Advanced Artificial Intelligence

1. Advanced Artificial Intelligence Lecture 2A: Probability Theory Review

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

3. 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}

5. 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)

6. Set operations

7. Conditional Probability

8. Conditional Probability

9. 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)=?

10. Conditional Probability

11. Conditional Probability

12. 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)=?

13. Joint Probability • Multiple events: cancer, test result

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

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

16. Conditional Probability • We have: • We can now calculate joint probabilities

17. Conditional Probability • “Diagnostic” question: How likely do is cancer given a positive test?

18. Bayes Theorem

19. Bayes Theorem Posterior Probability Prior Probability Likelihood Normalizing Constant

20. Bayes Theorem

21. Random Variables

22. Cumulative Distribution Functions

23. Probability Density Functions

24. Probability Density Functions

25. Probability Density Functions

26. Probability Density Functions f(X) X

27. Probability Density Functions f(X) X

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

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

30. Expectation

31. Expectation

32. Variance

33. Gaussian Distributions