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CHAPTER 4 PROBABILITY THEORY SEARCH FOR GAMES
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Representing Knowledge
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Uncertainty
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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
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Probabilistic Models
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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)
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Distributions on Random Vars
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Examples
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Marginalization
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Conditional Probabilities
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Inference by Enumeration
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The Chain Rule I
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Lewis Carroll's Pillow Problem
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Independence
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Example: Independence N fair, independent coins:
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Conditional Independence
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The Chain Rule II
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The Chain Rule III
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Expectations
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Expectations
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Estimation
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Estimation
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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.
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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
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Deterministic Single-Player?
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Approximating Node Value
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Stochastic Single-Player
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Deterministic Two-Player
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Tic-tac-toe Game Tree
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Minimax Example
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Minimax Search
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Stochastic Two-Player
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Evaluation Functions
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Function Approximation
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