Left quadratic decision boundaries obtained naive

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Unformatted text preview: mes the gaussian densities have the same variance in each class – shrinks the class centroids towards the overall mean in each class • More general models have less bias but are typically hard to estimate in high dimensions, so the independence assumption may not hurt too much. 60 ESL Chapter 4 — Linear Methods for Classification Trevor Hastie and Rob Tibshirani Naive Bayes vs Quadratic Discriminant Analysis 1 2 2 2 2 1 2 1 1 2 2 1 2 3 1 2 2 3 22 2 3 3 22 2 3 33 2 33 2 12 3 22 33 2 3 2 2 22 3 2 222 1 22 2 22 3 22 2 1 2 2 22 33 2 2 22 222 22 2 2 3 33 22 2 22 2 22 2 2 2 2 2 3 33 22 2 22 22 22 222 2 1 3 3 333 3 2222 2 2 22 22 2 2 222 222 2222 22 2 3 33 3 3 3 3 2 22 2 33 2 22 2 2 22 33 3 3 3 2 13 3 22 22 22 2 1 22 222 22 22 22 22 333 3 3 3 2 1 2 222 22 2 2 222 2 2 2 2 1 2 3 2 3333 3 22 2 2 1 2 2 1 1 2 21 2 3 1 3 33 3 1 22 2 2 3 2 1 22 22 22 1 1 1 1 2 1 1 1 3 33 3 1 22 1 2 2 2 21 1 2 1 11 1 3 33 1 3 333 3 3 3 2 2 11 22 11 1 1 1 1 1 33 1 2 1 222 3 3 11 1 1 12 1 1 1 33 33 2 1 11 1 21 22 1 31 3 11 1 1 11 1 13 33 3 3 3 11 11 3 2 33 3 1 2 11 1 1 1 11 1 1 33 333333 1 21 11 1 1 1 33 333 3 1 11 1 1 1 1 1 33 1 1 11 1 11 33 3 3 1 3 11 11 1 1 1 1 11 1 1 1 11 1 1 1 1 1 1 1 1 111 33 1 3 3 3 33 3 3 11 3 1 3 1 11 33 3 33 11 1 1 1 1 11 33 11 13 33 3 3 1 1 33 11 1 11 1 333 3333 11 1 1 1 1 1 1 1 1 1 1 3 33 1 1 1 1 1 11 1 1 3 33 3 1 1 3 3 33 3 3 333 333 3 1 111 1 1 3 1 33 111 33 3 1 11 33 33 33 3 3 1 3 1 2 2 3 22 2 3 3 22 2 3 33 2 33 2 12 3 22 33 2 3 2 2 22 3 2 222 1 22 2 22 3 22 2 1 2 2 22 33 2 2 22 222 22 2 2 3 33 22 2 22 2 22 2 2 2 2 2 3 33 22 2 22 22 22 222 2 1 3 3 333 3 2222 2 2 22 22 2 2 222 222 2222 22 2 3 33 3 3 3 3 2 22 2 33 2 22 2 2 22 33 3 3 3 2 13 3 22 22 22 2 1 22 222 22 22 22 22 333 3 3 3 2 1 2 222 22 2 2 222 2 2 2 2 1 2 3 2 3333 3 22 2 2 1 2 2 1 1 2 21 2 3 1 3 33 3 1 22 2 2 3 2 1 22 22 22 1 1 1 1 2 1 1 1 3 33 3 1 22 1 2 2 2 21 1 2 1 11 1 3 33 1 3 333 3 3 3 2 2 11 22 11 1 1 1 1 1 33 1 2 1 222 3 3 11 1 1 12 1 1 1 33 33 2 1 11 1 21 22 1 31 3 11 1 1 11 1 13 33 3 3 3 11 11 3 2 33 3 1 2 11 1 1 1 11 1 1 33 333333 1 21 11 1 1 1 33 333 3 1 11 1 1 1 1 1 33 1 1 11 1 11 33 3 3 1 3 11 11 1 1 1 1 11 1 1 1 11 1 1 1 1 1 1 1 1 111 33 1 3 3 3 33 3 3 11 3 1 3 1 11 33 3 33 11 1 1 1 1 11 33 11 13 33 3 3 1 1 33 11 1 11 1 333 3333 11 1 1 1 1 1 1 1 1 1 1 3 33 1 1 1 1 1 11 1 1 3 33 3 1 1 3 3 33 3 3 333 333 3 1 111 1 1 3 1 33 111 33 3 1 11 33 33 33 2 2 2 1 1 2 2 2 3 3 Two methods for fitting quadratic boundaries. [Left] Quadratic decision boundaries, obtained Naive Bayes. [Right] Quadratic decision boundaries found by QDA. 61 ESL Chapter 4 — Linear Methods for Classification Trevor Hastie and Rob Tibshirani Naive Bayes and GAMs • Note that p f1 (X )π1 gm ( X m ) , =α+ log f2 (X )π2 m=1 a generalized additive logistic regression model. • GAMs are fit by binomial maximum likelihood. • Naive Bayes models are fit using the full likelihood. • Generalizes the logistic regression-LDA...
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