Machine Learning 10 701 Tom M Mitchell Machine Learning Department Carnegie Mellon University January 18 2011 Today Bayes Rule Estimating parameters maximum likelihood max a posteriori Readings Probability review Bishop Ch 1 thru 1 2 3 Bishop Ch 2 thru 2 2 Andrew Moore s online tutorial many of these slides are derived from William Cohen Andrew Moore Aarti Singh Eric Xing Carlos Guestrin Thanks Visualizing Probabilities Sample space of all possible worlds A B B A Its area is 1 1 Definition of Conditional Probability P A B P A B P B B A Definition of Conditional Probability P A B P A B P B Corollary The Chain Rule P A B P A B P B P C A B P C A B P A B P B 2 Independent Events Definition two events A and B are independent if P A B P A P B Intuition knowing A tells us nothing about the value of B and vice versa Picture A independent of B 3 Bayes Rule let s write 2 expressions for P A B A B A B P A B P B A P A P B Bayes rule we call P A the prior and P A B the posterior Bayes Thomas 1763 An essay towards solving a problem in the doctrine of chances Philosophical Transactions of the Royal Society of London 53 370 418 by no means merely a curious speculation in the doctrine of chances but necessary to be solved in order to a sure foundation for all our reasonings concerning past facts and what is likely to be hereafter necessary to be considered by any that would give a clear account of the strength of analogical or inductive reasoning 4 Other Forms of Bayes Rule Applying Bayes Rule P A B P B A P A P B A P A P B A P A A you have the flu B you just coughed Assume P A 0 05 P B A 0 80 P B A 0 2 what is P flu cough P A B 5 what does all this have to do with function approximation The Joint Distribution Recipe for making a joint distribution of M variables Example Boolean variables A B C A B C Prob 0 0 0 0 30 0 0 1 0 05 0 1 0 0 10 0 1 1 0 05 1 0 0 0 05 1 0 1 0 10 1 1 0 0 25 1 1 1 0 10 A 0 05 0 25 0 30 B 0 10 0 05 0 10 0 05 C 0 10 6 The Joint Distribution Recipe for making a joint distribution of M variables 1 Make a truth table listing all combinations of values of your variables if there are M Boolean variables then the table will have 2M rows Example Boolean variables A B C A B C Prob 0 0 0 0 30 0 0 1 0 05 0 1 0 0 10 0 1 1 0 05 1 0 0 0 05 1 0 1 0 10 1 1 0 0 25 1 1 1 0 10 A 0 05 0 25 B 0 30 The Joint Distribution Recipe for making a joint distribution of M variables 1 Make a truth table listing all combinations of values of your variables if there are M Boolean variables then the table will have 2M rows 2 For each combination of values say how probable it is 0 10 0 05 0 10 0 05 C 0 10 Example Boolean variables A B C A B C Prob 0 0 0 0 30 0 0 1 0 05 0 1 0 0 10 0 1 1 0 05 1 0 0 0 05 1 0 1 0 10 1 1 0 0 25 1 1 1 0 10 A 0 05 0 25 0 30 B 0 10 0 05 0 10 0 05 C 0 10 7 The Joint Distribution Recipe for making a joint distribution of M variables 1 Make a truth table listing all combinations of values of your variables if there are M Boolean variables then the table will have 2M rows 2 For each combination of values say how probable it is 3 If you subscribe to the axioms of probability those numbers must sum to 1 Example Boolean variables A B C A B C Prob 0 0 0 0 30 0 0 1 0 05 0 1 0 0 10 0 1 1 0 05 1 0 0 0 05 1 0 1 0 10 1 1 0 0 25 1 1 1 0 10 A 0 05 0 25 0 30 B 0 10 0 05 0 10 0 05 C 0 10 Using the Joint One you have the JD you can ask for the probability of any logical expression involving your attribute 8 Using the Joint P Poor Male 0 4654 Using the Joint P Poor 0 7604 9 Inference with the Joint P Male Poor 0 4654 0 7604 0 612 Learning and the Joint Distribution Suppose we want to learn the function f G H W Equivalently P W G H Solution learn joint distribution from data calculate P W G H e g P W rich G female H 40 5 10 sounds like the solution to learning F X Y of P Y X Are we done C Guestrin 11 C Guestrin C Guestrin 12 Maximum Likelihood Estimate for C Guestrin C Guestrin 13 C Guestrin C Guestrin 14 Beta prior distribution P C Guestrin Beta prior distribution P C Guestrin 15 C Guestrin C Guestrin 16 Conjugate priors A Singh Conjugate priors A Singh 17 Estimating Parameters Maximum Likelihood Estimate MLE choose that maximizes probability of observed data Maximum a Posteriori MAP estimate choose that is most probable given prior probability and the data Dirichlet distribution number of heads in N flips of a two sided coin follows a binomial distribution Beta is a good prior conjugate prior for binomial what it s not two sided but k sided follows a multinomial distribution Dirichlet distribution is the conjugate prior 18 You should know Probability basics random variables events sample space conditional probs independence of random variables Bayes rule Joint probability distributions calculating probabilities from the joint distribution Estimating parameters from data maximum likelihood estimates maximum a posteriori estimates distributions binomial Beta Dirichlet conjugate priors Extra slides 19 Expected values Given discrete random variable X the expected value of X written E X is We also can talk about the expected value of functions of X Covariance Given two discrete r v s X and Y we define the covariance of X and Y as e g X gender Y playsFootball or X gender Y leftHanded Remember 20 21
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