CHAPTER 4 PROBABILITY THEORY SEARCH FOR GAMES

Representing Knowledge

Uncertainty

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

Probabilistic Models

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)

Distributions on Random Vars

Examples

Marginalization

Conditional Probabilities

Inference by Enumeration

The Chain Rule I

Lewis Carroll's Pillow Problem

Independence

Example: Independence N fair, independent coins:

Conditional Independence

The Chain Rule II

The Chain Rule III

Expectations

Expectations

Estimation

Estimation

Game Playing State-of-the-Art Checkers: Chinook ended 40-year-reign of human world champion Marion Tinsley in 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 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.

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

Deterministic Single-Player?

Approximating Node Value

Stochastic Single-Player

Deterministic Two-Player

Tic-tac-toe Game Tree

Minimax Example

Minimax Search

Stochastic Two-Player

Evaluation Functions

Function Approximation