1. CS 188: Artificial Intelligence Fall 2008
Lecture 17: Bayes Nets IV
10/28/2008
Dan Klein – UC Berkeley

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

3. Mini-Examples: VE Mini-Alarm Network B: there is a burglary
A: the alarm sounds
C: neighbor calls B B P b
0.1 b
0.9 A C B A P b a
0.8 a
0.2 b
0.1
0.9 A C P a c
0.7 c
0.3 a
0.5

4. Basic Operation: Join First basic operation: join factors
Combining two factors:
Just like a database join
Build a factor over the union of the variables involved
Example:
Computation for each entry: pointwise products

5. Basic Operation: Eliminate
Second basic operation: marginalization
Take a factor and sum out a variable
Shrinks a factor to a smaller one
A projection operation
Example:
Definition:

6. Example: P(A) Start Join on B Sum out B B A, B A A B P b 0.1 b 0.9 A
0.08 b
0.09 a
0.02
0.81 A P a
0.17 a
0.83 B A P b a
0.8 a
0.2 b
0.1
0.9

7. Example: Multiple JoinsB P b
0.1 b
0.9
Join on B B A P b a
0.8 a
0.2 b
0.1
0.9 B
B, A B A P b a
0.08 a
0.02 b
0.09
0.81 A C c A C P a c
0.7 c
0.3 a
0.5 A C P a c
0.7 c
0.3 a
0.5

8. Example: Multiple Joins
B, A
Example: Multiple Joins C
Join on A
B, A, C B A P b a
0.08 a
0.02 b
0.09
0.81 B A C P b a c
0.056 c
0.024 a
0.010 b
0.063
0.027
0.405 A C P a c
0.7 c
0.3 a
0.5

9. Query: P(C) Sum out B Sum out A A, C B, A, C C A C P a c 0.119 c
0.051 a
0.415 B A C P b a c
0.056 c
0.024 a
0.010 b
0.063
0.027
0.405
Sum out A C C P c
0.534 c
0.466

10. Variable Elimination Why is inference by enumeration so slow?
You join up the whole joint distribution before you sum out the hidden variables
You do lots of computations irrelevant to the evidence
You end up repeating a lot of work
Idea: interleave joining and marginalizing!
Marginalize variables as soon as possible
Called “Variable Elimination”
Still NP-hard, but usually much faster than inference by enumeration

11. P(C) : Marginalizing Early
B, A
P(C) : Marginalizing Early C
Sum out B A B A P b a
0.08 a
0.02 b
0.09
0.81 A P a
0.17 a
0.83 C A C P a c
0.7 c
0.3 a
0.5 A C P a c
0.7 c
0.3 a
0.5

12. Marginalizing Early Join on A Sum out A A A, C C C A C P a c 0.119 c
0.051 a
0.415 A P a
0.17 a
0.83 C
Sum out A A C P a c
0.7 c
0.3 a
0.5 C C P c
0.534 c
0.466

13. 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
Sum out H
Join all remaining factors and normalize

14. Example: P(B|a) Start / Select Join on B Normalize B a, B a B P b 0.1
0.9 a A B P a b
0.08 b
0.09 A B P a b
8/17 b
9/17 B A P b a
0.8 a
0.2 b
0.1
0.9

15. Example: P(B|c) Join on A B B A, c A c B P b 0.1 b 0.9 B P b 0.1
0.8 a
0.2 b
0.1
0.9 B
A, c A B A C P b a c
0.24 a
0.10 b
0.03
0.45 c A C P a c
0.7 c
0.3 a
0.5

16. Example: P(B|c) Sum out A B B A, c c B P b 0.1 b 0.9 B P b 0.1 b
0.24 a
0.10 b
0.03
0.45 B C P b c
0.34 b
0.48

17. Example: P(B|c) Join on B Normalize B, c B c B P b 0.1 b 0.9 B C P
0.034 b
0.423 c
Normalize B C P b c
0.34 b
0.48 B C P b c
0.073 b
0.927

18. Variable Elimination What you need to know:
Should be able to run it on small examples, understand the factor creation / reduction flow
Better than enumeration: VE caches intermediate computations
Saves time by marginalizing variables as soon as possible rather than at the end
Polynomial time for tree-structured graphs – sound familiar?
We will see special cases of VE later
You’ll have to implement the special cases
Approximations
Exact inference is slow, especially with a lot of hidden nodes
Approximate methods give you a (close, wrong?) 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 for conditional probabilities (i.e., 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