1. CHAPTER 4 PROBABILITY THEORY SEARCH FOR GAMES

2. Representing Knowledge

3. Uncertainty

4. Probabilities Probabilistic approach.- With no other information, A60 will get me there on time with probability 0.6 - P(A60) = 0.6 Probabilities change with new evidence: - P(A60 | 5 am) = 0.9 - P(A60 | 9 am) = 0.4 - P(A60 | accident report, 5 am) = 0.3 - P(A60 | accident report) = 0.1 I.e., observing evidence causes beliefs to be updated

5. Probabilistic Models

6. What Are Probabilities? Objectivist / frequentist answer: - Averages over repeated experiments - E.g. estimating P(rain) from historical observation - Assertion about future experiments (in the limit) - New evidence changes the reference class - Makes one think of inherently random events, like rolling dice Subjectivist / Bayesian answer: - Degrees of belief about unobserved variables - E.g., an agent’s belief that it’s raining, given the temp - Often estimate probabilities from past experience - New evidence updates beliefs Unobserved variables still have fixed assignments (we just don’t know what they are)

7. Distributions on Random Vars

8. Examples

9. Marginalization

10. Conditional Probabilities

11. Inference by Enumeration

13. The Chain Rule I

14. Lewis Carroll's Pillow Problem

15. Independence

16. Example: Independence N fair, independent coins:

17. Conditional Independence

19. The Chain Rule II

20. The Chain Rule III

21. Expectations

22. Expectations

23. Estimation

24. Estimation

25. Game Playing State-of-the-Art Checkers: Chinook ended 40-year-reign of human world champion Marion Tinsley in 1994. Used an endgame database defining perfect play for all positions involving 8 or fewer pieces on the board, a total of 443,748,401,247 positions. Exact solution imminent. Chess: Deep Blue defeated human world champion Gary Kasparov in a six-game match in 1997. Deep Blue examined 200 million positions per second, used very sophisticated evaluation and undisclosed methods for extending some lines of search up to 40 ply. Othello: human champions refuse to compete against computers, which are too good. Go: human champions refuse to compete against computers, which are too bad. In go, b > 300, so most programs use pattern knowledge bases to suggest plausible moves.

26. Game Playing Axes: –Deterministic or stochastic –One, two or more players –Perfect information (can you see the state) Want algorithms for calculating a strategy (policy) which recommends a move in each state

27. Deterministic Single-Player?

28. Approximating Node Value

29. Stochastic Single-Player

30. Deterministic Two-Player

31. Tic-tac-toe Game Tree

32. Minimax Example

33. Minimax Search

34. Stochastic Two-Player

36. Evaluation Functions

37. Function Approximation