1. Belief Networks
CS121 – Winter 2003
Belief Networks

2. Other Names Bayesian networks Probabilistic networks Causal networks
Belief Networks

3. Probabilistic Belief
There are several possible worlds that are indistinguishable to an agent given some prior evidence.
The agent believes that a logic sentence B is True with probability p and False with probability 1-p. B is called a belief
In the frequency interpretation of probabilities, this means that the agent believes that the fraction of possible worlds that satisfy B is p
The distribution (p,1-p) is the strength of B
Belief Networks

4. Problem
At a certain time t, the KB of an agent is some collection of beliefs
At time t the agent’s sensors make an observation that changes the strength of one of its beliefs
How should the agent update the strength of its other beliefs?
Belief Networks

5. Toothache Example
A certain dentist is only interested in two things about any patient, whether he has a toothache and whether he has a cavity
Over years of practice, she has constructed the following joint distribution:
Toothache
Toothache
Cavity
0.04
0.06
Cavity
0.01
0.89
Belief Networks

6. Toothache Example
Toothache
Toothache
Cavity
0.04
0.06
Cavity
0.01
0.89
Using the joint distribution, the dentist can compute the strength of any logic sentence built with the proposition Toothache and Cavity
In particular, this distribution implies that
the prior probability of Toothache is 0.05 P(T) = P((TC)v(TC)) = P(TC) + P(TC)
Belief Networks

7. New Evidence
Toothache
Toothache
Cavity
0.04
0.06
Cavity
0.01
0.89
She now makes an observation E that indicates that a specific patient x has high probability (0.8) of having a toothache, but is not directly related to whether he has a cavity
Belief Networks

8. Adjusting Joint Distribution
Toothache|E
Toothache|E
Cavity|E
0.04
0.06
Cavity|E
0.01
0.89
0.64
0.0126
0.16
0.1874
She now makes an observation E that indicates that a specific patient x has high probability (0.8) of having a toothache, but is not directly related to whether he has a cavity
She can use this additional information to create a joint distribution (specific for x) conditional to E, by keeping the same probability ratios between Cavity and Cavity
The probability of Cavity that was 0.1 is now (knowing E)
Belief Networks

9. Corresponding Calculus
Toothache
Toothache
Cavity
0.04
0.06
Cavity
0.01
0.89
P(C|T) = P(CT)/P(T) = 0.04/0.05
Belief Networks

10. Corresponding Calculus
Toothache|E
Toothache|E
Cavity|E
0.04
0.06
Cavity|E
0.01
0.89
P(C|T) = P(CT)/P(T) = 0.04/0.05
P(CT|E) = P(C|T,E) P(T|E) = P(C|T) P(T|E)
C and E are independent given T
Belief Networks

11. Corresponding Calculus
Toothache|E
Toothache|E
Cavity|E
0.04
0.06
Cavity|E
0.01
0.89
0.64
0.0126
0.16
0.1874
P(C|T) = P(CT)/P(T) = 0.04/0.05
P(CT|E) = P(C|T,E) P(T|E) = P(C|T) P(T|E) = (0.04/0.05) = 0.64
Belief Networks

12. Generalization n beliefs X1,…,Xn
The joint distribution can be used to update probabilities when new evidence arrives
But:
The joint distribution contains 2n probabilities
Useful independence is not made explicit
Belief Networks

13. Purpose of Belief Networks
Facilitate the description of a collection of beliefs by making explicit causality relations and conditional independence among beliefs
Provide a more efficient way (than by using joint distribution tables) to update belief strengths when new evidence is observed
Belief Networks

14. Alarm Example Five beliefs A: Alarm B: Burglary E: Earthquake
J: JohnCalls
M: MaryCalls
Belief Networks

15. A Simple Belief Network
Burglary
Earthquake
Alarm
MaryCalls
JohnCalls
causes
effects
Intuitive meaning of arrow
from x to y: “x has direct influence on y”
Directed acyclic graph (DAG)
Nodes are beliefs
Belief Networks

16. Assigning Probabilities to Roots
P(B)
0.001
P(E)
0.002
Burglary
Earthquake
Alarm
MaryCalls
JohnCalls
Belief Networks

17. Conditional Probability Tables
P(B)
0.001
P(E)
0.002
Burglary
Earthquake
Alarm
MaryCalls
JohnCalls B E
P(A|…)
TTFF
TFTF
Size of the CPT for a node with k parents: 2k
Belief Networks

18. Conditional Probability Tables
P(B)
0.001
P(E)
0.002
Burglary
Earthquake
Alarm
MaryCalls
JohnCalls B E
P(A|…)
TTFF
TFTF A
P(J|…) TF A
P(M|…) TF
Belief Networks

19. What the BN Means P(x1,x2,…,xn) = Pi=1,…,nP(xi|Parents(Xi)) Burglary
P(B)
0.001
P(E)
0.002
Burglary
Earthquake
Alarm
MaryCalls
JohnCalls B E
P(A|…)
TTFF
TFTF
P(x1,x2,…,xn) = Pi=1,…,nP(xi|Parents(Xi)) A
P(J|…) TF A
P(M|…) TF
Belief Networks

20. Calculation of Joint Probability
P(B)
0.001
P(E)
0.002
Burglary
Earthquake
Alarm
MaryCalls
JohnCalls B E
P(A|…)
TTFF
TFTF
P(JMABE) = P(J|A)P(M|A)P(A|B,E)P(B)P(E) = 0.9 x 0.7 x x x = A
P(J|…) TF A
P(M|…) TF
Belief Networks

21. What The BN Encodes
Burglary
Earthquake
Alarm
MaryCalls
JohnCalls
For example, John does not observe any burglaries directly
Each of the beliefs JohnCalls and MaryCalls is independent of Burglary and Earthquake given Alarm or Alarm
The beliefs JohnCalls and MaryCalls are independent given Alarm or Alarm
Belief Networks

22. What The BN Encodes
Burglary
Earthquake
Alarm
MaryCalls
JohnCalls
For instance, the reasons why John and Mary may not call if there is an alarm are unrelated
Note that these reasons could be other beliefs in the network. The probabilities summarize these non-explicit beliefs
Each of the beliefs JohnCalls and MaryCalls is independent of Burglary and Earthquake given Alarm or Alarm
The beliefs JohnCalls and MaryCalls are independent given Alarm or Alarm
Belief Networks

23. Distribution conditional to the observations made
Inference In BN
Set E of evidence variables that are observed with new probability distribution, e.g., {JohnCalls,MaryCalls}
Query variable X, e.g., Burglary, for which we would like to know the posterior probability distribution P(X|E)
Distribution conditional to the observations made
? ? ? ?
TFTF
TTFF
P(B|…) M J
Belief Networks

24. Inference Patterns Basic use of a BN: Given new
Burglary
Earthquake
Alarm
MaryCalls
JohnCalls
Diagnostic
Burglary
Earthquake
Alarm
MaryCalls
JohnCalls
Causal
Basic use of a BN: Given new
observations, compute the new strengths of some (or all) beliefs
Other use: Given the strength of
a belief, which observation should
we gather to make the greatest
change in this belief’s strength
Burglary
Earthquake
Alarm
MaryCalls
JohnCalls
Intercausal
Burglary
Earthquake
Alarm
MaryCalls
JohnCalls
Mixed
Belief Networks

25. Applications http://excalibur.brc.uconn.edu/~baynet/researchApps.html
Medical diagnosis, e.g., lymph-node deseases
Fraud/uncollectible debt detection
Troubleshooting of hardware/software systems
Belief Networks

26. Neural Networks
CS121 – Winter 2003
Belief Networks

27. Function-Learning Formulation
Goal function f
Training set: (xi, f(xi)), i = 1,…,n
Inductive inference: find a function h that fits the point well
Issues:
Representation
Incremental learning
Neural nets
Belief Networks

28. S Unit (Neuron) y = g(Si=1,…,n wi xi) xi x0 xn y wi g
g(u) = 1/[1 + exp(-a u)]
Belief Networks

29. Particular Case: Perceptron+ - S g xi x0 xn y wi
y = g(Si=1,…,n wi xi)
Belief Networks

30. Particular Case: Perceptron+ - ? S g xi x0 xn y wi
y = g(Si=1,…,n wi xi)
Belief Networks

31. S S Neural Network Network of interconnected neurons g xi x0 xn y g xiwi S g xi x0 xn y wi
Acyclic (feed-forward) vs. recurrent networks
Belief Networks

32. Two-Layer Feed-Forward Neural Network
Inputs
Hidden
layer
Output
Belief Networks

33. Backpropagation (Principle)
New example Yk = f(xk)
Error function:
E(w) = ||yk – Yk||2
wij(k) = wij(k-1) – e E/wij
Backprojection: Update the weights of the inputs to the last layer, then the weights of the inputs to the previous layer, etc.
Belief Networks

34. Issues How to choose the size and structure of networks?
If network is too large, risk of over-fitting (data caching)
If network is too small, representation may not be rich enough
Role of representation: e.g., learn the concept of an odd number
Belief Networks

35. What is AI?
Discipline that systematizes and automates intellectual tasks to create machines that:
Act like humans
Act rationally
Think like humans
Think rationally
Belief Networks

36. What Have We Learned? Collection of useful methods
Connection between fields
Relation between high-level (e.g., logic) and low-level (e.g., neural networks) representations
Impact of hardware
What is intelligence?
Our techniques are better than our understanding
Belief Networks