probabilistic reasoning; network-based reasoning

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Probabilistic Reasoning; Network-based reasoning. Set 7 ICS 179, Spring 2010. = A. = B. = A. = C. Propositional Reasoning. Example: party problem. If Alex goes, then Becky goes: If Chris goes, then Alex goes: Question: - PowerPoint PPT Presentation

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  • Probabilistic Reasoning;Network-based reasoning

    Set 7 ICS 179, Spring 2010

  • Chavurah 5/8/2010Propositional Reasoning

    If Alex goes, then Becky goes:

    If Chris goes, then Alex goes:

    Question: Is it possible that Chris goes to the party but Becky does not?Example: party problem

    Chavurah 5/8/2010

  • Chavurah 5/8/2010Probabilistic ReasoningParty example: the weather effectAlex is-likely-to-go in bad weatherChris rarely-goes in bad weatherBecky is indifferent but unpredictable

    Questions:Given bad weather, which group of individuals is most likely to show up at the party? What is the probability that Chris goes to the party but Becky does not?

    P(W,A,C,B) = P(B|W) P(C|W) P(A|W) P(W)

    P(A,C,B|W=bad) = 0.9 0.1 0.5

    WAP(A|W)good0.01good1.99bad0.1bad1.9

    Chavurah 5/8/2010

  • Chavurah 5/8/2010

    Mixed Probabilistic and Deterministic networksQuery:Is it likely that Chris goes to the party if Becky does not but the weather is bad?PNCNSemantics?

    Algorithms?

    Chavurah 5/8/2010

  • *The problemQ: Does T fly?P(Q)?True propositionsUncertain propositionsLogic?....but how we handle exceptionsProbability: astronomical

    All men are mortalTAll penguins are birdsTSocrates is a manMen are kindp1Birds flyp2T looks like a penguinTurn key > car startsP_n

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  • Alpha and beta are events

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  • Burglary is independent of Earthquake

  • Earthquake is independent of burglary

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  • * Bayesian Networks: Representation = P(S) P(C|S) P(B|S) P(X|C,S) P(D|C,B)lung CancerSmokingX-rayBronchitisDyspnoeaP(S, C, B, X, D)

  • Bayesian networks

    Chapter 14 , Russel and NorvigSection 1 2

  • OutlineSyntaxSemantics

  • ExampleTopology of network encodes conditional independence assertions:

    Weather is independent of the other variablesToothache and Catch are conditionally independent given Cavity

  • ExampleI'm at work, neighbor John calls to say my alarm is ringing, but neighbor Mary doesn't call. Sometimes it's set off by minor earthquakes. Is there a burglar?

    Variables: Burglary, Earthquake, Alarm, JohnCalls, MaryCalls

    Network topology reflects "causal" knowledge:A burglar can set the alarm offAn earthquake can set the alarm offThe alarm can cause Mary to callThe alarm can cause John to call

  • Example contd.

  • CompactnessA CPT for Boolean Xi with k Boolean parents has 2k rows for the combinations of parent values

    Each row requires one number p for Xi = true (the number for Xi = false is just 1-p)

    If each variable has no more than k parents, the complete network requires O(n 2k) numbers

    I.e., grows linearly with n, vs. O(2n) for the full joint distribution

    For burglary net, 1 + 1 + 4 + 2 + 2 = 10 numbers (vs. 25-1 = 31)

  • SemanticsThe full joint distribution is defined as the product of the local conditional distributions:

    P (X1, ,Xn) = i = 1 P (Xi | Parents(Xi))

    e.g., P(j m a b e)

    = P (j | a) P (m | a) P (a | b, e) P (b) P (e)

    n

  • Constructing Bayesian networks1. Choose an ordering of variables X1, ,Xn2. For i = 1 to nadd Xi to the networkselect parents from X1, ,Xi-1 such thatP (Xi | Parents(Xi)) = P (Xi | X1, ... Xi-1)

    This choice of parents guarantees:

    P (X1, ,Xn) = i =1 P (Xi | X1, , Xi-1)(chain rule)= i =1P (Xi | Parents(Xi))(by construction)

  • Suppose we choose the ordering M, J, A, B, E

    P(J | M) = P(J)?

    Example

  • Suppose we choose the ordering M, J, A, B, E

    P(J | M) = P(J)?NoP(A | J, M) = P(A | J)? P(A | J, M) = P(A)?

    Example

  • Suppose we choose the ordering M, J, A, B, E

    P(J | M) = P(J)?NoP(A | J, M) = P(A | J)? P(A | J, M) = P(A)? NoP(B | A, J, M) = P(B | A)? P(B | A, J, M) = P(B)?Example

  • Suppose we choose the ordering M, J, A, B, E

    P(J | M) = P(J)?NoP(A | J, M) = P(A | J)? P(A | J, M) = P(A)? NoP(B | A, J, M) = P(B | A)? YesP(B | A, J, M) = P(B)? NoP(E | B, A ,J, M) = P(E | A)?P(E | B, A, J, M) = P(E | A, B)?Example

  • Suppose we choose the ordering M, J, A, B, E

    P(J | M) = P(J)?No P(A | J, M) = P(A | J)? P(A | J, M) = P(A)? NoP(B | A, J, M) = P(B | A)? YesP(B | A, J, M) = P(B)? NoP(E | B, A ,J, M) = P(E | A)? NoP(E | B, A, J, M) = P(E | A, B)? YesExample

  • Example contd.

    Deciding conditional independence is hard in noncausal directions

    (Causal models and conditional independence seem hardwired for humans!)

    Network is less compact: 1 + 2 + 4 + 2 + 4 = 13 numbers needed

  • SummaryBayesian networks provide a natural representation for (causally induced) conditional independence

    Topology + CPTs = compact representation of joint distribution

    Generally easy for domain experts to construct

    **To illustrate these concepts let me consider a simple exampleConsider a simple scenario involving people going to parties. Lets assume that we have three individuals,Alex, Becky and Chris. And we have some knowledge about their social relationships in regard to party-going.ShowQuery.How can we do this kind of simple reasoning by a computer? We have to first formalize the information in the sentencesand represent this sentences in some computer language .

    We can associate symbols with fragments in the sentence which can either be true or false(we call those propositions) and then describe the information In the sentence by logical rules (knowledge-representation).

    Subsequently we need to be able to derive answers to queries, and to do it automatically, namely, to engage in automated reasoning.

    Automated reasoning is the study of algorithms and computer programs that answer such queries .

    There are many disciplines that contributed, ideas, viewpoints and techniques to AI. Mathematical logic had a tremendous influence on AI in the early days and even today.In general to become a formal science AI relied heavily on three fundamental areas: logic, probability and computation.

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    *Lets go back now to the party example but have a factor we did not have before.Lets say that on the day of the party the weather was bad. How would this affect our party-goers?(Remember Drews holiday party and the weather that day? I think it affected some of us)Lets assumeQueryWell, in order to handle this we may use the field of probability theory and express the information in those sentences probabilistically.

    We can put this together into a probabilistic network that has a node for each proposition and directed arcs signifyingCausal/probabilistic relationships. Although we express information just between several variables, we can argue that the information we have is sufficient to capture all the probabilistic relationship that the simple domain has. In other words, assuming that the individuals in the new story are affected only by the weather, we dont need to specify P(a|b) because, when we know theWeather a and b are conditionally independent. The local collection of function represent a joint probability on all the variablesThat can be obtained by a product of all these functions.

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