Corpora and Statistical Methods Lecture 12

Language modelling using N-GramsCorpora and Statistical MethodsLecture 7In this lectureWe consider one of the basic tasks in Statistical NLP:language models are probabilistic representations of allowable sequences This part:methodological overviewfundamental statistical estimation modelsNext part:smoothing techniquesAssumptions, definitions, methodology, algorithmsPart 1Example taskThe word-prediction task (Shannon game)

Given:a sequence of words (the history)a choice of next word

Predict:the most likely next word

Generalises easily to other problems, such as predicting the POS of unknowns based on history.Applications of the Shannon gameAutomatic speech recognition (cf. tutorial 1):given a sequence of possible words, estimate its probability

Context-sensitive spelling correction:Many spelling errors are real wordsHe walked for miles in the dessert. (resp. desert)

Identifying such errors requires a global estimate of the probability of a sentence.

Applications of N-gram models generallyPOS Tagging (cf. lecture 3):predict the POS of an unknown word by looking at its history

Statistical parsing:e.g. predict the group of words that together form a phrase

Statistical NL Generation:given a semantic form to be realised as text, and several possible realisations, select the most probable one.A real-world example: Googles did you meanGoogle uses an n-gram model (based on sequences of characters, not words).

In this case, the sequence apple desserts is much more probable than apple deserts

How it worksDocuments provided by the search engine are added to:An index (for fast retrieval)A language model (based on probability of a sequence of characters)

A submitted query (apple deserts) can be modified (using character insertions, deletions, substitutions and transpositions) to yield a query that fits the language model better (apple desserts).

Outcome is a context-sensitive spelling correction:apple deserts apple dessertsfrod baggins frodo bagginsfrod ford

The noisy channel modelAfter Jurafsky and Martin (2009), Speech and Language Processing (2nd Ed). Prentice Hall p. 198

The assumptions behind n-gram modelsThe Markov AssumptionMarkov models:probabilistic models which predict the likelihood of a future unit based on limited history

in language modelling, this pans out as the local history assumption:the probability of wn depends on a limited number of prior words

utility of the assumption:we can rely on a small n for our n-gram models (bigram, trigram)long n-grams become exceedingly sparseProbabilities become very small with long sequencesThe structure of an n-gram modelThe task can be re-stated in conditional probabilistic terms:

Limiting n under the Markov Assumption means:greater chance of finding more than one occurrence of the sequence w1wn-1more robust statistical estimations

N-grams are essentially equivalence classes or binsevery unique n-gram is a type or bin

Structure of n-gram models (II)If we construct a model where all histories with the same n-1 words are considered one class or bin, we have an (n-1)th order Markov Model

Note terminology:n-gram model = (n-1)th order Markov ModelMethodological considerationsWe are often concerned with:building an n-gram modelevaluating it

We therefore make a distinction between training and test dataYou never test on your training dataIf you do, youre bound to get good results.N-gram models tend to be overtrained, i.e.: if you train on a corpus C, your model will be biased towards expecting the kinds of events in C.Another term for this: overfittingDividing the dataGiven: a corpus of n units (words, sentences, depending on the task)A large proportion of the corpus is reserved for training.A smaller proportion for testing/evaluation (normally 5-10%)Held-out (validation) dataHeld-out estimation:during training, we sometimes estimate parameters for our model empirically

commonly used in smoothing (how much probability space do we want to set aside for unseen data)?

therefore, the training set is often split further into training data and validation datanormally, held-out data is 10% of the size of the training data

Development dataA common approach:train an algorithm on training dataa. (estimate further parameters on held-out data if required)evaluate itre-tune it go back to Step 1 until no further finetuning necessaryCarry out final evaluation

For this purpose, its useful to have:training data for step 1development set for steps 2-4final test set for step 5

Significance testingOften, we compare the performance of our algorithm against some baseline.

A single, raw performance score wont tell us much. We need to test for significance (e.g. using t-test).

Typical method:Split test set into several small test sets, e.g. 20 samplesevaluation carried out separately on eachmean and variance estimated based on 20 different samplestest for significant difference between algorithm and a predefined baselineSize of n-gram modelsIn a corpus of vocabulary size N, the assumption is that any combination of n words is a potential n-gram. For a bigram model: N2 possible n-grams in principleFor a trigram model: N3 possible n-grams.

Size (continued)Each n-gram in our model is a parameter used to estimate probability of the next possible word.too many parameters make the model unwieldytoo many parameters lead to data sparseness: most of them will have f = 0 or 1

Most models stick to unigrams, bigrams or trigrams.estimation can also combine different order modelsFurther considerationsWhen building a model, we tend to take into account the start-of-sentence symbol:the girl swallowed a large green caterpillar thethe girl

Also typical to map all tokens w such that count(w) < k to :usually, tokens with frequency 1 or 2 are just considered unknown or unseenthis reduces the parameter spaceBuilding models using Maximum Likelihood EstimationMaximum Likelihood Estimation ApproachBasic equation:

In a unigram model, this reduces to simple probability.

MLE models estimate probability using relative frequency.

Limitations of MLEMLE builds the model that maximises the probability of the training data.

Unseen events in the training data are assigned zero probability.Since n-gram models tend to be sparse, this is a real problem.

Consequences:seen events are given more probability mass than they haveunseen events are given zero massSeen/unseenAAProbability mass of events in training dataProbability massof events not in training dataThe problem with MLE is that it distributes A among members of A.The solutionSolution is to correct MLE estimation using a smoothing technique.

More on this in the next part

But cf. Tutorial 1, which introduced the simplest method of smoothing known.

Adequacy of different order modelsManning/Schutze `99 report results for n-gram models of a corpus of the novels of Austen.

Task: use n-gram model to predict the probability of a sentence in the test data.

Models:unigram: essentially zero-context markov model, uses only the probability of individual wordsbigramtrigram4-gramExample test caseTraining Corpus: five Jane Austen novels Corpus size = 617,091 words Vocabulary size = 14,585 unique types Task: predict the next word of the trigram inferior to ________from test data, Persuasion: [In person, she was] inferior to both [sisters.]Selecting an n

Vocabulary (V) = 20,000 wordsnNumber of bins(i.e. no. of possible unique n-grams) 2 (bigrams)400,000,0003 (trigrams)8,000,000,000,0004 (4-grams)1.6 x 1017Adequacy of unigramsProblems with unigram models:not entirely hopeless because most sentences contain a majority of highly common wordsignores syntax completely:P(In person she was inferior) = P(inferior was she person in)Adequacy of bigramsBigrams:improve situation dramaticallysome unexpected results:p(she|person) decreases compared to the unigram model. Though she is very common, it is uncommon after person

Adequacy of trigramsTrigram models will do brilliantly when theyre useful.They capture a surprising amount of contextual variation in text.Biggest limitation:most new trigrams in test data will not have been seen in training data.

Problem carries over to 4-grams, and is much worse!Reliability vs. Discriminationlarger n: more information about the context of the specific instance (greater discrimination)

smaller n: more instances in training data, better statistical estimates (more reliability)Backing offPossible way of striking a balance between reliability and discrimination:backoff model:where possible, use a trigramif trigram is unseen, try and back off to a bigram modelif bigrams are unseen, try and back off to a unigramEvaluating language modelsPerplexityRecall: Entropy is a measure of uncertainty: high entropy = high uncertainty

perplexity:if Ive trained on a sample, how surprised am I when exposed to a new sample?a measure of uncertainty of a model on new data

Entropy as expected valueOne way to think of the summation part is as a weighted average of the information content.

We can view this average value as an expectation: the expected surprise/uncertainty of our model.

Comparing distributionsWe have a language model built from a sample. The sample is a probability distribution q over n-grams. q(x) = the probability of some n-gram x in our model.

The sample is generated from a true population (the language) with probability distribution p.p(x) = the probability of x in the true distributionEvaluating a language modelWed like an estimate of how good our model is as a model of the languagei.e. wed like to compare q to p

We dont have access to p. (Hence, cant use KL-Divergence)

Instead, we use our test data as an estimate of p.Cross-entropy: basic intuitionMeasure the number of bits needed to identify an event coming from p, if we code it according to q:We draw sequences according to p;but we sum the log of their probability according to q.

This estimate is called cross-entropy H(p,q)Cross-entropy: p vs. qCross-entropy is an upper bound on the entropy of the true distribution p:H(p) H(p,q)if our model distribution (q) is good, H(p,q) H(p)

We estimate cross-entropy based on our test data.Gives an estimate of the distance of our language model from the distribution in the test sample.Estimating cross-entropy

Probability accordingto p (test set)Entropy accordingto q (language model)PerplexityThe perplexity of a language model with probability distribution q, relative to a test set with probability distribution p is:

A perplexity value of k (obtained on a test set) tells us:our model is as surprised on average as it would be if it had to make k guesses for every sequence (n-gram) in the test data.

The lower the perplexity, the better the language model (the lower the surprise on our test data).

Perplexity example (Jurafsky & Martin, 2000, p. 228)Trained unigram, bigram and trigram models from a corpus of news text (Wall Street Journal)applied smoothing38 million wordsVocab of 19,979 (low-frequency words mapped to UNK).Computed perplexity on a test set of 1.5 million words.J&Ms resultsTrigrams do best of all.Value suggests the extent to which the model can fit the data in the test set.Note: with unigrams, the model has to make lots of guesses!N-Gram modelPerplexityUnigram962Bigram170Trigram109SummaryMain point about Markov-based language models:data sparseness is always a problemsmoothing techniques are required to estimate probability of unseen events

Next part discusses more refined smoothing techniques than those seen so far.