124.11.lec3

124.11.lec3 - CS 124/LINGUIST 180: From Click to edit...

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Click to edit Master subtitle style 1/10/09 Dan Jurafsky Lecture 3: Intro to Probability, Language Modeling IP notice: some slides for today from: Jim Martin, Sandiway Fong, Dan Klein
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1/10/09 Outline Probability Basic probability Conditional probability Language Modeling (N-grams) N-gram Intro The Chain Rule The Shannon Visualization Method Evaluation: Perplexity Smoothing: Laplace (Add-1) Add-prior
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1/10/09 1. Introduction to Probability Experiment (trial) Repeatable procedure with well-defined possible outcomes Sample Space (S) the set of all possible outcomes finite or infinite Example coin toss experiment possible outcomes: S = {heads, tails} Example die toss experiment possible outcomes: S = {1,2,3,4,5,6} Slides from
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1/10/09 Introduction to Probability Definition of sample space depends on what we are asking Sample Space (S): the set of all possible outcomes Example die toss experiment for whether the number is even or odd possible outcomes: {even,odd} not {1,2,3,4,5,6}
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1/10/09 More definitions Events an event is any subset of outcomes from the sample space Example die toss experiment let A represent the event such that the outcome of the die toss experiment is divisible by 3 A = {3,6} A is a subset of the sample space S= {1,2,3,4,5,6} Example Draw a card from a deck suppose sample space S = {heart,spade,club,diamond} ( four suits ) let A represent the event of drawing a heart let B represent the event of drawing a red card A = {heart}
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1/10/09 Introduction to Probability Some definitions Counting suppose operation oi can be performed in ni ways, then a sequence of k operations o1o2. ..ok .. 0 nk ways Example die toss experiment, 6 possible outcomes two dice are thrown at the same time number of sample points in sample space = 6 & 6 = 36
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1/10/09 Definition of Probability The probability law assigns to an event a nonnegative number Called P(A) Also called the probability A That encodes our knowledge or belief about the collective likelihood of all the elements of A Probability law must satisfy certain properties
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1/10/09 Probability Axioms Nonnegativity P(A) >= 0, for every event A Additivity If A and B are two disjoint events, then the probability of their union satisfies: P(A U B) = P(A) + P(B) Normalization The probability of the entire sample space S is equal to 1, I.e. P(S) = 1.
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1/10/09 An example An experiment involving a single coin toss There are two possible outcomes, H and T Sample space S is {H,T} If coin is fair, should assign equal probabilities to 2 outcomes Since they have to sum to 1 P({H}) = 0.5 P({T}) = 0.5 P({H,T}) = P({H})+P({T}) = 1.0
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1/10/09 Another example Experiment involving 3 coin tosses Outcome is a 3-long string of H or T
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124.11.lec3 - CS 124/LINGUIST 180: From Click to edit...

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