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Natural Language Processing COMPSCI 423/723 Rohit Kate 1 Probability Theory Language Models 2 Probability Theory 3 What is Probability? • In ordinary language, probability is the degree of certainty of an event: It is very probable that it will rain today. • Probability theory gives a formal mathematical framework to work with numerical estimates of certainty of events, using this one can: – Predict likelihood of combinations of events – Predict most likely outcome – Predict something given that something else 4 Why is Probability Theory Important for NLP? • To obtain estimates for various outcomes of an ambiguity, for example: • Predict most likely parse or an interpretation of an ambiguous sentence given the context or background knowledge – Time flies like an arrow. • • • • Time goes by fast: 0.9 A particular type of flies “time flies” like an arrow: 0.05 Measure speed of flies like you will measure speed of an arrow: 0.03 … • Instead of using some ad hoc arithmetic to encode estimates of certainty, probability theory is preferable because it has sound mathematics behind it, for example rules for combining probabilities 5 Definitions • Sample Space (Ω): Space of possible outcomes – Outcomes of throwing a dice: {1,2,3,4,5,6} • Event: Subset of a sample space – An even number will show up: {2,4,6} – Number 2 will show up: {2} • Probability function (or probability distribution): Mapping from events to a real number in [0,1], such that: P(Ω) = 1 For any events α and β if α П β = ø (disjoint) then P(α U β) = P(α) + P(β) 6 An Example • For throwing a dice P({1,2,3,4,5,6}) = 1 (some outcome shows up) • Suppose each basic outcome is equally likely, since they are all disjoint and add up to 1, we will have P({1}) = 1/6, P({2})=1/6, P({3})=1/6… • P(an even number shows up) = P({2,4,6}) = P({2}) + P({4}) + P({6}) = 1/6 + 1/6 + 1/6 = 1/2 • P(an even number or 3 shows up) = 1/2 + 1/6= 2/3 • P(an even number or 6 shows up) ≠ 1/2 + 1/6 why? P({2,4,6}) = 1/2 7 Interpretation of Probability • Frequentist interpretation: P({3})=1/6: If a dice is thrown multiple times then 1/6th of the times 3 will show up P(It will rain tomorrow) = 1/2 ?? • Subjective interpretation: One’s degree of belief that the event will happen The mathematical rules should hold for both interpretations. 8 Estimating Probabilities • For well defined sample spaces and events, they can be analytically estimated: P({3}) = 1/6 (assuming fair dice) • For many other sample spaces it is not possible to analytically estimate, for example P(A teenager will drink and drive). • For these cases they can be empirically estimated from a good sample, P(A teenager will drink and drive) = # of Teenagers who drink and drive/# of teenagers 9 Conditional Probability • Updated probability of an event given that some event happened P({2}) = 1/6 P({2} given an even number showed up) = ? Represented as: P({2}|{2,4,6}) or P(A|B) P(A|B) = P(AПB)/P(B) (for P(B) > 0) P({2}|{2,4,6}) = P({2})/P({2,4,6}) = 1/6/(1/2) = 1/3 10 Multiplicative and Chain Rules • Multiplicative rule: P(AПB) = P(B)P(A|B) = P(A)P(B|A) • Generalization of the rule, chain rule: P(A1ПA2П…ПAn) = P(A1)P(A2|A1)P(A3|A1ПA2)…P(An|A1П..ПAn-1) Or in any order of As 11 Independence • Two events A and B are independent of each other if P(AПB) = P(A)P(B) or equivalently P(A)=P(A|B) or P(B)=P(B|A), i.e. happening of an event B does not change the probability of A or vice versa. Otherwise the events are dependent. Example: • P({1,2}) = 1/3 P(Even) = 1/2 • P({1,2} П Even) = P({2}) = 1/6 = P({1,2})*P(Even) • P({1,2}|Even) = P({1,2}ПEven)/P(Even) = 1/6/(1/2) = 1/3 Hence , P({1,2}|Even)=P({1,2}) – Given that an even number showed up does not change the probability of whether one or two showed up. Hence these are independent events. 12 Independence • P({2}|Even) = P({2}ПEven)/P(Even) = P({2})/P{2,4,6} = 1/6/(1/2) = 1/3 P({2}) ≠P({2}|Even) – Given that even number showed up increases the probability that the number was 2, hence these are not independent events • Outcome of second throw of dice is supposed to be independent of the first throw of dice P(consecutive {2}) = P({2})*P({2}) = 1/6*1/6 = 1/36 13 Independence • If A1, A2..An are independent then, P(A1ПA2П…ПAn) = P(A1)P(A2|A1)P(A3|A1ПA2)…P(An|A1П..ПAn-1) (chain rule) = P(A1)P(A2)P(A3)…P(An) Independence assumption is often used in NLP to simplify computations of complicated probabilities. S For example, probability of a parse tree is often simplified as the product of probabilities of generating individual productions. NP VP Article NN The girl Verb ate NP Article NN the cake 14 Conditional Independence • Two events A and B are conditionally independent given C if P(AПB|C) = P(A|C)P(B|C) Conditional independence is encountered more often than unconditional independence. 15 Maximum Likelihood Estimate • P(A teenager will drink and drive) = # of Teenagers who drink and drive/# of teenagers • Suppose out of a sample of 5 teenagers one drinks and drives P(A teenager will drink and drive) = 1/5 • Relative frequency estimates can be proven to be maximum likelihood estimates (MLE) because they maximize the probability that it will generate the sampled data • Any other probability value will explain the data with less probability 16 Maximum Likelihood Estimates • If P(TDD) = 1/5 then P(not TDD)=4/5 – Probability of the sampled data making assumption that each teenager is independent • P(Data)=1/5*(4/5)4=0.08192 – It will be less for any other value of P(TDD), for P(TDD)=1/6, • P(Data) = 1/6*(5/6)4 = 0.0803. – For P(TDD)=1/4 • P(Data) = 1/4*(3/4)4 = 0.07091 • Whenever possible to compute, simple frequency counts are not only intuitive but theoretically also the best probability estimates 17 Bayes’ Theorem • Lets us calculate P(B|A) in terms of P(A|B) • For example, using Bayes’ theorem we can calculate P(Hypothesis|Evidence) in terms of P(Evidence|Hypothesis) which is usually easier to estimate. 18 Bayes Theorem P( E | H ) P( H ) P( H | E ) P( E ) Simple proof from definition of conditional probability: P( H E ) P( H | E ) P( E ) (Def. cond. prob.) P( H E ) (Def. cond. prob.) P( E | H ) P( H ) P( H E) P( E | H ) P( H ) QED: P( H | E ) P( E | H ) P( H ) P( E ) 19 Random Variable • Represents a measurable value associated with events, for example, the number that showed up on the dice, sum of the numbers of consecutive throws of a dice • Let X represent the number that showed up P(X=2) is the probability that 2 showed up • Let Z represent sum of the numbers that showed up on throwing the dice twice P(Z > 5) is the probability that the sum was greater than 5 20 Probability Distribution • Probability distribution: Specification of probabilities for all the values of a random variable Example: X represents the number that shows up when a dice is thrown, a probability distribution (should add to 1) X P(X) 1 1/6 2 1/6 3 2/6 4 1/6 5 1/12 6 1/12 Given a probability distribution, probability of any event over the random variable can be computed, P(X>5), P(X=2 or 3) 21 Joint Probability Distribution • The joint probability distribution for a set of random variables, X1,…,Xn gives the probability of every combination of values: P(X1,…,Xn) • Given a joint probability distribution, probability of any event over the random variables can be computed • Example: S: shape (circle, square) C: color (red, blue) L: label (positive, negative) 22 Joint Probability Distribution • Joint distribution of P(S,C,L): S,C,L P(S,C,L) Circle, Red, Positive 0.2 Circle, Red, Negative 0.05 Circle, Blue, Positive 0.02 Circle, Blue, Negative 0.2 Square, Red, Positive 0.02 Square, Red, Negative 0.3 Square, Blue, Positive 0.01 Square, Blue, Negative 0.2 23 Marginal Probability Distributions • The probability of all possible conjunctions (assignments of values to some subset of variables) can be calculated by summing the appropriate subset of values from the joint distribution P(red circle) 0.20 0.05 0.25 P(red ) 0.20 0.02 0.05 0.3 0.57 • In general, distribution of subset of random variables, for e.g. P(C) or P(C^S), can be computed from joint distribution, these are called marginal probability distributions 24 Conditional Probability Distributions • Once marginal probabilities are computed, conditional probabilities can also be calculated P( positive| red circle) P( positive red circle) 0.20 0.80 P(red circle) 0.25 • In general, distributions of subsets of random variables with conditions, for e.g. P(L|C^S), can also be computed, these are called conditional probability distributions 25 Language Models Most of these slides have been adapted from Raymond Mooney’s slides from his NLP course at UT Austin. 26 What is a Language Model (LM)? • Given a sentence how likely is it a sentence of the language? The dog bit the man. => very likely or 0.75 Dog man the the bit. => very unlikely or 0.002 The dog bit man. => likely or 0.15 • A probabilistic model is better than a formal grammar model which will only give a binary decision • To specify a correct probability distribution, the probability of all sentences in a language must sum to 1 27 What are the Uses of an LM? • 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.” 28 What are the Uses of an LM? • 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. 29 What is the probability of a sentence? P(A1ПA2П…ПAn) = P(A1)P(A2|A1)P(A3|A1ПA2)…P(An|A1П..ПAn-1) (chain rule) P(Please,turn,off,your,cell,phone) = P(Please)P(turn|Please)P(off|Please,turn)P(your|Please,t urn,off)P(cell|Please,turn,off,your)P(phone|Please,turn,o ff,your,cell) Estimate the above probabilities from a large corpus – – Too many probabilities (parameters) to estimate They become sparse, cannot be estimated well. 30 What is the probability of a sentence? Approximate the probability by making independence assumptions P(Please,turn,off,your,cell,phone) = P(Please)P(turn|Please)P(off|Please,turn)P(your|Please,turn,off)P(c ell|Please,turn,off,your)P(phone|Please,turn,off,your,cell) Bigram approximation: P(Please,turn,off,your,cell,phone) =P(Please) P(turn|Please)P(off|turn)P(your|off)P(cell|your)P(phone|cell) Trigram approximation: P(Please,turn,off,your,cell,phone) = P(Please)P(turn|Please)P(off|Please,turn)P(your|turn,off)P(cell|off, your)P(phone|your,cell) 31 N-Gram Model Formulas • Word sequences w1n w1...wn • Chain rule of probability n P( w ) P( w1 ) P( w2 | w1 ) P(w3 | w )...P( wn | w ) P( wk | w1k 1 ) n 1 2 1 • Bigram approximation n 1 1 k 1 n P( w ) P( wk | wk 1 ) n 1 k 1 • N-gram approximation n P( w ) P( wk | wkk1N 1 ) n 1 k 1 32 Estimating Probabilities • N-gram conditional probabilities can be estimated from raw text based on the relative frequency of word sequences, they are the maximum likelihood estimates Bigram: N-gram: C ( wn 1wn ) P( wn | wn 1 ) C ( wn 1 ) n 1 n N 1 P(wn | w C (wnn1N 1wn ) ) C (wnn1N 1 ) • To have a consistent probabilistic model, append a unique start (<s>) and end (</s>) symbol to every sentence and treat these as additional words 33 Generative Model of a Language • An N-gram model can be seen as a probabilistic automata for generating sentences. Start with an <s> symbol Until </s> is generated do: Stochastically pick the next word based on the conditional probability of each word given the previous N 1 words. 34 Training and Testing a Language Model • A language model must be trained on a large corpus of text to estimate good parameter (probability) values • Model can be evaluated based on its ability to predict a high probability for a disjoint (held-out) test corpus • Ideally, the training (and test) corpus should be representative of the actual application data 35 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>. 36 Evaluation of an LM • Ideally, evaluate use of model in end application (extrinsic) – Realistic – Expensive • Evaluate on ability to model test corpus (intrinsic) – Less realistic – Cheaper 37 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 PP (W ) N 1 P ( w1w2 ...wN ) • Measures the weighted average branching factor in predicting the next word (lower is better) 38 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 39 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 (also know as sparse data problem). • If a new combination occurs during testing, it is given a probability of zero and the entire sequence gets a probability of zero • In practice, parameters are smoothed (also know as 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. 40 Laplace (Add-0ne) Smoothing • “Hallucinate” additional training data in which each word occurs exactly once in every possible (N1)gram context and adjust estimates accordingly. C ( wn1wn ) 1 C ( wn1 ) V Bigram: P( wn | wn1 ) N-gram: n 1 C ( w n N 1wn ) 1 P(wn | wnn1N 1 ) C (wnn1N 1 ) V where V is the total number of possible words (i.e. the vocabulary size) • Tends to reassign too much mass to unseen events, so can be adjusted to add 0<<1 (normalized by V instead of V) • More advanced smoothing techniques have also 41 been developed 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). 42 Interpolation • Linearly combine estimates of N-gram models of increasing order Interpolated Trigram Model: Pˆ (wn | wn2, wn1 ) 1P(wn | wn2, wn1 ) 2 P(wn | wn1 ) 3 P(wn ) Where: i 1 i • Learn proper values for i by training to (approximately) maximize the likelihood of an independent development (also known as tuning) corpus 43 Backoff • Only use lower-order model when data for higher-order model is unavailable (i.e. count is zero). • Recursively back-off to weaker models until data is available. n 1 n 1 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 44 A Problem for N-Grams LMs: 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 that use syntax and semantics are needed to handle such dependencies 45 Domain-specific LMs • Using a domain-specific corpus one can build a domain-specific language model • For example, train a language model for each difficulty level (grades 1 to 12) • Then automatically predict difficulty level of a new document and recommend for the grade level 46 A Large LM • Google had released a 5-gram LM trained on one trillion words in 2006 • Available through Linguistic Data Consortium (LDC) (not free) http://www.ldc.upenn.edu/Catalog/CatalogEntry.jsp?catalogId=LDC2006T13 • Data Sizes • • • • • • • File sizes: approx. 24 GB compressed (gzip'ed) text files Number of tokens: 1,024,908,267,229 Number of sentences: 95,119,665,584 Number of unigrams: 13,588,391 Number of bigrams: 314,843,401 Number of trigrams: 977,069,902 Number of fourgrams: 1,313,818,354 • Number of fivegrams: 1,176,470,663 47 Homework 1, Due next week in class Find marginal distributions for P(C^S) and P(C) from the joint distribution shown in slide #23. Compute the conditional distribution P(S|C) from this. 48