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# bayes-4up - 1 MachineLearning CS6375-Fall 2010 a Bayesian...

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Unformatted text preview: 1 MachineLearning CS6375---Fall 2010 a Bayesian Learning Reading: Sections 13.1-13.6, 20.1-20.2, R&N Sections 6.1-6.3, 6.7, 6.9, Mitchell 2 Uncertainty •Most real-world problems deal with uncertain information –Diagnosis: Likely disease given observed symptoms –Equipment repair: Likely component failure given sensor reading –Help desk: Likely operation based on past operations –Cannot be represented by deterministic rules Headache => Fever •Correct framework for representing uncertainty: Probability 3 Probability •P( A ) = Probability of event A = fraction of all possible worlds in which A is true. 4 Probability 5 Probability •Immediately derived properties More general version of the Theorem of Total Probability : IF we know that exactly one of B1 , B2 ..., Bnare true (i.e. P(B1 or B2 or ... Bn) = 1, and for all i,junequal, P(Biand Bj) = 0) THEN we know: P(A) = P(A, B1) + P(A, B2) + ... P(A, Bn) 6 Probability •A random variable is a variable X that can take values x 1 ,.., x n with a probability P( X = x i ) attached to each i = 1,.., n 7 Example My mood can take one of two values: Happy, Sad. The weather can take one of three values: Rainy, Sunny Cloudy. Given P(Mood=Happy ^ Weather=Rainy ) = 0.2 P(Mood=Happy ^ Weather=Sunny ) = 0.1 P(Mood=Happy ^ Weather=Cloudy ) = 0.4 Can I compute P(Mood=Happy)? Can I compute P(Mood=Sad)? Can I compute P(Weather=Rainy) ? 8 What’s so great about the axioms of probability? The axioms of probability mean you may not represent the following knowledge: P(A) = 0.4 P(B) = 0.3 P(A ^ B) = 0.0 P(A ∨ B) = 0.8 Would you ever want to do that? Difficult philosophical question. Pragmatic answer: if you disobey the laws of probability in your knowledge representation, you can make suboptimal decisions. 9 Conditional Probability •P( A | B ) = Fraction of those worlds in which B is true for which A is also true. 10 Conditional Probability Example • H = Headache P( H ) = 1/2 • F = Flu P( F) = 1/8 P( H | F ) = 1/2 11 Conditional Probability Example • H = Headache P( H ) = 1/2 • F = Flu P( F) = 1/8 P( H | F ) = (Area of “ H and F ”region) (Area of F region) P( H | F ) = P( H , F )/P( F ) P( H | F ) = 1/2 12 Conditional Probability •Definition: •Chain rule: Can you prove that P(A, B) <= P(A) for any events A and B? 13 Conditional Probability •Other useful relations: 14 Probabilistic Inference •What is the probability that F is true given H is true? Given •P( H ) = 1/2 •P( F) = 1/8 •P( H | F ) = 0.5 15 Probabilistic Inference •Correct reasoning: •We know P( H ), P( F ), P( H | F ) and the two chain rules: •Substituting the values: 16 BayesRule 17 BayesRule 18 BayesRule •What if we do not know P( A )??? •Use the relation: •More general Bayesrule: 19 BayesRule •Same rule for a non-binary random variable, except we need to sum over all the possible events 20 Generalizing BayesRule If we know that exactly one of A 1 , A 2 , ..., A n are true, then: P(B) = P(B|A 1 )P(A 1 ) + P(B|A...
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bayes-4up - 1 MachineLearning CS6375-Fall 2010 a Bayesian...

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