Daphne Koller Overview Conditional Probability Queries Probabilistic Graphical Models Inference.

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Presentation transcript:

Daphne Koller Overview Conditional Probability Queries Probabilistic Graphical Models Inference

Daphne Koller Inference in a PGM PGM encodes P(X 1,…,X n ) Can be used to answer any query over joint distribution

Daphne Koller Conditional Probability Queries Evidence: E = e Query: a subset of variables Y Task: compute P(Y | E=e) Applications – Medical/fault diagnosis – Pedigree analysis

Daphne Koller NP-Hardness The following are all NP-hard Given a PGM P , a variable X and a value x  Val(X), compute P  (X=x) – Or even decide if P  (X=x) > 0 Let  < 0.5. Given a PGM P , a variable X and a value x  Val(X), and observation e  Val(E), find a number p that has |P  (X=x|E=e) – p| < 

Daphne Koller Sum-Product C D I SG L J H D

Daphne Koller Sum-Product BD C A

Daphne Koller Evidence: Reduced Factors BD C A

Daphne Koller Evidence: Reduced Factors C D I SG L J H D P(J,I=i, H=h) =

Daphne Koller Sum-Product Computeand renormalize

Daphne Koller Algorithms: Conditional Probability Push summations into factor product – Variable elimination Message passing over a graph – Belief propagation – Variational approximations Random sampling instantiations – Markov chain Monte Carlo (MCMC) – Importance sampling

Daphne Koller Summary Conditional probability queries of subset of variables given evidence on others Summing over factor product Evidence simply reduces the factors Many exact and approximate algorithms

Daphne Koller END END END