Ilp 07

• Complementary advantages • logic can represent domain knowledge in a natural and expressive way • inductive logic programming (ILP) can generate theories and explanations • probabilities allow us to deal with uncertainty and noise in the data • Fixed unknown density ρ on X × Y • The optimal decision would be ρ(Y = 1|x) > 0.5 • Generative direction: model the class conditional densities p(x|y), then Bayes' theorem gives p(y |x) – SCFG: assign a probability to each production and infer the probability that string x was generated by the grammar – Stochastic logic programs generalize SCFGs: assign a probability to each definite clause and infer the probability that a goal is refuted • Discriminant direction: model p(y |x) directly or even more simply, learn f (x) = p(Y = 1|x) − 0.5 – Logistic regression: p(Y = 1|x) = 1 Z(x) e w φ(x) where φ(x) is a feature vector of x.