For a full tree balanced the sum of nt over all the ts

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Unformatted text preview: ion Trees (I) Denote the prior probability of class j by j . The priors j can be estimated from the data by Nj /N. Sometimes priors are given before-hand. The estimated probability of a sample in class j going to node t is p(t | j) = Nj (t)/Nj . p(tL | j) + p(tR | j) = p(t | j). For a full tree, the sum of p(t | j) over all t's at the same level is 1. Jia Li http://www.stat.psu.edu/jiali Classification/Decision Trees (I) The joint probability of a sample being in class j and going to node t is thus: p(j, t) = j p(t | j) = j Nj (t)/Nj . The probability of any sample going to node t is: p(t) = K j=1 p(j, t) = K j=1 j Nj (t)/Nj . Note p(tL ) + p(tR ) = p(t). Jia Li http://www.stat.psu.edu/jiali The probability of a sample being in class j given that it goes to node t is: p(j | t) = p(j, t)/p(t) . K For any t, j=1 p(j | t) = 1. Classification/Decision Trees (I) When j = Nj /N, we have the following simplification: p(j | t) = Nj (t)/N(t). p(t) = N(t)/N. p(j, t) = Nj (t)/N. Jia Li http://www.stat.psu.edu/jiali Classification/Decision Trees (I) Stopping Criteria A simple criteria: stop splitting a node t when max I (s, t) < , sS where is a chosen threshold. The above stopping criteria is unsatisfactory. A node with a small decrease of impurity after one step of splitting may have a large decrease after multiple levels of splits. Jia Li http://www.stat.psu.edu/jiali Classification/Decision Trees (I) Class Assignment Rule A class assignment rule assigns a class j = {1, ..., K } to every ~ ~ terminal node t T . The class as...
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