Probabilistic Graphical Models and Bayesian Networks. Artificial Intelligence Bert Huang Virginia Tech

Transcription

1 Probabilistic Graphical Models and Bayesian Networks Artificial Intelligence Bert Huang Virginia Tech

2 Concept Map for Segment Probabilistic Graphical Models Probabilistic Time Series Models Particle Filters (Neural Networks)

3 Outline Probabilistic graphical models Bayesian networks Inference in Bayes nets

4 Probabilistic Graphical Models PGMs represent probability distributions They encode conditional independence structure with graphs They enable graph algorithms for inference and learning

5 Probability Identities Random variables in caps (A) values in lowercase: A = a or just a for shorthand P(a b) = P(a, b) / P(b) conditional probability P(a, b) = P(a b) P(b) joint probability P(b a) = P(a b) P(b) / P(a)

6 Probability via Counting

7 Probability via Counting P(circle, red) =2/8 =

8 Probability via Counting P(circle red) = P(circle, red) / P(red) 2/3 2/8 3/8

9 Probability via Counting P(circle red) P(red) = P(circle, red) 2/3 3/8 2/8

10 Probability Identities Random variables in caps (A) values in lowercase: A = a or just a for shorthand P(a b) = P(a, b) / P(b) P(a, b) = P(a b) P(b) P(b a) = P(a b) P(b) / P(a)

11 Bayesian Networks P(L, R, W) conditional Win Lottery independence structure = P(L) P(R) P(W R) Rain Wet Ground Slip P(L, R, W, S) = P(L) P(R) P(W R) P(S W) P(S W, R)

12 Bayesian Networks P(R, W, S, C) = P(R) P(C) P(W C, R) P(S W) P(X Parents(X)) Rain Wet Ground Slip Car Wash

13 Independence in Bayes Nets A B Each variable is conditionally independent of its non-descendents given its parents Each variable is conditionally independent of any other variable given its Markov blanket C D Parents, children, and children s parents E

14 Independence in Bayes Nets A B Each variable is conditionally independent of its non-descendents given its parents Each variable is conditionally independent of any other variable given its Markov blanket C D Parents, children, and children s parents E

15 Independence in Bayes Nets A B Each variable is conditionally independent of its non-descendents given its parents Each variable is conditionally independent of any other variable given its Markov blanket C D Parents, children, and children s parents E

16 Independence in Bayes Nets A B Each variable is conditionally independent of its non-descendents given its parents Each variable is conditionally independent of any other variable given its Markov blanket C D Parents, children, and children s parents E

17 Independence in Bayes Nets A B Each variable is conditionally independent of its non-descendents given its parents Each variable is conditionally independent of any other variable given its Markov blanket C D Parents, children, and children s parents E

18 Inference Given a Bayesian Network describing P(X, Y, Z), what is P(Y) First approach: enumeration

19 P(R, W, S, C) = P(R) P(C) P(W C, R) P(S W) X P(r s) = X w P(r, w, s, c)/p(s) c X P(r s) / X w P(r)P(c)P(w c, r)p(s w) c P(r s) / P(r) X w P(s w) X c P(c)P(w c, r) O(2 n )

20 Second Approach: Variable Elimination X P(r s) / X w P(r)P(c)P(w c, r)p(s w) c f C (w) = X c P(c)P(w c, r) P(r s) / X w P(r)P(s w)f c (w)

21 P(W, X, Y, Z) =P(W )P(X W )P(Y X )P(Z Y ) P(Y )? X X P(Y )= X w P(w)P(x w)p(y x)p(z Y ) x z f w (x) = X w P(w)P(x w) X P(Y )= X x f w (x)p(y x)p(z Y ) z f x (Y )= X x f w (x)p(y x) P(Y )= X z f x (Y )P(z Y )

22 W X Y Z X X P(Y )= X w P(w)P(x w)p(y x)p(z Y ) x z f w (x) = X w P(w)P(x w) X P(Y )= X x f w (x)p(y x)p(z Y ) z f x (Y )= X x f w (x)p(y x) P(Y )= X z f x (Y )P(z Y )

23 Variable Elimination Every variable that is not an ancestor of a query variable or evidence variable is irrelevant to the query Iterate: choose variable to eliminate sum terms relevant to variable, generate new factor until no more variables to eliminate Exact inference is #P-Hard in tree-structured BNs, linear time (in number of table entries)

24 Learning in Bayes Nets Super easy! Estimate each conditional probability by counting