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Introduction LING 572 Fei Xia Week 1: 1/3/06

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Outline Course overview Problems and methods Mathematical foundation –Probability theory –Information theory

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Course overview

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Course objective Focus on statistical methods that produce state- of-the-art results Questions: for each algorithm –How the algorithm works: input, output, steps –What kind of tasks an algorithm can be applied to? –How much data is needed? Labeled data Unlabeled data

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General info Course website: –Syllabus (incl. slides and papers): updated every week. –Message board –ESubmit Office hour: W: 3-5pm. Prerequisites: –Ling570 and Ling571. –Programming: C, C++, or Java, Perl is a plus. –Introduction to probability and statistics

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Expectations Reading: –Papers are online: who don’t have access to printers? –Reference book: Manning & Schutze (MS) –Finish reading before class. Bring your questions to class. Grade: –Homework (3): 30% –Project (6 parts): 60% –Class participation: 10% –No quizzes, exams

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Assignments Hw1: FSA and HMM Hw2: DT, DL, and TBL. Hw3: Boosting No coding Bring the finished assignments to class.

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Project P1: Method 1 (Baseline): Trigram P2: Method 2: TBL P3: Method 3: MaxEnt P4: Method 4: choose one of four tasks. P5: Presentation P6: Final report Methods 1-3 are supervised methods. Method 4: bagging, boosting, semi-supervised learning, or system combination. P1 is an individual task, P2-P6 are group tasks. A group should have no more than three people. Use ESubmit Need to use others’ code and write your own code.

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Summary of Ling570 Overview: corpora, evaluation Tokenization Morphological analysis POS tagging Shallow parsing N-grams and smoothing WSD NE tagging HMM

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Summary of Ling571 Parsing Semantics Discourse Dialogue Natural language generation (NLG) Machine translation (MT)

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570/571 vs. 572 572 focuses more on statistical approaches. 570/571 are organized by tasks; 572 is organized by learning methods. I assume that you know –The basics of each task: POS tagging, parsing, … –The basic concepts: PCFG, entropy, … –Some learning methods: HMM, FSA, …

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An example 570/571: –POS tagging: HMM –Parsing: PCFG –MT: Model 1-4 training 572: –HMM: forward-backward algorithm –PCFG: inside-outside algorithm –MT: EM algorithm All special cases of EM algorithm, one method of unsupervised learning.

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Course layout Supervised methods –Decision tree –Decision list –Transformation-based learning (TBL) –Bagging –Boosting –Maximum Entropy (MaxEnt)

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Course layout (cont) Semi-supervised methods –Self-training –Co-training Unsupervised methods –EM algorithm Forward-backward algorithm Inside-outside algorithm EM for PM models

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Outline Course overview Problems and methods Mathematical foundation –Probability theory –Information theory

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Problems and methods

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Types of ML problems Classification problem Estimation problem Clustering Discovery … A learning method can be applied to one or more types of ML problems. We will focus on the classification problem.

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Classification problem Given a set of classes and data x, decide which class x belongs to. Labeled data: –(x i, y i ) is a set of labeled data. –x i is a list of attribute values. –y i is a member of a pre-defined set of classes.

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Examples of classification problem Disambiguation: –Document classification –POS tagging –WSD –PP attachment given a set of other phrases Segmentation: –Tokenization / Word segmentation –NP Chunking

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Learning methods Modeling: represent the problem as a formula and decompose the formula into a function of parameters Training stage: estimate the parameters Test (decoding) stage: find the answer given the parameters

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Modeling Joint vs. conditional models: –P(data, model) –P(model | data) –P(data | model) Decomposition: –Which variable conditions on which variable? –What independent assumptions?

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An example of different modeling

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Training Objective functions: –Maximize likelihood: –Minimize error rate –Maximum entropy –…. Supervised, semi-supervised, unsupervised: –Ex: Maximize likelihood Supervised: simple counting Unsupervised: EM

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Decoding DP algorithm –CYK for PCFG –Viterbi for HMM –…–… Pruning: –TopN: keep topN hyps at each node. –Beam: keep hyps whose weights >= beam * max_weight –Threshold: keep hyps whose weights >= threshold –…–…

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Outline Course overview Problems and methods Mathematical foundation –Probability theory –Information theory

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Probability Theory

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Probability theory Sample space, event, event space Random variable and random vector Conditional probability, joint probability, marginal probability (prior)

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Sample space, event, event space Sample space (Ω): a collection of basic outcomes. –Ex: toss a coin twice: {HH, HT, TH, TT} Event: an event is a subset of Ω. –Ex: {HT, TH} Event space (2 Ω ): the set of all possible events.

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Random variable The outcome of an experiment need not be a number. We often want to represent outcomes as numbers. A random variable is a function that associates a unique numerical value with every outcome of an experiment. Random variable is a function X: Ω R. Ex: toss a coin once: X(H)=1, X(T)=0

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Two types of random variable Discrete random variable: X takes on only a countable number of distinct values. –Ex: Toss a coin 10 times. X is the number of tails that are noted. Continuous random variable: X takes on uncountable number of possible values. –Ex: X is the lifetime (in hours) of a light bulb.

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Probability function The probability function of a discrete variable X is a function which gives the probability p(x i ) that the random variable equals x i : a.k.a. p(x i ) = p(X=x i ).

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Random vector Random vector is a finite-dimensional vector of random variables: X=[X 1,…,X k ]. P(x) = P(x 1,x 2,…,x n )=P(X 1 =x 1,…., X n =x n ) Ex: P(w 1, …, w n, t 1, …, t n )

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Three types of probability Joint prob: P(x,y)= prob of x and y happening together Conditional prob: P(x|y) = prob of x given a specific value of y Marginal prob: P(x) = prob of x for all possible values of y

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Common equations

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More general cases

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Information Theory

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Information theory It is the use of probability theory to quantify and measure “information”. Basic concepts: –Entropy –Joint entropy and conditional entropy –Cross entropy and relative entropy –Mutual information and perplexity

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Entropy Entropy is a measure of the uncertainty associated with a distribution. The lower bound on the number of bits it takes to transmit messages. An example: –Display the results of horse races. –Goal: minimize the number of bits to encode the results.

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An example Uniform distribution: p i =1/8. Non-uniform distribution: (1/2,1/4,1/8, 1/16, 1/64, 1/64, 1/64, 1/64) (0, 10, 110, 1110, 111100, 111101, 111110, 111111)

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Entropy of a language The entropy of a language L: If we make certain assumptions that the language is “nice”, then the cross entropy can be calculated as:

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Joint and conditional entropy Joint entropy: Conditional entropy:

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Cross Entropy Entropy: Cross Entropy: Cross entropy is a distance measure between p(x) and q(x): p(x) is the true probability; q(x) is our estimate of p(x).

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Cross entropy of a language The cross entropy of a language L: If we make certain assumptions that the language is “nice”, then the cross entropy can be calculated as:

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Relative Entropy Also called Kullback-Leibler distance: Another distance measure between prob functions p and q. KL distance is asymmetric (not a true distance):

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Relative entropy is non-negative

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Mutual information It measures how much is in common between X and Y: I(X;Y)=KL(p(x,y)||p(x)p(y))

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Perplexity Perplexity is 2 H. Perplexity is the weighted average number of choices a random variable has to make.

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Summary Course overview Problems and methods Mathematical foundation –Probability theory –Information theory M&S Ch2

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Next time FSA HMM: M&S Ch 9.1 and 9.2

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