1. Lecture 4A: Probability Theory Review Advanced Artificial Intelligence

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

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

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 13 Has cancer?Test positive?P(C,TP) yes 0.018 yesno0.002 noyes0.196 no 0.784

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

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

16. 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

17. 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

18. Bayes Theorem

19. Posterior Probability Likelihood Normalizing Constant Prior Probability

20. Bayes Theorem

21. Random Variables

22. Cumulative Distribution Functions

23. Probability Density Functions

26. 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

32. Variance

33. Gaussian Distributions