CS 388: Natural Language Processing: N-Gram Language Models Raymond J. Mooney University of Texas at Austin 1 Language Models Formal grammars (e.g. regular, context free) give a hard binary model of the legal sentences in a language. For NLP, a probabilistic model of a language that gives a probability that a string is a member of a language is more useful. To specify a correct probability distribution, the probability of all sentences in a

language must sum to 1. Uses of Language Models Speech recognition I ate a cherry is a more likely sentence than Eye eight uh Jerry OCR & Handwriting recognition More probable sentences are more likely correct readings. Machine translation More likely sentences are probably better translations. Generation More likely sentences are probably better NL generations. Context sensitive spelling correction Their are problems wit this sentence.

Completion Prediction A language model also supports predicting the completion of a sentence. Please turn off your cell _____ Your program does not ______ Predictive text input systems can guess what you are typing and give choices on how to complete it. N-Gram Models Estimate probability of each word given prior context. P(phone | Please turn off your cell) Number of parameters required grows exponentially with the number of words of prior context. An N-gram model uses only N1 words of prior context.

Unigram: P(phone) Bigram: P(phone | cell) Trigram: P(phone | your cell) The Markov assumption is the presumption that the future behavior of a dynamical system only depends on its recent history. In particular, in a kth-order Markov model, the next state only depends on the k most recent states, therefore an N-gram model is a (N1)-order Markov model. N-Gram Model Formulas Word sequences w1n w1...wn Chain rule of probability n 1

2 1 n P( w ) P( w1 ) P( w2 | w1 ) P( w3 | w )...P( wn | w ) P( wk | w1k 1 ) Bigram approximation n 1 n P( w ) P( wk | wk 1 ) k 1 N-gram approximation n

1 n P( w ) P( wk | wkk 1N 1 ) k 1 n 1 1 k 1 Estimating Probabilities N-gram conditional probabilities can be estimated from raw text based on the relative frequency of word sequences. Bigram:

C ( wn 1wn ) P ( wn | wn 1 ) C ( wn 1 ) N-gram: n 1 C ( w n N 1 wn ) P( wn | wnn 1N 1 ) C ( wnn 1N 1 ) To have a consistent probabilistic model, append a unique start (~~) and end (~~) symbol to every sentence and treat these as additional words.

Generative Model & MLE An N-gram model can be seen as a probabilistic automata for generating sentences. Initialize sentence with N1 ~~ symbols Until ~~ is generated do: Stochastically pick the next word based on the conditional probability of each word given the previous N 1 words. Relative frequency estimates can be proven to be maximum likelihood estimates (MLE) since they maximize the probability that the model M will generate the training corpus T. argmax P(T | M ( )) Example from Textbook P(~~ i want english food ~~) = P(i | ~~) P(want | i) P(english | want) ~~

~~
~~ ~~
~~~~
~~
~~P(food | english) P(~~ | food) = .25 x .33 x .0011 x .5 x .68 = .000031 P(~~ i want chinese food ~~) = P(i | ~~) P(want | i) P(chinese | want) P(food | chinese) P(~~ | food) = .25 x .33 x .0065 x .52 x .68 = .00019 Train and Test Corpora A language model must be trained on a large corpus of text to estimate good parameter values. Model can be evaluated based on its ability to predict a high probability for a disjoint (held-out) test corpus (testing on the training corpus would give an optimistically biased estimate). Ideally, the training (and test) corpus should be representative of the actual application data. May need to adapt a general model to a small

amount of new (in-domain) data by adding highly weighted small corpus to original training data. Unknown Words How to handle words in the test corpus that did not occur in the training data, i.e. out of vocabulary (OOV) words? Train a model that includes an explicit symbol for an unknown word (). Choose a vocabulary in advance and replace other words in the training corpus with . Replace the first occurrence of each word in the training data with . Evaluation of Language Models Ideally, evaluate use of model in end application (extrinsic, in vivo)

Realistic Expensive Evaluate on ability to model test corpus (intrinsic). Less realistic Cheaper Verify at least once that intrinsic evaluation correlates with an extrinsic one. Perplexity Measure of how well a model fits the test data. Uses the probability that the model assigns to the test corpus. Normalizes for the number of words in the test corpus and takes the inverse. 1

PP (W ) N P ( w1w2 ...wN ) Measures the weighted average branching factor in predicting the next word (lower is better). Sample Perplexity Evaluation Models trained on 38 million words from the Wall Street Journal (WSJ) using a 19,979 word vocabulary. Evaluate on a disjoint set of 1.5 million WSJ words. Unigram Perplexity 962 Bigram 170

Trigram 109 Smoothing Since there are a combinatorial number of possible word sequences, many rare (but not impossible) combinations never occur in training, so MLE incorrectly assigns zero to many parameters (a.k.a. sparse data). If a new combination occurs during testing, it is given a probability of zero and the entire sequence gets a probability of zero (i.e. infinite perplexity). In practice, parameters are smoothed (a.k.a. regularized) to reassign some probability mass to unseen events. Adding probability mass to unseen events requires removing it from seen ones (discounting) in order to

maintain a joint distribution that sums to 1. Laplace (Add-One) Smoothing Hallucinate additional training data in which each possible N-gram occurs exactly once and adjust estimates accordingly. C ( wn 1wn ) 1 C ( wn 1 ) V Bigram: P( wn | wn 1 ) N-gram: n 1 C (

w n N 1 wn ) 1 P( wn | wnn 1N 1 ) C ( wnn 1N 1 ) V where V is the total number of possible (N1)-grams (i.e. the vocabulary size for a bigram model). Tends to reassign too much mass to unseen events, so can be adjusted to add 0<<1 (normalized by V instead of V). Advanced Smoothing Many advanced techniques have been developed to improve smoothing for language models.

Good-Turing Interpolation Backoff Kneser-Ney Class-based (cluster) N-grams Model Combination As N increases, the power (expressiveness) of an N-gram model increases, but the ability to estimate accurate parameters from sparse data decreases (i.e. the smoothing problem gets worse). A general approach is to combine the results of multiple N-gram models of increasing complexity (i.e. increasing N).

Interpolation Linearly combine estimates of N-gram models of increasing order. Interpolated Trigram Model: P ( wn | wn 2, wn 1 ) 1 P ( wn | wn 2, wn 1 ) 2 P ( wn | wn 1 ) 3 P ( wn ) Where: i 1 i Learn proper values for i by training to

(approximately) maximize the likelihood of an independent development (a.k.a. tuning) corpus. Backoff Only use lower-order model when data for higherorder model is unavailable (i.e. count is zero). Recursively back-off to weaker models until data is available. n 1 n P * ( w | w )

if C ( w n 1 n n N 1 n N 1 ) 1 Pkatz ( wn | wn N 1 ) n 1 n 1 ( w ) P ( w

| w otherwise n N 1 katz n n N 2 ) Where P* is a discounted probability estimate to reserve mass for unseen events and s are back-off weights s are back-off weights (see text for details). A Problem for N-Grams: Long Distance Dependencies Many times local context does not provide the most useful predictive clues, which instead are provided by long-distance dependencies.

Syntactic dependencies The man next to the large oak tree near the grocery store on the corner is tall. The men next to the large oak tree near the grocery store on the corner are tall. Semantic dependencies The bird next to the large oak tree near the grocery store on the corner flies rapidly. The man next to the large oak tree near the grocery store on the corner talks rapidly. More complex models of language are needed to handle such dependencies. Summary Language models assign a probability that a sentence is a legal string in a language.

They are useful as a component of many NLP systems, such as ASR, OCR, and MT. Simple N-gram models are easy to train on unsupervised corpora and can provide useful estimates of sentence likelihood. MLE gives inaccurate parameters for models trained on sparse data. Smoothing techniques adjust parameter estimates to account for unseen (but not impossible) events.