IAI Chapter 7 Reasoning under Uncertainty
Questions about the lecture 'Introduction to Artificial Intelligence' of the RWTH Aachen Chapter 7 Reasoning under Uncertainty
Questions about the lecture 'Introduction to Artificial Intelligence' of the RWTH Aachen Chapter 7 Reasoning under Uncertainty
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Cartes-fiches
What does the statement P(A) mean for probability theory?
How likely is event A
What is the definition of an event A for probability theory?
A is a boolean combination of atomic propositions
What are the components of an event A for probability theory? [3]
1. X=n is also an atomic proposition with P(X=n) also written as P(n)
2. Random variable X with probability distribution of all sequences
3. Discrete element n
What are the components of decision theory? [2]
1. Probability theory and 2. utility theory
What are the axioms for probability theory? [3]
1. P is a real number between 0 and 1
2. P(true)=1 and P(false)=0
3. P(A \lor B) = P(A) + P(B) – P(A \land B)
What is the statement of de Finetti (1930s)?
“Someone who does not follow the axioms of probability theory will lose his or her bet!”
What is the idea of conditional probabilities?
Specify an evidence E representing background knowledge
What is the product rule of conditional probabilities? [2]
1. P(A \land B) = P(A|B)*P(B) = P(B|A)*P(A)
2. P(A|B) = [P(B|A)*P(A)]/P(B) // Bayes rule with 1/P(B) seen as normalization coefficient (set later)
What are the rules for multiple evidences of conditional probabilities? [3]
1. P(A|B,C) = [P(B|A,C)*P(A|C)]/P(B|C) // Switch A and B
2. P(A|B,C) = [P(B,C|A)*P(A)]/P(B,C) // Switch A with B,C but 2n possibilites for P(B,C|A)
2. P(A|B,C) = alpha*P(A)*P(B|A)*(C|A) // General
What is the Bayesian update of conditional probabilities? [2]
1. P(A|B,C) = P(A|B) with A,C are independent given C
2. P(A|B,C) = [P(C|A)*P(A|B)]/P(C|B) with B,C independent given A
What is the definition of an atomic event of conditional probabilities? [2]
1. Assignment of values to all random variables
2. Sum of all assignment is 1
What are the characteristics of a belief network? [4]
1. Acyclic graph
2. Labels are called conditional probability table (CPT)
3. Each node must be conditionally independent of its predecessors given its parents
4. NP-hard for multiply connected networks and linear for singly connected networks
What are the components of a belief network? [2]
1. Nodes are labeled with P(X|Parents(X)) with random variable X
2. Edges indicates parental random variables
What is the definition of a joint distribution (JD)? [2]
1. P(X1=x1 \land … \land Xn=xn) = P(x1,… ,xn)
2. P(x1,… ,xn) = P(xn|x1,… ,xn-1)*P(x1,… ,xn-1) = Prod_i P(xi|x1,… ,xi-1)
How is a belief network constructed? [3]
1. Choose relevant random variables
2. Choose ordering of those variables
3. For all variables create a node and add edges for the JD
What is the definition of d-Separation for belief networks?
Set of nodes E d-separates set of nodes X and Y
What are the types of blocking for d-Separation for belief networks? [3]
1. Z has one in and one out going edges
2. Z has two out going edges
3. Z has two in going edges and out going edges are not in E
What is the theorem of Pearl regarding d-Separation for belief networks?
“If E exists for X and Y → X is independent of Y given E”
What are the characteristics of d-Separation for belief networks? [3]
1. Time complexity is polynomial
2. Method is incomplete
3. Sufficient for certain inference algorithms
Name four types of inference in belief networks? [4]
1. Diagnostic if first Q then E
2. Causal if first E then Q
3. Intercausal or explaining away if E and Q into next node
4. Mixed if first E then Q then E
When can two nodes be clustered in a belief network? [2]
1. If two nodes are on same height in network
2. Cover all cases tt tf ft ff with // E.g. P(S)*(1-P(R)) for tf
