CS 541: Artificial Intelligence

1. CS 541: Artificial Intelligence Lecture VII: Inference in Bayesian Networks

2. Schedule • TA: Vishakha Sharma • Email Address: vsharma1@stevens.edu

3. Re-cap of Lecture VI – Part I • Probability is a rigorous formalism for uncertain knowledge • Joint probability distribution species probability of every atomic event • Queries can be answered by summing over atomic events • For nontrivial domains, we must find a way to reduce the joint size • Independence and conditional independence provide the tools

4. Re-cap of Lecture VI – Part II • Bayes nets provide a natural representation for (causally induced) conditional independence • Topology + CPTs = compact representation of joint distribution • Generally easy for (non)experts to construct • Canonical distributions (e.g., noisy-OR) = compact representation of CPTs • Continuous variables parameterized distributions (e.g., linear Gaussian)

5. Outline • Exact inference by enumeration • Exact inference by variable elimination • Approximate inference by stochastic simulation • Approximate inference by Markov chain Monte Carlo

6. Exact inference by enumeration

7. Inference tasks • Simple queries: compute posterior marginal P( Xi | E=e ) • e.g., P( NoGas| Gauge=empty, Lights=on, Starts=false ) • Conjunctive queries: P( Xi, Xj| E=e ) = P( Xi | E=e )P( Xj| Xi, E=e ) • Optimal decisions: decision networks include utility information; • probabilistic inference required for P( outcome | action, evidence) • Value of information: which evidence to seek next? • Sensitivity analysis: which probability values are most critical? • Explanation: why do I need a new starter motor?

8. Inference by enumeration • Slightly intelligent way to sum out variables from the joint without actually • Simple query on the burglary network: • Rewrite full joint entries using product of CPT entries: • Recursive depth-first enumeration: O(n) space, O(dn) time

9. Enumeration algorithm

10. Evaluation tree • Enumeration is inefficient: repeated computation • E.g., computes for every value of e

11. Inference by variable elimination

12. Inference by variable elimination • Variable elimination: carry out summations right-to-left, storing intermediate results (factors) to avoid recomputation

13. Variable elimination: basic operations • Summing out a variable from a product of factors: • Move any constant factors outside the summation • Add up submatrices in pointwise product of remaining factors • Assuming do not depend on • Pointwise product of factors and : • E.g.,

14. Variable elimination algorithm

15. Irrelevant variables • Consider the query • Sum over m is identically 1,M is irrelevant of to the query • Theorem 1: is irrelevant unless • Here , and • so is irrelevant • (Compare this to backward chaining from the query in Horn clause KBs)

16. Irrelevant variables • Definition: moral graph of Bayesian network: marry all parents and drop arrows • Definition: A is m-separated from B by Ciff separated by C in the moral graph • Theorem 2: Y is irrelevant if m-separated from X by E • For , both Burglary and Earthquake are irrelevant

17. Complexity of exact inference • Singly connected networks (or polytree): • any two nodes are connected by at most one (undirected) path • time and space cost of variable elimination are O(dkn) • Multiply connected networks: • Can reduce 3SAT to exact inference  NP-hard • Equivalent to counting 3SAT models  #P-complete

18. Inference by stochastic simulation

19. Inference by stochastic simulation • Basic idea: • Draw N samples from a sampling distribution S • Compute an approximate posterior probability • Show this converges to the true probability P • Outline: • Direct sampling from a Bayesian network • Rejection sampling: reject samples disagreeing with evidence • Likelihood weighting: use evidence to weight samples • Markov chain Monte Carlo (MCMC): sample from a stochastic process, whose stationary distribution is the true posterior

20. Direct sampling

21. Example

22. Example

23. Example

24. Example

25. Example

26. Example

27. Example

28. Direct sampling • Probability that PRIORSAMPLE generates a particular event • i.e., the true prior probability • E.g., • Let be the number of samples generated for events • Then we have • That is, estimates from PRIORSAMPLE are consistent • Shorthand:

29. Rejection sampling

30. Rejection sampling • estimated from samples agreeing with e • E.g, estimate using 100 samples • 27 samples have • Of these, 8 have , 19 have . • Similar to a basic real-world empirical estimation procedure

31. Analysis of rejection sampling • Hence rejection sampling returns consistent posterior estimates • Problem: hopelessly expensive if P(e) is small • P(e) drops o exponentially with number of evidence variables!

32. Likelihood weighting

33. Likelihood weighting Idea: x evidence variables, sample only nonevidence variables, and weight each sample by the likelihood it accords the evidence

34. Likelihood weighting example

35. Likelihood weighting example

36. Likelihood weighting example

37. Likelihood weighting example

38. Likelihood weighting example

39. Likelihood weighting example

40. Likelihood weighting example

41. Likelihood weighting analysis • Sampling probability for WEIGHTEDSAMPLE is • Note: pays attention to evidence in ancestors only • Somewhere "between" prior and posterior distribution • Weight for a given sample z, e is • Weighted sampling probability is • Hence likelihood weighting returns consistent estimates • but performance still degrades with many evidence variables because a few samples have nearly all the total weight

42. Markov chain Monte Carlo

43. Approximate inference using MCMC • "State" of network = current assignment to all variables. • Generate next state by sampling one variable given Markov blanket • Sample each variable in turn, keeping evidence fixed • Can also choose a variable to sample at random each time

44. The Markov chain • With Sprinkler =true , WetGrass=true, there are four states: • Wander about for a while, average what you see

45. MCMC example • Estimate • Sample Cloudy or Rain given its Markov blanket, repeat. • Count number of times Rain is true and false in the samples. • E.g., visit 100 states • 31 have Rain=true, 69 have Rain=false • Theorem: chain approaches stationary distribution, i.e., long-run fraction of time spent in each state is exactly proportional to its posterior probability

46. Markov blanket sampling • Markov blanket of Cloudy is • Sprinklerand Rain • Markov blanket of Rain is • Cloudy, Sprinkler, and WetGrass • Probability given the Markov blanket is calculated as follows: • Easily implemented in message-passing parallel systems, brains • Main computational problems: • Difficult to tell if convergence has been achieved • Can be wasteful if Markov blanket is large: • P(Xi|mb(Xi)) won't change much (law of large numbers)

47. Summary • Exact inference by variable elimination: • polytime on polytrees, NP-hard on general graphs • space = time, very sensitive to topology • Approximate inference by LW, MCMC: • LW does poorly when there is lots of (downstream) evidence • LW, MCMC generally insensitive to topology • Convergence can be very slow with probabilities close to 1 or 0 • Can handle arbitrary combinations of discrete and continuous variables

48. HW#3 Probabilistic reasoning • Mandatory • Problem 14.1 (1 points) • Problem 14.4 (2 points) • Problem 14.5 (2 points) • Problem 14.21 (5 points) • Programming is needed for question e) • You need to submit your executable along with source code, with detailed readme file on how we can run it. • You lose 3 points if no program. • Bonus • Problem 14.19 (2 points)