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Lecture 4A: Probability Theory Review Advanced Artificial Intelligence

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Outline Axioms of Probability Product and chain rules Bayes Theorem Random variables PDFs and CDFs Expected value and variance

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Introduction Sample space - set of all possible outcomes of a random experiment – Dice roll: {1, 2, 3, 4, 5, 6} – Coin toss: {Tails, Heads} Event space - subsets of elements in a sample space – Dice roll: {1, 2, 3} or {2, 4, 6} – Coin toss: {Tails}

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examples Coin flip – P(H) – P(T) – P(H,H,H) – P(x1=x2=x3=x4) – P({x1,x2,x3,x4} contains more than 3 heads)

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Set operations

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Conditional Probability

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examples Coin flip – P(x1=H)=1/2 – P(x2=H|x1=H)=0.9 – P(x2=T|x1=T)=0.8 – P(x2=H)=?

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Conditional Probability

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P(A, B)0.005 P(B)0.02 P(A|B)0.25

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Quiz P(D1=sunny)=0.9 P(D2=sunny|D1=sunny)=0.8 P(D2=rainy|D1=sunny)=? P(D2=sunny|D1=rainy)=0.6 P(D2=rainy|D1=rainy)=? P(D2=sunny)=? P(D3=sunny)=?

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Joint Probability Multiple events: cancer, test result 13 Has cancer?Test positive?P(C,TP) yes 0.018 yesno0.002 noyes0.196 no 0.784

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Joint Probability The problem with joint distributions It takes 2 D -1 numbers to specify them! 14

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Conditional Probability Describes the cancer test: Put this together with: Prior probability 15

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Has cancer?Test positive?P(TP, C) yes no yes no Has cancer?Test positive?P(TP, C) yes 0.018 yesno0.002 noyes0.196 no 0.784 Conditional Probability We have: We can now calculate joint probabilities 16

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Conditional Probability “Diagnostic” question: How likely do is cancer given a positive test? 17 Has cancer?Test positive?P(TP, C) yes 0.018 yesno0.002 noyes0.196 no 0.784

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Bayes Theorem

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Posterior Probability Likelihood Normalizing Constant Prior Probability

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Bayes Theorem

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Random Variables

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Cumulative Distribution Functions

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Probability Density Functions

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f(X) X

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Probability Density Functions f(X) X

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Probability Density Functions f(x) x F(x) 1 x

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Probability Density Functions f(x) x F(x) 1 x

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Expectation

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Variance

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Gaussian Distributions

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Random Variables If is an outcome space with a probability measure and X is a real-valued function defined over the elements of , then X is a random.

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