1. Probabilistic Reasoning Course 8
Artificial Intelligence
Probabilistic Reasoning
Course 8

2. Logic Propositional Logic Predicate Logic
The study of statements and their connectivity structure
Predicate Logic
The study of individuals and their properties

3. Uncertainty
Agents almost never have access to the hole truth about their environment
Agents have to act under uncertainty
Rational decisions depends
Relative importance of several goals
Likelihood
Degree to which they will be achived
Example toofache

4. Uncertainty Complete description of environment problems
Laziness – too much work to describe all facts
Theoretical ignorance – no complete theory of the domain (medicine)
Practical Ignorance – not all situations are analyzed
Degree of belief of affirmations
Probability theory
Dempster-Shafer theory
Fuzzy logic
Truth maintenance systems
Nonmonotonic reasoning

5. Probabilities
Provides a way of summarizing the uncertainty that cones from laziness and ignorance

6. Uncertainty
Abduction is a reasoning process that tries to form plausible explanations for abnormal observations
Abduction is distinctly different from deduction and induction
Abduction is inherently uncertain
Uncertainty is an important issue in abductive reasoning
Definition (Encyclopedia Britannica): reasoning that derives an explanatory hypothesis from a given set of facts
The inference result is a hypothesis that, if true, could explain the occurrence of the given facts

7. Comparing abduction, deduction, and induction
Deduction: major premise: All balls in the box are black
minor premise: These balls are from the box
conclusion: These balls are black
Abduction: rule: All balls in the box are black
observation: These balls are black
explanation: These balls are from the box
Induction: case: These balls are from the box
hypothesized rule: All ball in the box are black
A => B A B
A => B B
Possibly A
Whenever A then B
Possibly
A => B
Deduction reasons from causes to effects
Abduction reasons from effects to causes
Induction reasons from specific cases to general rules

8. Uncertainty Uncertain inputs Uncertain knowledge Uncertain outputs
Missing data
Noisy data
Uncertain knowledge
Multiple causes lead to multiple effects
Incomplete enumeration of conditions or effects
Incomplete knowledge of causality in the domain
Probabilistic/stochastic effects
Uncertain outputs
Abduction and induction are inherently uncertain
Default reasoning, even in deductive fashion, is uncertain
Incomplete deductive inference may be uncertain
Probabilistic reasoning only gives probabilistic results (summarizes uncertainty from various sources)

9. Probabilities
Kolmogorov showed that three simple axioms lead to the rules of probability theory
De Finetti, Cox, and Carnap have also provided compelling arguments for these axioms
All probabilities are between 0 and 1:
0 ≤ P(a) ≤ 1
Valid propositions (tautologies) have probability 1, and unsatisfiable propositions have probability 0:
P(true) = 1 ; P(false) = 0
The probability of a disjunction is given by:
P(a  b) = P(a) + P(b) – P(a  b) a
ab b

10. Paradox
“Monty Hall” problem

11. Paradox

12. Conditional Probabilities
P(A|B) – the part of the environment in which B is true and A is also true
Probability of A, conditioned by B
D = headache, P(D)=1/10
G = flue, P(G)=1/40
P(D|G) = ½
If someone has flue, the probability of also having headache is 50%
P(D|G)=P(D^G)/P(G)

13. Bayesian Theorem P(A|B) = P(A^B) / P(B) P(A^B) = P(A|B) * P(B)
P(A^B) = P(B|A) * P(A)
=>P(B|A) = P(A|B) * P(B) / P(A)

14. Example Diagnosis Known probabilities
Meningitis: P(M)=0.002%
Stiffed neck: P(N)=5%
Meningitis causes in half of cases stiffed neck: P(N|M)=50%
If a patient has stiffed neck, what is the probability to have meningitis?
P(M|N) = P(G|M)*P(M)/P(G) = 0.02%

15. Independence
Variables A and B are independent if any of the following hold:
P(A,B) = P(A) P(B)
P(A | B) = P(A)
P(B | A) = P(B)
This says that knowing the outcome of A does not tell me anything new about the outcome of B.

16. Independence How is independence useful?
Suppose you have n coin flips and you want to calculate the joint distribution P(C1, …, Cn)
If the coin flips are not independent, you need 2n values in the table
If the coin flips are independent, then

17. Conditional Independence
Variables A and B are conditionally independent given C if any of the following hold:
P(A, B | C) = P(A | C) P(B | C)
P(A | B, C) = P(A | C)
P(B | A, C) = P(B | C)
Knowing C tells me everything about B. I don’t gain anything by knowing A (either because A doesn’t influence B or because knowing C provides all the information knowing A would give)

18. Bayesian Network
Directed acyclic graphs (DAF) where the nodes represent random variables and directed edges capture their dependence
Each node in the graph is a random variable
A node X is a parent of another node Y if there is an arrow from node X to node Y eg. A is a parent of B A B C D
Informally, an arrow from node X to node Y means X has a direct influence on Y

19. Bayesian Networks Two important properties:
Encodes the conditional independence relationships between the variables in the graph structure
Is a compact representation of the joint probability distribution over the variables

20. Conditional Independence
The Markov condition: given its parents (P1, P2), a node (X) is conditionally independent of its non-descendants (ND1, ND2) P1 P2
ND1 X
ND2 C1 C2

21. The Joint Probability Distribution
Due to the Markov condition, we can compute the joint probability distribution over all the variables X1, …, Xn in the Bayesian net using the formula:
Where Parents(Xi) means the values of the Parents of the node Xi with respect to the graph

22. Example
Variables:
weather can have three states: sunny, cloudy, or rainy
grass can be wet or dry
sprinkler can be on or off
Causal links in this world:
If it is rainy, then it will make the grass wet directly.
But if it is sunny for a long time, that too can make the grass wet, indirectly, by causing us to turn on the sprinkler.

23. Links in bayesian networks
The links may form loops, but they may not form cycles

24. Dempster-Shafer Theory
It is based on the work of Dempster who attempted to model uncertainty by a range of probabilities rather than a single probabilistic number.
Belife [bel,pl] interval
Bel = belief bel(A)
Pl = plausibility pl(A)=1-bel(¬A)
If no information about A and ¬A is present the belief interval is [0,1]
(not 0.5)
In the knowledge acquisition process the interval becomes smaller
bel(A)<=P(A)<=pl(A)

25. Example Sue tells the true 90% of the times
P(M)=0.9, P(¬M)=0.1
Bill tells the true 80% of the times
P(B)=0.8, P(¬B)=0.2
Case 1: Sue and Bill tell George that his car has been stolen
Probability that non of them to be trustable is 0.02
Probability that at least one them is trustable is =0.98
Belief interval is [0.98,1]

26. Example 2
Case 2. Sue says that the car is stolen and Bill says that not
Can be both affirmations trustable (contradictions)
Sue is trustable (the car was stolen)
0.9 x (1-0.8) = 0.18
Bill is trustable (the car was not stolen)
0.8 x (1-0.9) = 0.08
Both of them are not trustable (no concrete information)
(1-0.8) x (1-0.9) = 0.02
All non null probabilities
=0.28
Belief that the car was stolen
0.18/0.28=0.64
Belief that the car was not stolen
0.o8/0.28=0.29
The belief interval that the car was not stolen
[0.64,1-0.29]=[0.64,0.71]

27. Next Course Laboratory Planning and Reasoning in real world
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