1. QUIZ!! T/F: Probability Tables PT(X) do not always sum to one. FALSE
T/F: Conditional Probability Tables CPT(X|Y=y) always sum to one. TRUE
T/F: Conditional Probability Tables CPT(X|Y) always sum to one. FALSE
T/F: Marginal distr. can be computed from joint distributions. TRUE
T/F: P(X|Y)P(Y)=P(X,Y)=P(Y|X)P(X). TRUE
T/F: A probabilistic model is a joint distribution over a set of r.v.’s. TRUE
T/F: Probabilistic inference = compute conditional probs. from joint. TRUE
What is the power of Bayes Rule?
Name the three steps of Inference by Enumeration.
Power of Bayes Rule --- compute one condition probability from the “reverse” conditional probability + relevant priors
Inference by Enumeration (on next slide!!).

2. Inference by Enumeration
General case:
Evidence variables:
Query* variable:
Hidden variables:
We want:
First, select the entries consistent with the evidence
Second, sum out H to get joint of Query and evidence:
Third, normalize the remaining entries to conditionalize
Obvious problems:
Worst-case time complexity O(dn)
Space complexity O(dn) to store the joint distribution
All variables
d^n if you have n variables, each of which can have d different data values.
Hidden variables: --- those for which you don’
* Works fine with multiple query variables, too

3. CSE 511a: Artificial Intelligence Spring 2012
Lecture 14: Bayes’Nets
/ Graphical Models
10/25/2010
Kilian Q. Weinberger
Many slides adapted from Dan Klein – UC Berkeley

4. Last Lecture ... Probabilistic Models Inference by Enumeration
Inference with Bayes Rule
Works well for small problems,
but what if we have many random variables?

5. This Lecture: Bayes Nets

6. Probabilistic Models
Distribution over T,W
A probabilistic model is a joint distribution over a set of random variables
Probabilistic models:
(Random) variables with domains Assignments are called outcomes
Joint distributions: say whether assignments (outcomes) are likely
Normalized: sum to 1.0
Ideally: only certain variables directly interact T W P
hot
sun
0.4
rain
0.1
cold
0.2
0.3

7. Probabilistic Models
Models describe how (a portion of) the world works
Models are always simplifications
May not account for every variable
May not account for all interactions between variables
“All models are wrong; but some are useful.” – George E. P. Box
What do we do with probabilistic models?
We (or our agents) need to reason about unknown variables, given evidence
Example: explanation (diagnostic reasoning)
Example: prediction (causal reasoning)
Example: value of information
George Box: All models are wrong; some of them are useful

8. Probabilistic Models
A probabilistic model is a joint distribution over a set of variables
Given a joint distribution, we can reason about unobserved variables given observations (evidence)
General form of a query:
This kind of posterior distribution is also called the belief function of an agent which uses this model
Stuff you care about
Stuff you already know

9. Model for Ghostbusters
Reminder: ghost is hidden, sensors are noisy
T: Top sensor is red B: Bottom sensor is red G: Ghost is in the top
Queries:
P( +g) = ?? P( +g | +t) = ?? P( +g | +t, -b) = ??
Problem: joint
distribution too
large / complex
Joint Distribution T B G
P(T,B,G) +t +b +g
0.16 g b
0.24
0.04
t t
0.06

10. Independence
Two variables are independent in a joint distribution if and only if:
Says the joint distribution factors into a product of two simple ones
Usually variables aren’t independent!
Can use independence as a modeling assumption
Independence can be a simplifying assumption
Empirical joint distributions: at best “close” to independent
What could we assume for {Weather, Traffic, Cavity}?
Independence is like something from CSPs: what?

11. Example: Independence
N fair, independent coin flips: h
0.5 t h
0.5 t h
0.5 t
What about UNFAIR, independent coin flips?

12. Example: Independence?T P
warm
0.5
cold T W P
warm
sun
0.4
rain
0.1
cold
0.2
0.3 T W P
warm
sun
0.3
rain
0.2
cold
Start with the Joint Table, and here are the marginalized tables for Temperature and Weather.
Is this Joint Table independent? W P
sun
0.6
rain
0.4

13. Conditional Independence
P(Toothache, Cavity, Catch)
If I have a cavity, the probability that the probe catches in it doesn't depend on whether I have a toothache:
P(+catch | +toothache, +cavity) = P(+catch | +cavity)
The same independence holds if I don’t have a cavity:
P(+catch | +toothache, cavity) = P(+catch| cavity)
Catch is conditionally independent of Toothache given Cavity:
P(Catch | Toothache, Cavity) = P(Catch | Cavity)
Equivalent statements:
P(Toothache | Catch , Cavity) = P(Toothache | Cavity)
P(Toothache, Catch | Cavity) = P(Toothache | Cavity) P(Catch | Cavity)
One can be derived from the other easily

14. Conditional Independence
Unconditional (absolute) independence very rare
Conditional independence is our most basic and robust form of knowledge about uncertain environments:
What about this domain:
Traffic
Umbrella
Raining
Traffic Umbrella Raining. On the next slide!
Fire  Smoke  Alarm

15. The Chain Rule Trivial decomposition:
With assumption of conditional independence:
Bayes’ nets / graphical models help us express conditional independence assumptions

16. Ghostbusters Chain Rule
Each sensor depends only on where the ghost is
That means, the two sensors are conditionally independent, given the ghost position
T: Top square is red B: Bottom square is red G: Ghost is in the top
Givens:
P( +g ) = 0.5
P( +t | +g ) = 0.8 P( +t | g ) = 0.4 P( +b | +g ) = 0.4 P( +b | g ) = 0.8
P(T,B,G) = P(G) P(T|G) P(B|G) T B G
P(T,B,G) +t +b +g
0.16 g b
0.24
0.04
t t
0.06

17. Bayes’ Nets: Big Picture
Two problems with using full joint distribution tables as our probabilistic models:
Unless there are only a few variables, the joint is WAY too big to represent explicitly
Hard to learn (estimate) anything empirically about more than a few variables at a time
Bayes’ nets: a technique for describing complex joint distributions (models) using simple, local distributions (conditional probabilities)
More properly called graphical models
We describe how variables locally interact
Local interactions chain together to give global, indirect interactions
For about 10 min, we’ll be vague about how these interactions are specified

18. Example Bayes’ Net: Insurance

19. Graphical Model Notation
Nodes: variables (with domains)
Can be assigned (observed) or unassigned (unobserved)
Arcs: interactions
Similar to CSP constraints
Indicate “direct influence” between variables
Formally: encode conditional independence (more later)
For now: imagine that arrows mean direct causation (in general, they don’t!)

20. Example Bayes’ Net: Car

21. Example: Coin Flips N independent coin flips
No interactions between variables: absolute independence X1 X2 Xn
A graph can have no edges. A bayes net graph of independent random variables also has no edges.

22. Example: Traffic Variables: R Model 1: independence
R: It rains
T: There is traffic
Model 1: independence
Model 2: rain causes traffic
Why is an agent using model 2 better? R T
More accurate model of the dependence

23. Example: Traffic II Let’s build a causal graphical model Variables
T: Traffic
R: It rains
L: Low pressure
D: Roof drips
B: Ballgame
C: Cavity

24. Example: Alarm Network
Variables
B: Burglary
A: Alarm goes off
M: Mary calls
J: John calls
E: Earthquake!
Story --- you live in LA, where there are both burglaries and earthquakes.
You have an alarm that is sensitive to vibration of your front door.
The alarm company autimatlcally notifiies you and john and mary as a precaution if the alarm goes off.
John and Mary like you, but don’t know each other, so they may call to check on you.

25. Does smoking cause cancer?
In 1950s, suspicion:
Smoking
Cancer
Correlation discovered by Ernst Wynder, at WashU 1948.

26. Does smoking cause cancer?
Explanation of the
Tobacco Research Council:
Unknown
Gene
Smoking
Cancer
P(cancer | smoking, gene)=P(cancer | gene)
Correlation discovered by Ernst Wynder, at WashU 1948.
Link between smoking and cancer finally established in 1998.
(22 Million deaths due to tobacco in those 50 years.)

27. Global Warming Human Activity Green House Gases Climate Change Model:
Explains data
Makes verifiable predictions

28. Global Warming Human Activity Unknown Cause X Green House Gases
Climate
Change
Model:
Undefined (mystery) variables
Does not explain data
Makes no predictions

29. Bayes’ Net Semantics Let’s formalize the semantics of a Bayes’ net
A set of nodes, one per variable (A’s and X’s)
A directed, acyclic graph
A conditional distribution for each node
A collection of distributions over X, one for each combination of parents’ values
CPT: conditional probability table
Description of a noisy “causal” process A1 An X
A Bayes net = Topology (graph) + Local Conditional Probabilities

30. Probabilities in BNs Bayes’ nets implicitly encode joint distributions
As a product of local conditional distributions
To see what probability a BN gives to a full assignment, multiply all the relevant conditionals together:
Example:
This lets us reconstruct any entry of the full joint
Not every BN can represent every joint distribution
The topology enforces certain conditional independencies
Equation assumes that I have specific labels for *EVERY* node (I have lower case x’s), and I am just trying to compute overall probability of that assignment.
What example have we already seen where a BN cannot represent a joint distribution (indendence early on of weather/temperature).

31. Example: Coin Flips X1 X2 Xn h 0.5 t h 0.5 t h 0.5 t
Only distributions whose variables are absolutely independent can be represented by a Bayes’ net with no arcs.

32. Example: Traffic R T +r 1/4 r 3/4 +r +t 3/4 t 1/4 r +t 1/2 t
P(+r) * P(~t | +r)
= ¼ * ¼ = 1/16 r +t
1/2 t

33. Example: Alarm Network
P(E) +e
0.002 e
0.998 B
P(B) +b
0.001 b
0.999
Burglary
Earthqk
Alarm B E A
P(A|B,E) +b +e +a
0.95 a
0.05 e
0.94
0.06 b
0.29
0.71
0.001
0.999
John calls
Mary calls
This let’s up compute:
P(b,e,a,j,m) WITHOUT have the full joint distribution table (which would be how big?!) what is probably that you are popular, but nothing bad happened? P(j,m, and not everything else?)
P(~b) * P(~e) * P(~a | ~b, ~e) * P(j | ~a) * p(m | ~a) A J
P(J|A) +a +j
0.9 j
0.1 a
0.05
0.95 A M
P(M|A) +a +m
0.7 m
0.3 a
0.01
0.99

34. Bayes’ Nets So far: how a Bayes’ net encodes a joint distribution
Next: how to answer queries about that distribution
Key idea: conditional independence
Last class: assembled BNs using an intuitive notion of conditional independence as causality
Today: formalize these ideas
Main goal: answer queries about conditional independence and influence
After that: how to answer numerical queries (inference)

35. Example: Traffic Causal direction R T r 1/4 r 3/4 r t 3/16 t 1/16 r
6/16 r t
3/4 t
1/4 T r t
1/2 t

36. Example: Reverse Traffic
Reverse causality? T t
9/16 t
7/16 r t
3/16 t
1/16 r
6/16 t r
1/3 r
2/3 R t r
1/7 r
6/7

37. Causality? When Bayes’ nets reflect the true causal patterns:
Often simpler (nodes have fewer parents)
Often easier to think about
Often easier to elicit from experts
BNs need not actually be causal
Sometimes no causal net exists over the domain (especially if variables are missing)
E.g. consider the variables Traffic and Drips
End up with arrows that reflect correlation, not causation
What do the arrows really mean?
Topology may happen to encode causal structure
Topology really encodes conditional independence

38. Example: Naïve Bayes
Imagine we have one cause y and several effects x:
This is a naïve Bayes model
We’ll use these for classification later

39. Example: Alarm Network

40. The Chain Rule
Can always factor any joint distribution as an incremental product of conditional distributions
Why is the chain rule true?
This actually claims nothing…
What are the sizes of the tables we supply?

41. Example: Alarm Network
Burglary
Earthquake
Alarm
John calls
Mary calls