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Daphne Koller Overview Conditional Probability Queries Probabilistic Graphical Models Inference.

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Presentation on theme: "Daphne Koller Overview Conditional Probability Queries Probabilistic Graphical Models Inference."— Presentation transcript:

1 Daphne Koller Overview Conditional Probability Queries Probabilistic Graphical Models Inference

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

3 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

4 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| < 

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

6 Daphne Koller Sum-Product BD C A

7 Daphne Koller Evidence: Reduced Factors BD C A

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

9 Daphne Koller Sum-Product Computeand renormalize

10 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

11 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

12 Daphne Koller END END END

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