CS 388: Natural Language Processing: N-Gram Language ModelsLanguage ModelsUses of Language ModelsCompletion PredictionN-Gram ModelsN-Gram Model FormulasEstimating ProbabilitiesGenerative Model & MLEExample from TextbookTrain and Test CorporaUnknown WordsEvaluation of Language ModelsPerplexitySample Perplexity EvaluationSmoothingLaplace (Add-One) SmoothingAdvanced SmoothingModel CombinationInterpolationBackoffA Problem for N-Grams: Long Distance DependenciesSummary1CS 388: Natural Language Processing:N-Gram Language ModelsRaymond J. MooneyUniversity of Texas at AustinLanguage 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 N1 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 (N1)-order Markov model.N-Gram Model Formulas•Word sequences•Chain rule of probability•Bigram approximation•N-gram approximationnnwww ...11)|()|()...|()|()()(111112131211knkknnnwwPwwPwwPwwPwPwP)|()(1111kNknkknwwPwP)|()(111 knkknwwPwPEstimating Probabilities•N-gram conditional probabilities can be estimated from raw text based on the relative frequency of word sequences.•To have a consistent probabilistic model, append a unique start (<s>) and end (</s>) symbol to every sentence and treat these as additional words.)()()|(111nnnnnwCwwCwwP)()()|(111111nNnnnNnnNnnwCwwCwwPBigram:N-gram:Generative Model & MLE•An N-gram model can be seen as a probabilistic automata for generating sentences.•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.Initialize sentence with N1 <s> symbolsUntil </s> is generated do: Stochastically pick the next word based on the conditional probability of each word given the previous N 1 words.))(|(argmaxˆMTPExample from Textbook•P(<s> i want english food </s>) = P(i | <s>) P(want | i) P(english | want) P(food | english) P(</s> | food) = .25 x .33 x .0011 x .5 x .68 = .000031•P(<s> i want chinese food </s>) = P(i | <s>) P(want | i) P(chinese | want) P(food | chinese) P(</s> | food) = .25 x .33 x .0065 x .52 x .68 = .00019Train 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 (<UNK>).–Choose a vocabulary in advance and replace other words in the training corpus with <UNK>.–Replace the first occurrence of each word in the training data with <UNK>.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.NNwwwPWPP)...(1)(21•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 Bigram TrigramPerplexity 962 170 109Smoothing•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. where V is the total number of possible (N1)-grams (i.e. the vocabulary size for a