1. CSCI 121 Special Topics: Bayesian Networks Lecture #2: Bayes Nets

2. Earthquake Burglary Alarm JohnCalls MaryCalls P(B=F) P(B=T) .999 .001
P(E=F) P(E=T)
Earthquake
Burglary
B E P(A=F) P(A=T)
F F
T F
F T
T T
Alarm
JohnCalls
MaryCalls
A P(J=F) P(J=T) F T
A P(M=F) P(M=T) F T

3. Why Use Bayes Nets? T T T T T Pm
Why not just use joint prob. (can be used to answer any query)?
B E A J M P
F F F F F P1
...
T T T T T Pm
Table size is exponential (m=2n entries for n vars.)
Network (graph) is sparse; only encodes local dependencies.
n2k entries, where k = avg. # inputs to node.

4. Types of Inferences / Queries
Diagnostic: P(B | J) = 0.02
Causal: P(J | B) = 0.85
Intercausal: P(B | J & M & ~E) = 0.34
Explaining Away: A university accepts a student if s/he is a good scholar or good athlete. If an accepted student is a good scholar, is s/he a good athlete? (Berkson's Paradox / selection bias)

5. Answering Queries E.g., compute P(B|A)
Product Rule tells us that P(B|A) = P(B&A) / P(A)
Problem: the only values we can obtain directly from the probability tables are the priors and some joint conditionals. For P(B&A) and P(A), the values are confounded with other variables (E).
Solution: we can marginalize (sum out) the other variables to focus on the ones we care about
First, we must convert the tables to joint probabilities of all relevant variables:

6. Answering Queries: Marginalization
B E A Prob
T T T .001*.002*.95 =
T T F .001*.002*.05 =
T F T .001*.998*.94 =
T F F .001*.998*.06 =
F T T .999*.002*.29 =
F T F .999*.002*.71 =
F F T .999*.998*.001 =
F F F .999*.998*.999 =
P(B)
.001
P(E)
.002
B E P(A)
T T .95
T F .94
F T .29
F F .001

7. Answering Queries: Marginalization
Now we get P(A) by summing over rows where
A = True:
B E A Prob
T T T
T T F
T F T
T F F
F T T
F T F
F F T
F F F
P(A) =

8. Answering Queries: Marginalization
B E A Prob
T T T
T T F
T F T
T F F
F T T
F T F
F F T
F F F
Next, we need to compute P(B&A)
Again, we marginalize:
So P(B|A) = / =
P(B&A) =