1. CS 188: Artificial Intelligence Fall 2007
Lecture 18: Bayes Nets III
10/30/2007
Dan Klein – UC Berkeley

2. Announcements Project shift: Contest is live
Project 4 moved back a little
Instead, mega-mini-homework, worth 3x, graded
Contest is live

3. Inference
Inference: calculating some statistic from a joint probability distribution
Examples:
Posterior probability:
Most likely explanation: L R B D T T’

4. Reminder: Alarm Network

5. Normalization Trick
Normalize

6. Inference by Enumeration?

7. Nesting Sums Atomic inference is extremely slow!
Slightly clever way to save work:
Move the sums as far right as possible
Example:

8. Evaluation Tree View the nested sums as a computation tree:
Still repeated work: calculate P(m | a) P(j | a) twice, etc.

9. Variable Elimination: Idea
Lots of redundant work in the computation tree
We can save time if we cache all partial results
Join on one hidden variable at a time
Project out that variable immediately
This is the basic idea behind variable elimination

10. Basic Objects Track objects called factors
Initial factors are local CPTs
During elimination, create new factors
Anatomy of a factor:
4 numbers, one for each value of D and E
Argument variables, always non-evidence variables
Variables introduced
Variables summed out

11. Basic Operations First basic operation: join factors
Combining two factors:
Just like a database join
Build a factor over the union of the domains
Example:

12. Basic Operations Second basic operation: marginalization
Take a factor and sum out a variable
Shrinks a factor to a smaller one
A projection operation
Example:

13. Example

14. Example

15. General Variable Elimination
Query:
Start with initial factors:
Local CPTs (but instantiated by evidence)
While there are still hidden variables (not Q or evidence):
Pick a hidden variable H
Join all factors mentioning H
Project out H
Join all remaining factors and normalize

16. Example
Choose A

17. Example
Choose E
Finish
Normalize

18. Variable Elimination What you need to know:
VE caches intermediate computations
Polynomial time for tree-structured graphs!
Saves time by marginalizing variables ask soon as possible rather than at the end
We will see special cases of VE later
You’ll have to implement the special cases
Approximations
Exact inference is slow, especially when you have a lot of hidden nodes
Approximate methods give you a (close) answer, faster

19. Sampling Basic idea: Outline:
Draw N samples from a sampling distribution S
Compute an approximate posterior probability
Show this converges to the true probability P
Outline:
Sampling from an empty network
Rejection sampling: reject samples disagreeing with evidence
Likelihood weighting: use evidence to weight samples

20. Prior Sampling Cloudy Cloudy Sprinkler Sprinkler Rain Rain WetGrass

21. Prior Sampling This process generates samples with probability
…i.e. the BN’s joint probability
Let the number of samples of an event be
Then
I.e., the sampling procedure is consistent

22. Example We’ll get a bunch of samples from the BN:
c, s, r, w
c, s, r, w
c, s, r, w
c, s, r, w
If we want to know P(W)
We have counts <w:4, w:1>
Normalize to get P(W) = <w:0.8, w:0.2>
This will get closer to the true distribution with more samples
Can estimate anything else, too
What about P(C| r)? P(C| r, w)?
Cloudy
Sprinkler
Rain
WetGrass C S R W

23. Rejection Sampling Let’s say we want P(C) Let’s say we want P(C| s)
No point keeping all samples around
Just tally counts of C outcomes
Let’s say we want P(C| s)
Same thing: tally C outcomes, but ignore (reject) samples which don’t have S=s
This is rejection sampling
It is also consistent (correct in the limit)
Cloudy
Sprinkler
Rain
WetGrass C S R W
c, s, r, w
c, s, r, w
c, s, r, w
c, s, r, w

24. Likelihood Weighting Problem with rejection sampling:
If evidence is unlikely, you reject a lot of samples
You don’t exploit your evidence as you sample
Consider P(B|a)
Idea: fix evidence variables and sample the rest
Problem: sample distribution not consistent!
Solution: weight by probability of evidence given parents
Burglary
Alarm
Burglary
Alarm

25. Likelihood Sampling Cloudy Cloudy Sprinkler Sprinkler Rain Rain
WetGrass
WetGrass

26. Likelihood Weighting
Sampling distribution if z sampled and e fixed evidence
Now, samples have weights
Together, weighted sampling distribution is consistent
Cloudy
Rain C S R W

27. Likelihood Weighting
Note that likelihood weighting doesn’t solve all our problems
Rare evidence is taken into account for downstream variables, but not upstream ones
A better solution is Markov-chain Monte Carlo (MCMC), more advanced
We’ll return to sampling for robot localization and tracking in dynamic BNs
Cloudy
Rain C S R W

28. Decision Networks
MEU: choose the action which maximizes the expected utility given the evidence
Can directly operationalize this with decision diagrams
Bayes nets with nodes for utility and actions
Lets us calculate the expected utility for each action
New node types:
Chance nodes (just like BNs)
Actions (rectangles, must be parents, act as observed evidence)
Utilities (depend on action and chance nodes)
Umbrella U
Weather
Report

29. Decision Networks Action selection: Instantiate all evidence
Calculate posterior over parents of utility node
Set action node each possible way
Calculate expected utility for each action
Choose maximizing action
Umbrella U
Weather
Report

30. Example: Decision Networks
Umbrella U
Weather A W
U(A,W)
leave
sun
100
rain
take 20 70 W
P(W)
sun
0.7
rain
0.3

31. Example: Decision Networks
Umbrella W
P(W)
sun
0.7
rain
0.3 U
Weather A W
U(A,W)
leave
sun
100
rain
take 20 70 R
P(R|sun)
clear
0.5
cloudy
Report R
P(R|rain)
clear
0.2
cloud
0.8

32. Value of Information
Idea: compute value of acquiring each possible piece of evidence
Can be done directly from decision network
Example: buying oil drilling rights
Two blocks A and B, exactly one has oil, worth k
Prior probabilities 0.5 each, mutually exclusive
Current price of each block is k/2
Probe gives accurate survey of A. Fair price?
Solution: compute value of information
= expected value of best action given the information minus expected value of best action without information
Survey may say “oil in A” or “no oil in A,” prob 0.5 each
= [0.5 * value of “buy A” given “oil in A”] +
[0.5 * value of “buy B” given “no oil in A”] – 0
= [0.5 * k/2] + [0.5 * k/2] - 0 = k/2
DrillLoc U
OilLoc

33. General Formula Current evidence E=e, possible utility inputs s
Potential new evidence E’: suppose we knew E’ = e’
BUT E’ is a random variable whose value is currently unknown, so:
Must compute expected gain over all possible values
(VPI = value of perfect information)

34. VPI Properties Nonnegative in expectation
Nonadditive ---consider, e.g., obtaining Ej twice
Order-independent

35. VPI Example
Umbrella U
Weather
Report

36. VPI Scenarios Imagine actions 1 and 2, for which U1 > U2
How much will information about Ej be worth?
Little – we’re sure action 1 is better.
A lot – either could be much better
Little – info likely to change our action but not our utility