1. Today

2. Next week

3. Marginalization ) , ( y x P P(x1) = sum sum ) , ( y x P
Suppose you have some joint probability,
Involving observations, y, and hidden states, x.
Suppose you’re at x1, and you want to find the marginal
probability there, given the observations. Normally,
you would have to compute:
For N other hidden nodes, each of M states, that will
take MN additions. ) , ( 3 2 1 y x P
P(x1) =
sum
sum ) , ( 3 2 1 y x P x x 2 3

4. Special case: Markov network
But suppose the joint probability has a special structure, shown by this Markov network:
Then this sum:
can be computed with N M2 additions, as follows… y1 x1 y2 x2 y3 x3 ) , (
sum 3 2 1 y x P
P(x1) =

5. Derivation of belief propagationy1 y2 y3 x1 x2 x3
P(x1) =
sum
sum P ( x , x , x , y , y , y ) 1 2 3 1 2 3 x x 2 3

6. The posterior factorizes
P(x1) =
sum
sum P ( x , x , x , y , y , y ) 1 2 3 1 2 3 x x 2 3 =
sum
sum F ( x , y ) 1 1 x x 2 3 F ( x , y ) Y ( x , x ) 2 2 1 2 F ( x , y ) Y ( x , x ) 3 3 2 3 x =
mean F ( x , y ) 1
MMSE 1 1 x y1 y2 y3 1
sum F ( x , y ) Y ( x , x ) 2 2 1 2 x 2 x1 x2 x3
sum F ( x , y ) Y ( x , x ) 3 3 2 3 x 3

7. Propagation rules P(x1) = sum sum P ( x , x , x , y , y , y ) = sum2 3 1 2 3 x x 2 3 =
sum
sum F ( x , y ) 1 1 x x 2 3 F ( x , y ) Y ( x , x ) 2 2 1 2 F ( x , y ) Y ( x , x ) 3 3 2 3
P(x1) = F ( x , y ) 1 1 y1 y2 y3
sum F ( x , y ) Y ( x , x ) 2 2 1 2 x 2 x1 x2 x3
sum F ( x , y ) Y ( x , x ) 3 3 2 3 x 3

8. Propagation rules P(x1) = F ( x , y ) sum F ( x , y ) Y ( x , x ) sum2 2 1 2 x 2
sum F ( x , y ) Y ( x , x ) 3 3 2 3 x 3 y1 y2 y3 x1 x2 x3

9. Belief and message update rulesj = i j i

10. Belief propagation updates= i j i ( ) = * .* .*

11. Simple example
For the 3-node example, worked out in detail, see Sections 2.0, 2.1 of:

12. Optimal solution in a chain or tree: Belief Propagation
“Do the right thing” Bayesian algorithm.
For Gaussian random variables over time: Kalman filter.
For hidden Markov models: forward/backward algorithm (and MAP variant is Viterbi).

13. Other loss functions
The above rules let you compute the marginal probability at a node. From that, you can compute the mean estimate.
But you can also use a related algorithm to compute the MAP estimate for x1.

14. MAP estimate for a chain or a treey1 y2 y3 x1 x2 x3

15. The posterior factorizesy1 y2 y3 x1 x2 x3

16. Propagation rules y1 y2 y3 x1 x2 x3

17. Using conditional probabilities instead of compatibility functionsx1 y2 x2 y3 x3
By Bayes rule

18. Writing it as a factorizationy1 x1 y2 x2 y3 x3
By the fact that conditioning on x1 makes y1 and x2, x3, y2, y3 independent

19. Writing it as a factorizationy1 x1 y2 x2 y3 x3
Now use Bayes rule (with x2) for the rightmost term.

20. Writing it as a factorizationy1 x1 y2 x2 y3 x3
From the Markov structure, conditioning on x1 and x2 is the same as conditioning on x2.

21. Writing it as a factorizationy1 x1 y2 x2 y3 x3
Conditioning on x2 makes y2 independent of x3 and y3.

22. Writing it as a factorizationy1 x1 y2 x2 y3 x3
The same operations, once more, with the far right term.

23. A toy problem
10 nodes. 2 states for each node. Local evidence as shown below.

27. Classic 1976 paper

28. Relaxation labelling

29. Belief propagation
Relaxation labelling

30. Yair’s motion example

31. Yair’s figure/ground example