Learning Tree Structures
If we measured a distribution P, what is the tree-dependent distribution Pt that best approximates P? Which Pt will be closest to P? • Search Space: All possible trees • Goal: From all possible trees find the one closest to P • Distance Measurement: • Kullback–Leibler cross –entropy measure • Operators/Procedure
Problem definition • X1…Xn are random variables • P is unknown • Given independent samples x1,…,xs drawn from distribution P • Estimate P • Solution 1 - independence • Assume X1…Xn are independent • P(x) = Π P(xi) • Solution 2 - trees • P(x) = Π P(xi|xj) • xj- The parent of xi in some
Kullback–Leibler cross–enthropy measure • For probability distributions P and Q of a discrete random variable the K–L divergence of Q from P is defined to be • It can be seen from the definition of the Kullback-Leibler divergence that • where H(P,Q) is called the cross entropy of P and Q, and H(P) is the entropy of P. • Non negative measure (by Gibb’s inequality)
Entropy is a measure for Uncertainty • Fair coin: • H(½, ½) = – ½ log2(½) – ½ log2(½) = 1 bit • (ie, need 1 bit to convey the outcome of coin flip) • Biased coin: H( 1/100, 99/100) = – 1/100 log2(1/100) – 99/100 log2(99/100) = 0.08 bit • As P( heads ) 1, info of actual outcome 0 H(0, 1) = H(1, 0) = 0 bits ie, no uncertainty left in source (0 log2(0) = 0)
Optimization Task • Init: Fix the structure of some tree t • Assign Probabilities: What conditional probabilities Pt(x|y) would yield the best approximation of P? • Procedure: vary the structure of t over all possible spanning trees • Goal: among all trees with probabilities- which is the closest to P?
What Probabilities to assign? Pt(x|y) = P(x|y) Theorem 1 : If we Force the probabilities along the branches of the tree t to coincide with those computed from P => We get the best t-dependent approximation of P
How to vary over all trees? How to move in the search space? Maximum Weight Spanning Tree Theorem 2 : The distance measure(Kullback–Leibler) is minimized by assigning the best distribution on any maximum weight spanning tree, where the weight on the branch (x,y) is defined by the mutual information measurement
Mutual information • measures how much knowing one of these variables reduces our uncertainty about the other • the mutual information is the same as the uncertainty contained in Y (or X) alone, namely the entropy of Y or X • Mutual information is a measure of dependence • Mutual information is nonnegative (i.e. I(X;Y) ≥ 0) and symmetric (i.e. I(X;Y) = I(Y;X)).
The algorithm • Find Maximum spanning tree with weights given by : • Compute Pt • Select an arbitrary root node and compute
B D A B D A C B D A C C (0.56, 0.11, 0.02, 0.31) (0.51, 0.17, 0.17, 0.15) (0.53, 0.15, 0.19, 0.13) (0.44, 0.14, 0.23, 0.19) (0.46, 0.12, 0.26, 0.16) (0.64, 0.04, 0.08, 0.24) AB AC AD BC BD CD AB AC AD BC BD CD (0.56, 0.11, 0.02, 0.31) (0.51, 0.17, 0.17, 0.15) (0.53, 0.15, 0.19, 0.13) (0.44, 0.14, 0.23, 0.19) (0.46, 0.12, 0.26, 0.16) (0.64, 0.04, 0.08, 0.24) 0.3126 0.0229 0.0172 0.0230 0.0183 0.2603 Illustration of CL-Tree Learning 0.3126 0.0229 0.0172 0.0230 0.0183 0.2603
Theorem 1 :If we Force the probabilities along the branches of the tree t to coincide with those computed from P => We get the best t-dependent approximation of P
Theorem 1 :If we Force the probabilities along the branches of the tree t to coincide with those computed from P => We get the best t-dependent approximation of P By Gibb’s inequality the Expression is maximized when P’(x)=P(x) => all the expression is maximal when P’(xi|xj)=P(xi|xj) Q.E.D.
Theorem 1 :If we Force the probabilities along the branches of the tree t to coincide with those computed from P => We get the best t-dependent approximation of P Gibbs' inequality
Theorem 1 :If we Force the probabilities along the branches of the tree t to coincide with those computed from P => We get the best t-dependent approximation of P Gibbs' inequality
Theorem 2 : The distance measure(Kullback–Leibler) is minimized by assigning the best distribution on any maximum weight spanning tree, where the weight on the branch (x,y) is defined by the mutual information measurement: From theorem 1, we get: maximizes DKL After assignment, and Bayes rule:
Theorem 2 : The distance measure(Kullback–Leibler) is minimized by assigning the best distribution on any maximum weight spanning tree, where the weight on the branch (x,y) is defined by the mutual information measurement: From theorem 1, we get: maximizes DKL After assignment, and Bayes rule:
Theorem 2 : The distance measure(Kullback–Leibler) is minimized by assigning the best distribution on any maximum weight spanning tree, where the weight on the branch (x,y) is defined by the mutual information measurement: • The second and third term are independent of t • D(P,Pt) is nonnegative (Gibb’s inequality) • Thus, minimizing the distance D(P,Pt) is equivalent to maximizing the • sum of branch weights • Q.E.D.
Chow-Liu (CL) Results • If distribution P is tree-structured,CL finds CORRECT one • If distribution P is NOT tree-structured,CL finds tree structured Q that has min’l KL-divergence – argminQ KL(P; Q) • Even though 2(n log n) trees,CL finds BEST one in poly time O(n2 [m + log n])
Chow-Liu Trees -Summary • Approximation of a joint distribution with a tree-structured distribution [Chow and Liu 68] • Learning the structure and the probabilities • Compute individual and pairwise marginal distributions for all pairs of variables • Compute mutual information (MI) for each pair of variables • Build maximum spanning tree with for a complete graph with variables as nodes and MIs as weights • Properties • Efficient: • O(#samples×(#variables)2×(#values per variable)2) • Optimal
References • S. Kullback (1959) Information theory and statistics (John Wiley and Sons, NY). • Chow, C. K. and Liu, C. N. (1968). Approximating discrete probability distributions with dependence trees. IEEE Transactions on Information Theory, IT-14(3):462{467.
