m)$ to the whole sequence. Having a way to estimate the relative likelihood of different phrases is useful in many natural language processing applications, especially ones that generate text as an output. Language modeling is used in speech recognition, machine translation, part-of-speech tagging, parsing, handwriting recognition, information retrieval and other applications.

In speech recognition, the computer tries to match sounds with word sequences. The language model provides context to distinguish between words and phrases that sound similar. For example, in American English, the phrases "recognize speech" and "wreck a nice beach" are pronounced almost the same but mean very different things. These ambiguities are easier to resolve when evidence from the language model is incorporated with the pronunciation model and the acoustic model.

Language models are used in information retrieval in the query likelihood model. Here a separate language model is associated with each document in a collection. Documents are ranked based on the probability of the query Q in the document's language model $P(Q\mid M_d)$ . Commonly, the unigram language model is used for this purpose— otherwise known as the bag of words model.

Data sparsity is a major problem in building language models. Most possible word sequences will not be observed in training. One solution is to make the assumption that the probability of a word only depends on the previous n words. This is known as an n-gram model or unigram model when n = 1.

### 2013b

1. For any $\langle x_1 \cdots x_n \rangle \in \mathcal{V}^{\dagger} ,\; p(x_1, x_2, \cdots x_n) \geq 0$
$\displaystyle \sum_{\langle x_1 \cdots x_n \rangle \in \mathcal{V}^{\dagger}} p(x_1, x_2, \cdots x_n) = 1$
Hence $p(x_1, x_2, \cdots, x_n)$ is a probability distribution over the sentences in $\mathcal{V}^{\dagger}$ .