George F Luger ARTIFICIAL INTELLIGENCE 5th edition Structures and Strategies for Complex Problem Solving STOCHASTIC METHODS Luger: Artificial Intelligence,

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George F Luger ARTIFICIAL INTELLIGENCE 5th edition Structures and Strategies for Complex Problem Solving STOCHASTIC METHODS Luger: Artificial Intelligence, 5 th edition. © Pearson Education Limited, Introduction 5.1 The Elements of Counting 5.2Elements of Probability Theory 5.3Applications of the Stochastic Methodology 5.4Bayes’ Theorem 5.5Epilogue and References 5.6Exercises 1

Luger: Artificial Intelligence, 5 th edition. © Pearson Education Limited,

The probability of any event E from the sample space S is: The sum of the probabilities of all possible outcomes is 1 The probability of the compliment of an event is The probability of the contradictory or false outcome of an event Luger: Artificial Intelligence, 5 th edition. © Pearson Education Limited,

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The three Kolmogorov Axioms: Luger: Artificial Intelligence, 5 th edition. © Pearson Education Limited,

Luger: Artificial Intelligence, 5 th edition. © Pearson Education Limited, 2005 Table 5.1The joint probability distribution for the traffic slowdown, S, accident, A, and construction, C, variable of the example of Section Fig 5.1 A Venn diagram representation of the probability distributions of Table 5.1; S is traffic slowdown, A is accident, C is construction. 11

Luger: Artificial Intelligence, 5 th edition. © Pearson Education Limited,

Luger: Artificial Intelligence, 5 th edition. © Pearson Education Limited,

Luger: Artificial Intelligence, 5 th edition. © Pearson Education Limited, 2005 Fig 5.2A Venn diagram illustrating the calculations of P(d|s) as a function of p(s|d). 15

The chain rule for two sets: The generalization of the chain rule to multiple sets Luger: Artificial Intelligence, 5 th edition. © Pearson Education Limited, 2005 We make an inductive argument to prove the chain rule, consider the n th case: We apply the intersection of two sets of rules to get: And then reduce again, considering that: Until is reached, the base case, which we have already demonstrated. 16

Luger: Artificial Intelligence, 5 th edition. © Pearson Education Limited,

The general form of Bayes’ theorem where we assume the set of hypotheses H partition the evidence set E : Luger: Artificial Intelligence, 5 th edition. © Pearson Education Limited,