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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 l.jpg

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 l.jpg

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 c lassification l.jpg

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 l.jpg

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 l.jpg

    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 na ve bayes l.jpg

    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


    Na ve bayes l.jpg

    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 na ve bayes l.jpg

    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 l.jpg

    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 l.jpg

    1-parameter linear regression

    • Normal linear regression

      or equivalently,

    • MLEΘ?

    • MLE σ2 ?


    Multivariate linear regression l.jpg

    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 l.jpg

    Constant term?

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

    • Trick?


    The constant term l.jpg

    The constant term


    Regression another e xample l.jpg

    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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