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CS498-EA Reasoning in AI Lecture #5 - PowerPoint PPT Presentation

CS498-EA Reasoning in AI Lecture #5. Instructor: Eyal Amir Fall Semester 2009. Last Time. Propositional Logic Inference in different representations CNF: SAT hard; small representation sometimes DNF: SAT easy; large representation OBDDs: SAT easy; large representation sometimes

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CS498-EAReasoning in AILecture #5

Instructor: Eyal Amir

Fall Semester 2009

• Propositional Logic

• Inference in different representations

• CNF: SAT hard; small representation sometimes

• DNF: SAT easy; large representation

• OBDDs: SAT easy; large representation sometimes

• NNF: SAT hard; fewest large representations

• Applications:

• Circuit and program verification; computational biology

• Prove or disprove:

• Every CNF representation with n variables of propositional formulas takes O(2n) space to represent some propositional theories on n variables (hint: how many non-equivalent theories of n variables are there?)

• Probabilistic graphical models

• Treewidth methods:

• Variable elimination

• Clique tree algorithm

• Applications du jour: Sensor Networks

• A sample space Omega (O) is a set of outcomes of a random experiment

• A probability P is a function from a sigma-field (e.g., all measurable subsets) A on O (the events) to [0,1].

• A random variable X is a function X:OR such that for all B Borel set in R, X-1(B) is in A.

• Two variables X and Y are independent if

• P(X = x|Y = y) = P(X = x) for all values x,y

• That is, learning the values of Y does not change prediction of X

• If X and Y are independent then

• P(X,Y) = P(X|Y)P(Y) = P(X)P(Y)

• In general, if X1,…,Xp are independent, then P(X1,…,Xp)= P(X1)...P(Xp)

• Requires O(n) parameters

• Unfortunately, most of random variables of interest are not independent of each other

• A more suitable notion is that of conditional independence

• Two variables X and Y are conditionally independent given Z if

• P(X = x|Y = y,Z=z) = P(X = x|Z=z) for all values x,y,z

• That is, learning the values of Y does not change prediction of X once we know the value of Z

• notation: I ( X , Y | Z )

Homer

Lisa

Maggie

Bart

Example: Family trees

Noisy stochastic process:

Example: Pedigree

• A node represents an individual’sgenotype

• Modeling assumptions:

• Ancestors can effect descendants' genotype only by passing genetic materials through intermediate generations

Y1

Y2

X

Non-descendent

Markov Assumption

Ancestor

Parent

• We now make this independence assumption more precise for directed acyclic graphs (DAGs)

• Each random variable X, is independent of its non-descendents, given its parents Pa(X)

• Formally,I (X, NonDesc(X) | Pa(X))

Non-descendent

Descendent

Earthquake

Alarm

Call

Markov Assumption Example

• In this example:

• I ( E, B )

• I ( B, {E, R} )

• I ( R, {A, B, C} | E )

• I ( A, R | B,E )

• I ( C, {B, E, R} | A)

Y

X

Y

I-Maps

• A DAG G is an I-Map of a distribution P if all Markov assumptions implied by G are satisfied by P

(Assuming G and P both use the same set of random variables)

Examples:

Y

Factorization

• Given that G is an I-Map of P, can we simplify the representation of P?

• Example:

• Since I(X,Y), we have that P(X|Y) = P(X)

• Applying the chain ruleP(X,Y) = P(X|Y) P(Y) = P(X) P(Y)

• Thus, we have a simpler representation of P(X,Y)