1. Directed Graphical Probabilistic Models: the sequel
William W. Cohen
Machine Learning
Feb 2008

2. Summary of Monday(1): Bayes nets
Many problems can be solved using the joint probability P(X1,…,Xn).
Bayes nets describe a way to compactly write the joint.
For a Bayes net: A
P(A) 1
0.33 2 3
First guess
The money B
P(B) 1
0.33 2 3 A B
Stick or swap? C
The goat D A B C
P(C|A,B) 1 2
0.5 3
1.0 … E
Conditional independence:
Second guess A C D
P(E|A,C,D) …

3. Aside: Conditional Independence?
(Chain rule – always true)
(Fancy version of c.r.)
(Def’n of cond. Indep.)
Caveat divisor: we’ll usually assume no probabilities are zero, so division is safe….

4. Summary of Monday(2): d-separationX E Y
There are three ways paths from X to Y given evidence E can be blocked.
X is d-separated from Y given E iff all paths from X to Y given E are blocked…see there? 
If X is d-separated from Y given E, then I<X,E,Y> Z Z Z

5. d-separation continued…X Y
Question: is E X
P(X)
0.5 1 ? E Y
P(Y|E)
0.5 1
It depends…on the CPTs X E
P(E|X)
0.01 1
0.99
This is why d-separation => independence but not the converse… E Y
P(Y|E)
0.01 1
0.99

6. d-separation continued…X Y
Question: is E
Yes! ?

7. d-separation continued…X Y
Question: is E
Yes! ?
Bayes rule
Fancier version
of B.R.
From previous slide…

8. An aside: computations with Bayes NetsX Y
Question: what is ? E
Main point: inference has no preferred direction in a Bayes net

9. d-separation X E Y Z Z Z

10. d-separation X E Y ? Z ? Z Z

11. d-separation continued…X Y
Question: is E
Yes! ?

12. d-separation continued…X Y
Question: is E No ? E
P(E)
0.5 1

13. d-separation X E Y ? Z ? Z Z

14. d-separation continued…X Y
Question: is E
Yes! ?

15. d-separation continued…X Y
Question: is E ? X Y E
P(E|X,Y)
P(E,X,Y)
0.96
0.24 1
0.04
0.01

16. d-separation continued…X Y
Question: is E
No! ? X Y E
P(E|X,Y)
P(E,X,Y)
0.96
0.24 1
0.04
0.01

17. d-separation continued…X Y
Question: is E
No! ? X Y E
P(E|X,Y)
P(E,X,Y)
0.96
0.24 1
0.04
0.01

18. “Explaining away” NO X Y E YES This is “explaining away”:
E is common symptom of two causes, X and Y
After observing E=1, both X and Y become more probable
After observing E=1 and X=1, Y becomes less probable
since X alone is enough to “explain” E

19. “Explaining away” and common-sense
Historical note:
Classical logic is monotonic: the more you know, the more you deduce.
“Common-sense” reasoning is not monotonic
birds fly
but, not after being cooked for 20min/lb at 350o F
This led to numerous “non-monotonic logics” for AI
This examples shows that Bayes nets are not monotonic
If P(Y|E) is “your belief” in Y after observing E,
and P(Y|X,E) is “your belief” in Y after observing E,X
your belief in Y decreases after you discover X

20. A special case: linear chain networksX1 Xn
... Xj
...
d-separation
“backward”
“forward”

21. A special case: linear chain networksX1 Xn
... Xj
...
Fwd:
Recursion!
(fwd)
CPT entry

22. A special case: linear chain networksX1 Xn
... Xj
...
Back:
Chain rule
CPT
Recursion backward

23. A special case: linear chain networksX1 Xn
... Xj
...
“backward”
“forward”
Instead of recursion:
iteratively compute P(Xj|x1) from P(Xj-1|x1) – the forward probabilities
iteratively compute P(xn|Xj) from P(xn|Xj+1) – the backward probabilities
can view the forward computations as passing a “message” forward
and vice versa

24. Linear-chain message passing
How long is this line? 1 1 Xj
... Xn X1 E+ E-
j-1
n-j 2 … … ? ? …
How many ahead? …
How many behind?

25. Linear-chain message passing
P(Xj|E) = P(X|E+)P(X|E-) … true by d-separation Xj
... Xn X1 E+ E-
Pass forward: P(Xj|E+)…computed from P(Xj-1|E+) and CPT for Xj
Pass backward: P(Xj|E-)…computed from P(Xj+1|E-) and CPT for Xj+1

26. Inference in Bayes nets
General problem: given evidence E1,…,Ek compute P(X|E1,..,Ek) for any X
Big assumption: graph is “polytree”
<=1 undirected path between any nodes X,Y
Notation: U1 U2 X Z1 Z2 Y1 Y2

27. Inference in Bayes nets: P(X|E)

28. Inference in Bayes nets: P(X|E+)
d-sep – write as product
d-sep.

29. Inference in Bayes nets: P(X|E+)
d-sep – write as product
d-sep.
CPT table lookup
So far: simple way of propogating “belief due to causal evidence” up the tree
Recursive call to P(.|E+)

30. Inference in Bayes nets: P(E-|X)
recursion

31. Inference in Bayes nets: P(E-|X)
recursion

32. Inference in Bayes nets: P(E-|X)

33. Inference in Bayes nets: P(E-|X)
Recursive call to P(.|E)
CPT
Recursive call to P(E-|.)
where

34. More on Message Passing
We reduced P(X|E) to product of two recursively calculated parts:
P(X=x|E+)
i.e., CPT for X and product of “forward” messages from parents
P(E-|X=x)
i.e., combination of “backward” messages from parents, CPTs, and P(Z|EZ\Yk), a simpler instance of P(X|E)
This can also be implemented by message-passing (belief propogation)

35. Learning for Bayes nets
Input:
Sample of the joint:
Graph structure of the variables
for I=1,…,N, you know Xi and parents(Xi)
Output:
Estimated CPTs A B B
P(B) 1
0.33 2 3 C D
Method (discrete variables):
Estimate each CPT independently
Use a MLE or MAP A B C
P(C|A,B) 1 2
0.5 3
1.0 … E …

36. Learning for Bayes nets
Method (discrete variables):
Estimate each CPT independently
Use a MLE or MAP
MLE: A B B
P(B) 1
0.33 2 3 C D A B C
P(C|A,B) 1 2
0.5 3
1.0 … E …

37. MAP estimates The beta distribution:
“pseudo-data”: like hallucinating a few heads and a few tails

38. The Dirichlet distribution:
MAP estimates
The Dirichlet distribution:
“pseudo-data”: like hallucinating αi examples of X=i for each value of i

39. Learning for Bayes nets
Method (discrete variables):
Estimate each CPT independently
Use a MLE or MAP
MAP: A B B
P(B) 1
0.33 2 3 C D A B C
P(C|A,B) 1 2
0.5 3
1.0 … E …

40. Additional reading
ftp://ftp.research.microsoft.com/pub/tr/tr pdf A Tutorial on Learning With Bayesian Networks, Heckerman