Informa(on theory in ML
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1 Informa(on theory in ML
2 Feature Selec+on For efficiency of the classifier and to suppress noise choose subset of all possible features. Selected features should be frequent to avoid overfitting the classifier to the training data, but not too frequent in order to be characteristic. Features should be good discriminators between classes (i.e. frequent/characteristic in one class but infrequent in other classes). Approach: - compute measure of discrimination for each feature - select the top k most discriminative features in greedy manner tf*idf is usually not a good discrimination measure, and may give undue weight to terms with high idf value (leading to the danger of overfitting)
3 Entropy:idea Die Toten Hosen: Sieben fuhren nach Düsseldorf und einer fuhr nach Köln 7 P (D) = 8 P (K) = 8 ( log2 + 0 log2 0) = 0 X 7 7 H = pi log2 pi = log2 + log i Hwater = log2 + log2 = lim pi!0 pi log2 pi = 0 Hkoelsch = Halt =
4 Example for Feature Selec+on f f2 f3 f4 f5 f6 f7 f8 d: d2: d3: d4: d5: d6: d7: d8: d9: d0: d: d2: Class Tree: Entertainment Calculus Math training docs: d, d2, d3, d4 Entertainment d5, d6, d7, d8 Calculus d9, d0, d, d2 Algebra Algebra
5 Discrimina+ve Classifiers: Decision Trees given: a multiset of m-dimensional training data records dom(a)... dom(am) with numerical, ordinal, or categorial attributes Ai (e.g. term occurrence frequencies N 0... N 0 ) and with class labels wanted: a tree with attribute value conditions of the form Ai value for numerical or ordinal attributes or Ai value set or Ai value set = for categorial attributes or linear combinations of this type k i A i for several numerical attributes as inner nodes and labeled classes as leaf nodes value
6 Examples for Decision Trees () tf(homomorphism) 2 tf(vector) 3 tf(limit) 2 Lineare Algebra T T F T F F Algebra Calculus Other has read Tolkien T has read Eco T F intellectual uneducated F boring salary T credit worthy university degree & salary T F credit worthy F not credit worthy
7 Top- Down Construc+on of Decision Tree Input: decision tree node k that represents one partition D of dom(a)... dom(am) Output: decision tree with root k ) BuildTree (root, dom(a)... dom(am)) 2) PruneTree: reduce tree to appropriate size with: procedure BuildTree (k, D): if k contains only training data of the same class then terminate; determine split dimension Ai; determine split value x for most suitable partitioning of D into D = D {d d.ai x} and D2= D {d d.ai > x}; create children k and k2 of k; BuildTree (k, D); BuildTree (k2, D2);
8 Split Criterion: Informa+on Gain Goal is to split current node such that the resulting partitions are as pure as possible w.r.t. class labels of the corresponding training data. Thus we aim to minimize the impurity of the partitions. An approach to define impurity is via the entropy-based (statistical) information gain (referring to the distribution of class labels within a partition) G (k, k, k2) = H(k) ( p*h(k) + p2*h(k2) ) where: n k : # training data records in k n k,j : # training data records in k that belong to class j p = n k / n k and p2 = n k2 / n k nk, j nk, j H( k ) = log2 n n j k k
9 Example for Decision Tree for Text Classifica+on C: Algebra C2: Calculus C3: Stochastics f f2 f3 f4 f5 f6 f7 f8 d: d2: d3: d4: d5: d6: f2>0 Algebra f7> Stochastics Calculus G = H(k) ( 2/6*H(k) + 4/6*H(k2) ) H(k) = /3 log 3 + /3 log 3 + /3 log 3 H(k) = log H(k2) = 0 + /2 log 2 + /2 log 2 G = log 3 0 2/3*,6 0,66 = 0,94
10 Simple (Class- unspecific) Criteria for Feature Selec+on Document Frequency Thresholding: Consider for class Cj only terms ti that occur in at least δ training documents of Cj. Term Strength: For decision between classes C,..., Ck select (binary) features Xi with the highest value of s( X ) : P[ X occurs in doc d X occurs in similar doc d' i = i To this end the set of similar doc pairs (d, d ) is obtained by thresholding on pairwise similarity or by clustering/grouping the training docs. i + further possible criteria along these lines
11 Feature Selec+on Based on Informa+on Gain Information gain: For discriminating classes c,..., ck select the (binary) features Xi with the largest gain in entropy G( Xi ) = k P[ c j j= log2 P[ c j P[ P[ X i X i k P[ c j j= k P[ c j j= X i X i log2 log2 P[ c j P[ c j Xi Xi Generalization for non-binary features is straightforward..
12 Feature Selec+on Based on Mutual Informa+on Mutual information (Kullback-Leibler distance, relative entropy): for class cj select those (binary) features Xi with the largest value of MI( X i,c j ) = X { X i,x i } C { c j,c j } P[ X C log P[ X C P[ X P[ C and for discriminating classes c,..., ck: MI( X i ) = k j= P[ c j MI( X i, c j ) Generalization for non-binary features is straightforward
13 Condi(onal mutual informa(on Idea: construct the set of features itera+vely (e.g. by adding features one by one) For a new candidate, compute score s(n, k) which depends of ist individual discrimina+ve power, given class labels and previously chosen features. By taking the feature Xn with the maximum score s(n, k) we ensure that the new feature is both informa+ve and different than the preceding ones - at least in terms of predic+ng the class label Y. Instrument for construc+ng s(n, k): condi+onal mutual informa+on.
14 Comparing distribu(ons We are given: two probability distribu+ons P and Q. Ques+on: how (dis)similar are they to each other? Common example: distribu+ons of two documents over (some) latent topics, Expressed by means of mul+nomials.. We can compute MI (P,Q) or MI (Q,P) but they are both asymmetric Workaround: take as a dissimilarity measure between the value of DIS (P,Q) = DIS (Q,P) = 0.5 * MI (P,Q) * MI (Q,P) or JSD (P,Q) = JSD (Q,P) = 0.5 * MI (P, M) * MI (Q,M) M = 0.5 * (P+Q) JSD also known as Jensen- Shannon divergence Not yet a real distance (JS is not a proper metric), but indicator Becomes a metric with further regulariza+on (e.g. square root of JSD)..
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