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A comprehensive review of probability theory concepts including axioms, rules, Bayes Theorem, random variables, PDFs, CDFs, and expected value. Learn about sample and event spaces, set operations, conditional probability, joint probability, Bayes Theorem applications, and Gaussian distributions.
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Advanced Artificial Intelligence Lecture 2A: Probability Theory Review
Outline • Axioms of Probability • Product and chain rules • Bayes Theorem • Random variables • PDFs and CDFs • Expected value and variance
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}
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)
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)=?
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)=?
Joint Probability • Multiple events: cancer, test result
Joint Probability • The problem with joint distributions It takes 2D-1 numbers to specify them!
Conditional Probability • Describes the cancer test: • Put this together with: Prior probability
Conditional Probability • We have: • We can now calculate joint probabilities
Conditional Probability • “Diagnostic” question: How likely do is cancer given a positive test?
Bayes Theorem Posterior Probability Prior Probability Likelihood Normalizing Constant
Probability Density Functions f(x) x F(x) 1 x
Probability Density Functions f(x) x F(x) 1 x
