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Bayes Classifier , Linear Regression. 10701 /15781 Recitation Jan uary 29, 2008. Parts of the slides are from previous years’ recitation and lecture notes, and from Prof. Andrew Moore’s data mining tutorials. Classification and Regression. Classification

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Bayes Classifier , Linear Regression

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Bayes Classifier,Linear Regression

10701/15781 Recitation

January 29, 2008

Parts of the slides are from previous years’ recitation and lecture notes, and from Prof. Andrew Moore’s data mining tutorials.


Classification and Regression

  • Classification

    • Goal: Learn the underlying function

      f: X (features)Y (class, or category)

      e.g. words  “spam”, or “not spam”

  • Regression

    f: X (features)  Y (continuous values)

    e.g. GPA  salary


Supervised Classification

  • How to find an unknown function

    f: X Y

    (features  class)

    or equivalently P(Y|X)

    • Classifier:

    • Find P(X|Y), P(Y), and use Bayes rule - generative

    • Find P(Y|X) directly - discriminative


Classification

Learn P(Y|X)

1. Bayes rule:

P(Y|X) = P(X|Y)P(Y) / P(X) ~ P(X|Y)P(Y)

  • Learn P(X|Y), P(Y)

  • “Generative” classifier

    2. Learn P(Y|X) directly

  • “Discriminative”(to be covered later in class)

  • e.g. logistic regression


  • Generative Classifier: Bayes Classifier

    Learn P(X|Y), P(Y)

    • e.g. email classification problem

      • 3 classes for Y = { spam, not spam, maybe }

      • 10,000 binary features for X = {“Cash”, “Rolex”,…}

      • How many parameters do we have?

        • P(Y) :

        • P(X|Y) :


    Generative learning:Naïve Bayes

    • Introduce conditional independence

      P(X1,X2|Y) = P(X1 |Y) P(X2 |Y)

      P(Y|X) = P(X|Y)P(Y) / P(X) for X=(Xi,…,Xn)

      = P(X1|Y)…P(Xn|Y)P(Y) / P(X)

      = prodi P(Xi|Y) P(Y) / P(X)

      • Learn P(X1|Y), … P(Xn|Y), P(Y)

        instead of learning P(X1,…, Xn |Y) directly


    Naïve Bayes

    • 3 classes for Y = {spam, not spam, maybe}

    • 10,000 binary features for X = {“Cash”,”Rolex”,…}

    • Now, how many parameters?

      • P(Y)

      • P(X|Y)

    • fewer parameters

    • “simpler” – less likely to overfit


    Full Bayes vs. Naïve Bayes

    P(Y=1|(X1,X2)=(0,1))=?

    • Full Bayes:

      P(Y=1)=?

      P((X1,X2)=(0,1)|Y=1)=?

    • Naïve Bayes:

      P(Y=1)=?

      P((X1,X2)=(0,1)|Y=1)=?

    • XOR


    Regression

    • Prediction of continuous variables

      • e.g. I want to predict salaries from GPA.

        •  I can regress that …

    • Learn the mapping f: X  Y

      • Model is linear in the parameters (+ some noise)

         linear regression

      • Assume Gaussian noise

      • Learn MLE Θ


    1-parameter linear regression

    • Normal linear regression

      or equivalently,

    • MLEΘ?

    • MLE σ2 ?


    Multivariate linear regression

    • What if the inputs are vectors?

      • Write matrix X and Y :

        (n data points, k features for each data)

      • MLE Θ =


    Constant term?

    • We may expect linear data that does not go through the origin

    • Trick?


    The constant term


    Regression: another example

    • Assume the following model to fit the data. The model has one unknownparameter θ to be learned from data.

    • A maximum likelihood estimation of θ?


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