Language model

Unigram models

A unigram model used in information retrieval can be treated as the combination of several one-state finite automata. It splits the probabilities of different terms in a context, e.g. from P ( t 1 t 2 t 3 ) = P ( t 1 ) P ( t 2 ∣ t 1 ) P ( t 3 ∣ t 1 t 2 ) to P uni ( t 1 t 2 t 3 ) = P ( t 1 ) P ( t 2 ) P ( t 3 ) .

In this model, the probability to hit each word all depends on its own, so we only have one-state finite automata as units. For each automaton, we only have one way to hit its only state, assigned with one probability. Viewing from the whole model, the sum of all the one-state-hitting probabilities should be 1. Followed is an illustration of a unigram model of a document.

The probability generated for a specific query is calculated as

For different documents, we can build their own unigram models, with different hitting probabilities of words in it. And we use probabilities from different documents to generate different hitting probabilities for a query. Then we can rank documents for a query according to the generating probabilities. Next is an example of two unigram models of two documents.

In information retrieval contexts, unigram language models are often smoothed to avoid instances where P(term) = 0. A common approach is to generate a maximum-likelihood model for the entire collection and linearly interpolate the collection model with a maximum-likelihood model for each document to create a smoothed document model.

n-gram models

In an n-gram model, the probability P ( w 1 , … , w m ) of observing the sentence w 1 , … , w m is approximated as

Here, it is assumed that the probability of observing the i^th word w_i in the context history of the preceding i − 1 words can be approximated by the probability of observing it in the shortened context history of the preceding n − 1 words (n^th order Markov property).

The conditional probability can be calculated from n-gram model frequency counts:

The words bigram and trigram language model denote n-gram model language models with n = 2 and n = 3, respectively.

Typically, however, the n-gram model probabilities are not derived directly from the frequency counts, because models derived this way have severe problems when confronted with any n-grams that have not explicitly been seen before. Instead, some form of smoothing is necessary, assigning some of the total probability mass to unseen words or n-grams. Various methods are used, from simple "add-one" smoothing (assign a count of 1 to unseen n-grams) to more sophisticated models, such as Good-Turing discounting or back-off models.

Example

In a bigram (n = 2) language model, the probability of the sentence I saw the red house is approximated as

whereas in a trigram (n = 3) language model, the approximation is

Note that the context of the first n – 1 n-grams is filled with start-of-sentence markers, typically denoted <s>.

Additionally, without an end-of-sentence marker, the probability of an ungrammatical sequence *I saw the would always be higher than that of the longer sentence I saw the red house.

Exponential language models

Maximum entropy language models encode the relationship between a word and the n-gram history using feature functions. The equation is P ( w 1 , … , w m ) = 1 Z ( w 1 , … , w m − 1 ) exp ⁡ ( a T f ( w 1 , … , w m ) ) where Z ( w 1 , … , w m − 1 ) is the partition function, a is the parameter vector, and f ( w 1 , … , w m ) is the feature function. In the simplest case, the feature function is just an indicator of the presence of a certain n-gram. It is helpful to use a prior on a or some form of regularization.

The log-bilinear model is another example of an exponential language model.

Continuous space language models

Continuous space language models use continuous representations or embeddings of words to make their predictions. These models make use of Neural networks.

Continuous space embeddings help to alleviate the curse of dimensionality in language modeling: as language models are trained on larger and larger texts, the number of unique words (the vocabulary) increases and the number of possible sequences of words increases exponentially with the size of the vocabulary, causing a data sparsity problem because for each of the exponentially many sequences, statistics are needed to properly estimate probabilities. Neural networks avoid this problem by representing words in a distributed way, as non-linear combinations of weights in a neural net. The neural net architecture might be feed-forward or recurrent.

Typically, neural net language models are constructed and trained as probabilistic classifiers that learn to predict a probability distribution

I.e., the network is trained to predict a probability distribution over the vocabulary, given some linguistic context. This is done using standard neural net training algorithms such as stochastic gradient descent with backpropagation. The context might be a fixed-size window of previous words, so that the network predicts

from a feature vector representing the previous k words. Another option is to use "future" words as well as "past" words as features, so that the estimated probability is

A third option, that allows faster training, is to invert the previous problem and make a neural network learn the context, given a word. One then maximizes the log-probability

This is called a skip-gram language model, and is the basis of the popular word2vec program.

Instead of using neural net language models to produce actual probabilities, it is common to instead use the distributed representation encoded in the networks' "hidden" layers as representations of words; each word is then mapped onto an n-dimensional real vector called the word embedding, where n is the size of the layer just before the output layer. The representations in skip-gram models have the distinct characteristic that they model semantic relations between words as linear combinations, capturing a form of compositionality. For example, in some such models, if v is the function that maps a word w to its n-d vector representation, then

where ≈ is made precise by stipulating that its right-hand side must be the nearest neighbor of the value of the left-hand side.

Other models

A positional language model is one that describes the probability of given words occurring close to one another in a text, not necessarily immediately adjacent. Similarly, bag-of-concepts models leverage on the semantics associated with multi-word expressions such as buy_christmas_present, even when they are used in information-rich sentences like "today I bought a lot of very nice Christmas presents".