Probability Reasoning - Arizona State University
Probability Reasoning
Bayesian Nets
Will Cushing, Nov. 3rd
Instead of inventing a new logic, use probability
So, for example:
“Normally birds fly”
=> P(fly | bird) = 0.98
Judea Pearl published a book that laid the foundation for probability in AI.
After the advent of learning, probability became more important; previously, there was not much serious attention due to the difficulty of acquiring the probabilities themselves...in the example above, should it be 0.97, 0.96? 0.99?
One idea is a joint probability distribution:
Given several variables, specify probabilities of the combinations of values of the variables.
a b c P(a,b,c,)
T T T 0.1
T T F 0.2
T F T 0.1
T F F 0.3
F T T 0.05
F T F 0.05
F F T 0.2
F F F 0.0
Total= 1.0
P(A^B) = P(A) * P(B|A)
Or = P(B) * P(A|B)
P(A,B) = P(A^B) – just syntax
P(B|A) = P(A,B) / P(A)
P(b=t | a=t) = p(a=t,b=t) / p(a=t)
= (P(a=t,b=t,c=t) + P(a=t,b=t,c=f)) / (sum over b,c (P(a=t,b,c))) = 3/7
i.e. – sum over unknown values
Conditional probability is similar to implication:
P(A|B) if B => A is equal to 1
Another useful formula:
P(B,A) = [ P(B) / P(A) ] * P(A|B)
(i.e., convert from P(A|B) to P(B|A))
For example, B is a disease, and A is a symptom – measure P(A), P(B), and P(A|B), and then when A is observed, one can make an informed guess at B, using P(B|A)
This provides the early impetus in probability AI research – medical reasoning.
Representing joint probability distribution explicitly is far too cumbersome. So, need an efficient representation.
Knowledge is often given (communicated) in the form of relations between variables; knowledge is like P(B|A) = 0.7 – not in explicit joint probability distributions.
This is where Bayesian nets come in.
P(A1,A2,A3,...,An) =
P(A1|A2..An) P(A2|A3..An) P(A3|A4..An) ... P(An-1|An) P(An)
P(fly|bird,black) = P(fly|bird)
P(fly,bird,black) / P(bird,black) = P(fly,bird) / P(bird)
P(fly|bird,black) = P(fly|bird)
If
P(fly|black) = P(fly)
Or
P(black|bird) = 1.0
One can draw a graph (net) to representation relations between variables.
A markovian graph has the property that children of a parent are independent of one another.
More formally; we represent relations as a DAG (directed acyclic graph).
The markovian property is that the set S={yx | e=yx is an edge of the DAG }
Has the property that
P(x|S,z) = P(x|S) if z is not a descendant of x
Suppose we can form a markovian DAG from A1..An Let A1..An be taken in the order of a topological sort. Then:
The P(A1..An) can be simplified to:
P(A1..An) = product of (P(Ai|Par(Ai)))
Need probability of all nodes with no parents. For each node with a parent, we need only the joint distribution table for P(node | Par(node)).
To compute P(A2,A5|A1,A7)
= P(A1,A2,A5,A7) / P(A1,A7)
= sum over (P(A1=t,A2=t,...,A5=t,A7=t)) / sum over (P(A1=t,...,A7=t,...))
Example with tampering, fires, alarms, smoke, leaving, and reports
Fire,tampering => alarm
TT 0.5
TF 0.99
FT 0.85
FF 0.0001
Alarm => leaving
T 0.88
F 0.001
Fire => smoke
T 0.9
F 0.01
Leaving=>report
T 0.75
F 0.01
=>tamper
0.02
=>fire
0.01
P(leaving|alarm) = P(leaving | alarm, fire)
But
P(leaving|alarm) != P(leaving | alarm, report)
Descendant, vs., non-descendant
P(tampering|fire) = P(tampering)
But, the following is not true
P(tampering|alarm,fire) = P(tampering|fire)
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