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Learning Objectives At the end of the class you should be able to: identify a supervised learning problem characterize how the prediction is a function of the error measure avoid mixing the training and test sets D. Poole and A. Mackworth

  1. Learning Objectives At the end of the class you should be able to: identify a supervised learning problem characterize how the prediction is a function of the error measure avoid mixing the training and test sets � D. Poole and A. Mackworth 2010 c Artificial Intelligence, Lecture 7.2, Page 1

  2. Supervised Learning Given: a set of inputs features X 1 , . . . , X n a set of target features Y 1 , . . . , Y k a set of training examples where the values for the input features and the target features are given for each example a new example, where only the values for the input features are given predict the values for the target features for the new example. � D. Poole and A. Mackworth 2010 c Artificial Intelligence, Lecture 7.2, Page 2

  3. Supervised Learning Given: a set of inputs features X 1 , . . . , X n a set of target features Y 1 , . . . , Y k a set of training examples where the values for the input features and the target features are given for each example a new example, where only the values for the input features are given predict the values for the target features for the new example. classification when the Y i are discrete regression when the Y i are continuous � D. Poole and A. Mackworth 2010 c Artificial Intelligence, Lecture 7.2, Page 3

  4. Example Data Representations A travel agent wants to predict the preferred length of a trip, which can be from 1 to 6 days. (No input features). Two representations of the same data: — Y is the length of trip chosen. — Each Y i is an indicator variable that has value 1 if the chosen length is i , and is 0 otherwise. Example Y Example Y 1 Y 2 Y 3 Y 4 Y 5 Y 6 e 1 1 e 1 1 0 0 0 0 0 6 0 0 0 0 0 1 e 2 e 2 6 0 0 0 0 0 1 e 3 e 3 2 0 1 0 0 0 0 e 4 e 4 1 1 0 0 0 0 0 e 5 e 5 What is a prediction? � D. Poole and A. Mackworth 2010 c Artificial Intelligence, Lecture 7.2, Page 4

  5. Evaluating Predictions Suppose we want to make a prediction of a value for a target feature on example e : o e is the observed value of target feature on example e . p e is the predicted value of target feature on example e . The error of the prediction is a measure of how close p e is to o e . There are many possible errors that could be measured. Sometimes p e can be a real number even though o e can only have a few values. � D. Poole and A. Mackworth 2010 c Artificial Intelligence, Lecture 7.2, Page 5

  6. Measures of error E is the set of examples, with single target feature. For e ∈ E , o e is observed value and p e is predicted value: � absolute error L 1 ( E ) = | o e − p e | e ∈ E � D. Poole and A. Mackworth 2010 c Artificial Intelligence, Lecture 7.2, Page 6

  7. Measures of error E is the set of examples, with single target feature. For e ∈ E , o e is observed value and p e is predicted value: � absolute error L 1 ( E ) = | o e − p e | e ∈ E � sum of squares error L 2 ( o e − p e ) 2 2 ( E ) = e ∈ E � D. Poole and A. Mackworth 2010 c Artificial Intelligence, Lecture 7.2, Page 7

  8. Measures of error E is the set of examples, with single target feature. For e ∈ E , o e is observed value and p e is predicted value: � absolute error L 1 ( E ) = | o e − p e | e ∈ E � sum of squares error L 2 ( o e − p e ) 2 2 ( E ) = e ∈ E worst-case error : L ∞ ( E ) = max e ∈ E | o e − p e | � D. Poole and A. Mackworth 2010 c Artificial Intelligence, Lecture 7.2, Page 8

  9. Measures of error E is the set of examples, with single target feature. For e ∈ E , o e is observed value and p e is predicted value: � absolute error L 1 ( E ) = | o e − p e | e ∈ E � sum of squares error L 2 ( o e − p e ) 2 2 ( E ) = e ∈ E worst-case error : L ∞ ( E ) = max e ∈ E | o e − p e | number wrong : L 0 ( E ) = # { e : o e � = p e } � D. Poole and A. Mackworth 2010 c Artificial Intelligence, Lecture 7.2, Page 9

  10. Measures of error E is the set of examples, with single target feature. For e ∈ E , o e is observed value and p e is predicted value: � absolute error L 1 ( E ) = | o e − p e | e ∈ E � sum of squares error L 2 ( o e − p e ) 2 2 ( E ) = e ∈ E worst-case error : L ∞ ( E ) = max e ∈ E | o e − p e | number wrong : L 0 ( E ) = # { e : o e � = p e } A cost-based error takes into account costs of errors. � D. Poole and A. Mackworth 2010 c Artificial Intelligence, Lecture 7.2, Page 10

  11. Measures of error (cont.) With binary feature: o e ∈ { 0 , 1 } : likelihood of the data � p o e e (1 − p e ) (1 − o e ) e ∈ E � D. Poole and A. Mackworth 2010 c Artificial Intelligence, Lecture 7.2, Page 11

  12. Measures of error (cont.) With binary feature: o e ∈ { 0 , 1 } : likelihood of the data � p o e e (1 − p e ) (1 − o e ) e ∈ E log likelihood � ( o e log p e + (1 − o e ) log(1 − p e )) e ∈ E is negative of number of bits to encode the data given a code based on p e . � D. Poole and A. Mackworth 2010 c Artificial Intelligence, Lecture 7.2, Page 12

  13. Information theory overview A bit is a binary digit. 1 bit can distinguish 2 items k bits can distinguish 2 k items n items can be distinguished using log 2 n bits Can we do better? � D. Poole and A. Mackworth 2010 c Artificial Intelligence, Lecture 7.2, Page 13

  14. Information and Probability Consider a code to distinguish elements of { a , b , c , d } with P ( a ) = 1 2 , P ( b ) = 1 4 , P ( c ) = 1 8 , P ( d ) = 1 8 Consider the code: a 0 b 10 c 110 d 111 This code uses 1 to 3 bits. On average, it uses P ( a ) × 1 + P ( b ) × 2 + P ( c ) × 3 + P ( d ) × 3 1 2 + 2 4 + 3 8 + 3 8 = 13 = 4 bits. The string aacabbda has code � D. Poole and A. Mackworth 2010 c Artificial Intelligence, Lecture 7.2, Page 14

  15. Information and Probability Consider a code to distinguish elements of { a , b , c , d } with P ( a ) = 1 2 , P ( b ) = 1 4 , P ( c ) = 1 8 , P ( d ) = 1 8 Consider the code: a 0 b 10 c 110 d 111 This code uses 1 to 3 bits. On average, it uses P ( a ) × 1 + P ( b ) × 2 + P ( c ) × 3 + P ( d ) × 3 1 2 + 2 4 + 3 8 + 3 8 = 13 = 4 bits. The string aacabbda has code 00110010101110. The code 0111110010100 represents string � D. Poole and A. Mackworth 2010 c Artificial Intelligence, Lecture 7.2, Page 15

  16. Information and Probability Consider a code to distinguish elements of { a , b , c , d } with P ( a ) = 1 2 , P ( b ) = 1 4 , P ( c ) = 1 8 , P ( d ) = 1 8 Consider the code: a 0 b 10 c 110 d 111 This code uses 1 to 3 bits. On average, it uses P ( a ) × 1 + P ( b ) × 2 + P ( c ) × 3 + P ( d ) × 3 1 2 + 2 4 + 3 8 + 3 8 = 13 = 4 bits. The string aacabbda has code 00110010101110. The code 0111110010100 represents string adcabba � D. Poole and A. Mackworth 2010 c Artificial Intelligence, Lecture 7.2, Page 16

  17. Information Content To identify x , we need − log 2 P ( x ) bits. Give a distribution over a set, to a identify a member, the expected number of bits � − P ( x ) × log 2 P ( x ) . x is the information content or entropy of the distribution. The expected number of bits it takes to describe a distribution given evidence e : � I ( e ) = − P ( x | e ) × log 2 P ( x | e ) . x � D. Poole and A. Mackworth 2010 c Artificial Intelligence, Lecture 7.2, Page 17

  18. Information Gain Given a test that can distinguish the cases where α is true from the cases where α is false, the information gain from this test is: I ( true ) − ( P ( α ) × I ( α ) + P ( ¬ α ) × I ( ¬ α )) . I ( true ) is the expected number of bits needed before the test P ( α ) × I ( α ) + P ( ¬ α ) × I ( ¬ α ) is the expected number of bits after the test. � D. Poole and A. Mackworth 2010 c Artificial Intelligence, Lecture 7.2, Page 18

  19. Linear Predictions 8 L ∞ 7 6 5 4 3 2 1 0 0 1 2 3 4 5 � D. Poole and A. Mackworth 2010 c Artificial Intelligence, Lecture 7.2, Page 19

  20. Linear Predictions 8 L ∞ 7 L 2 6 2 5 4 3 L 1 2 1 0 0 1 2 3 4 5 � D. Poole and A. Mackworth 2010 c Artificial Intelligence, Lecture 7.2, Page 20

  21. Point Estimates To make a single prediction for feature Y , with examples E . The prediction that minimizes the sum of squares error on E is � D. Poole and A. Mackworth 2010 c Artificial Intelligence, Lecture 7.2, Page 21

  22. Point Estimates To make a single prediction for feature Y , with examples E . The prediction that minimizes the sum of squares error on E is the mean (average) value of Y . � D. Poole and A. Mackworth 2010 c Artificial Intelligence, Lecture 7.2, Page 22

  23. Point Estimates To make a single prediction for feature Y , with examples E . The prediction that minimizes the sum of squares error on E is the mean (average) value of Y . The prediction that minimizes the absolute error on E is � D. Poole and A. Mackworth 2010 c Artificial Intelligence, Lecture 7.2, Page 23

  24. Point Estimates To make a single prediction for feature Y , with examples E . The prediction that minimizes the sum of squares error on E is the mean (average) value of Y . The prediction that minimizes the absolute error on E is the median value of Y . � D. Poole and A. Mackworth 2010 c Artificial Intelligence, Lecture 7.2, Page 24

  25. Point Estimates To make a single prediction for feature Y , with examples E . The prediction that minimizes the sum of squares error on E is the mean (average) value of Y . The prediction that minimizes the absolute error on E is the median value of Y . The prediction that minimizes the number wrong on E is � D. Poole and A. Mackworth 2010 c Artificial Intelligence, Lecture 7.2, Page 25

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