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homework3sol

# homework3sol - CS573 Homework 3 Solution Part of the...

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CS573: Homework 3 Solution Part of the solution is compiled from Jaewoo L.’s and Pedro P.’s submissions December 11, 2010 Tree Augmented Naive Bayes In this assignment we investigate a graphical model that extends Naive Bayes (NB) for classification. NB uses a generative model which assumes conditional independence be- tween the attributes ( X = { X 1 , ..., X n } ) given the class ( C ). This can be represented using a directed graphical model where the nodes are V = { X i | 1 i n } ∪ { C } and the edges are E = { ( C, X i ) | 1 i n } . Tree Augmented Naive Bayes (TAN) augments this graphical model with a set of edges E 0 X × X . The restriction on E 0 is that every X i has exactly one parent from X (in addition to C ), except for one X i that has no parents other than C . Figure 1 gives a general description of Naive Bayes versus TAN. To estimate a TAN model, the learning algorithm needs to search over the structure of the model (i.e., which edges to add among the attributes) and then estimate the parameters of the conditional probability distributions (CPDs). Learning the structure of TAN model requires a procedure that consists of five main steps: 1. Compute I P ( X i ; X j | C ) between each pair of attributes, i 6 = j , where I P ( X ; Y | z ) = X x,y,z P ( x, y, z ) log P ( x, y | z ) P ( x | z ) P ( y | z ) 2. Build a complete undirected graph in which the vertices are the attributes X 1 , .... , X n . Annotate the weight of an edge connecting X i to X j by I P ( X i ; X j | C ).

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