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Data Mining with Naïve Bayesian Methods

Data Mining with Naïve Bayesian Methods. Instructor: Qiang Yang Hong Kong University of Science and Technology Qyang@cs.ust.hk Thanks: Dan Weld, Eibe Frank. How to Predict?. From a new day’s data we wish to predict the decision Applications Text analysis Spam Email Classification

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Data Mining with Naïve Bayesian Methods

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  1. Data Mining with Naïve Bayesian Methods Instructor: Qiang Yang Hong Kong University of Science and Technology Qyang@cs.ust.hk Thanks: Dan Weld, Eibe Frank

  2. How to Predict? • From a new day’s data we wish to predict the decision • Applications • Text analysis • Spam Email Classification • Gene analysis • Network Intrusion Detection

  3. Naïve Bayesian Models • Two assumptions: Attributes are • equally important • statistically independent (given the class value) • This means that knowledge about the value of a particular attribute doesn’t tell us anything about the value of another attribute (if the class is known) • Although based on assumptions that are almost never correct, this scheme works well in practice!

  4. Why Naïve? • Assume the attributes are independent, given class • What does that mean? play outlook temp humidity windy Pr(outlook=sunny | windy=true, play=yes)= Pr(outlook=sunny|play=yes)

  5. Weather data set

  6. Is the assumption satisfied? • #yes=9 • #sunny=2 • #windy, yes=3 • #sunny|windy, yes=1 Pr(outlook=sunny|windy=true, play=yes)=1/3 Pr(outlook=sunny|play=yes)=2/9 Pr(windy|outlook=sunny,play=yes)=1/2 Pr(windy|play=yes)=3/9 Thus, the assumption is NOT satisfied. But, we can tolerate some errors (see later slides)

  7. Probabilities for the weather data • A new day:

  8. Bayes’ rule • Probability of event H given evidence E: • A priori probability of H: • Probability of event before evidence has been seen • A posteriori probability of H: • Probability of event after evidence has been seen

  9. Naïve Bayes for classification • Classification learning: what’s the probability of the class given an instance? • Evidence E = an instance • Event H = class value for instance (Play=yes, Play=no) • Naïve Bayes Assumption: evidence can be split into independent parts (i.e. attributes of instance are independent)

  10. The weather data example Evidence E Probability for class “yes”

  11. The “zero-frequency problem” • What if an attribute value doesn’t occur with every class value (e.g. “Humidity = high” for class “yes”)? • Probability will be zero! • A posteriori probability will also be zero! (No matter how likely the other values are!) • Remedy: add 1 to the count for every attribute value-class combination (Laplace estimator) • Result: probabilities will never be zero! (also: stabilizes probability estimates)

  12. Modified probability estimates • In some cases adding a constant different from 1 might be more appropriate • Example: attribute outlook for class yes • Weights don’t need to be equal (if they sum to 1) Sunny Overcast Rainy

  13. Missing values • Training: instance is not included in frequency count for attribute value-class combination • Classification: attribute will be omitted from calculation • Example:

  14. Dealing with numeric attributes • Usual assumption: attributes have a normal or Gaussian probability distribution (given the class) • The probability density function for the normal distribution is defined by two parameters: • The sample mean : • The standard deviation : • The density function f(x):

  15. Statistics for the weather data • Example density value:

  16. Classifying a new day • A new day: • Missing values during training: not included in calculation of mean and standard deviation

  17. Probability densities • Relationship between probability and density: • But: this doesn’t change calculation of a posteriori probabilities because  cancels out • Exact relationship:

  18. Example of Naïve Bayes in Weka • Use Weka Naïve Bayes Module to classify • Weather.nominal.arff

  19. Discussion of Naïve Bayes • Naïve Bayes works surprisingly well (even if independence assumption is clearly violated) • Why? Because classification doesn’t require accurate probability estimates as long as maximum probability is assigned to correct class • However: adding too many redundant attributes will cause problems (e.g. identical attributes) • Note also: many numeric attributes are not normally distributed

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