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Bayesian Belief Network

Bayesian Belief Network

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Bayesian Belief Network

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  1. Bayesian Belief Network

  2. The decomposition of large probabilistic domains into weakly connected subsets via conditional independence is one of the most important developments in the recent history of AI • This can work well, even the assumption is not true!

  3. vNB • Naive Bayes assumption: • which gives

  4. Bayesian networks • Conditional Independence • Inference in Bayesian Networks • Irrelevant variables • Constructing Bayesian Networks • Aprendizagem Redes Bayesianas • Examples - Exercisos

  5. Naive Bayes assumption of conditional independence too restrictive • But it's intractable without some such assumptions... • Bayesian Belief networks describe conditional independence among subsets of variables • allows combining prior knowledge about (in)dependencies amongvariables with observed training data

  6. Bayesian networks • A simple, graphical notation for conditional independence assertions and hence for compact specification of full joint distributions • Syntax: • a set of nodes, one per variable • a directed, acyclic graph (link ≈ "directly influences") • a conditional distribution for each node given its parents: P (Xi | Parents (Xi)) • In the simplest case, conditional distribution represented as a conditional probability table (CPT) giving the distribution over Xi for each combination of parent values

  7. Y Z P Bayesian Networks • Bayesian belief network allows a subset of the variables conditionally independent • A graphical model of causal relationships • Represents dependency among the variables • Gives a specification of joint probability distribution • Nodes: random variables • Links: dependency • X,Y are the parents of Z, and Y is the parent of P • No dependency between Z and P • Has no loops or cycles X

  8. Conditional Independence • Once we know that the patient has cavity we do not expect the probability of the probe catching to depend on the presence of toothache • Independence between a and b

  9. Example • Topology of network encodes conditional independence assertions: • Weather is independent of the other variables • Toothache and Catch are conditionally independent given Cavity

  10. Bayesian Belief Network: An Example Family History Smoker (FH, ~S) (~FH, S) (~FH, ~S) (FH, S) LC 0.7 0.8 0.5 0.1 LungCancer Emphysema ~LC 0.3 0.2 0.5 0.9 The conditional probability table for the variable LungCancer: Shows the conditional probability for each possible combination of its parents PositiveXRay Dyspnea Bayesian Belief Networks

  11. Example • I'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 off • An earthquake can set the alarm off • The alarm can cause Mary to call • The alarm can cause John to call

  12. Belief Networks Earthquake Burglary P(B) 0.001 P(E) 0.002 Burg. Earth. P(A) t t .95 t f .94 f t .29 f f .001 Alarm JohnCalls MaryCalls A P(M) t .7 f .01 A P(J) t .90 f .05

  13. Full Joint Distribution

  14. Compactness • A 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)

  15. Inference in Bayesian Networks • How can one infer the (probabilities of) values of one or more network variables, given observed values of others? • Bayes net contains all information needed for this inference • If only one variable with unknown value, easy to infer it • In general case, problem is NP hard

  16. Example • In the burglary network, we migth observe the event in which JohnCalls=true and MarryCalls=true • We could ask for the probability that the burglary has occured • P(Burglary|JohnCalls=ture,MarryCalls=true)

  17. Remember - Joint distribution

  18. Normalization

  19. Normalization • X is the query variable • E evidence variable • Y remaining unobservable variable • Summation over all possible y (all possible values of the unobservable varables Y)

  20. P(Burglary|JohnCalls=ture,MarryCalls=true) • The hidden variables of the query are Earthquake and Alarm • For Burglary=true in the Bayesain network

  21. To compute we had to add four terms, each computed by multipling five numbers • In the worst case, where we have to sum out almost all variables, the complexity of the network with n Boolean variables is O(n2n)

  22. P(b) is constant and can be moved out, P(e) term can be moved outside summation a • JohnCalls=true and MarryCalls=true, the probability that the burglary has occured is aboud 28%

  23. Computation for Burglary=true

  24. Variable elimination algorithm • Eliminate repeated calculation • Dynamic programming

  25. Irrelevant variables • (X query variable, E evidence variables)

  26. Complexity of exact inference • The burglary network belongs to a family of networks in which there is at most one undiracted path between tow nodes in the network • These are called singly connected networks or polytrees • The time and space complexity of exact inference in polytrees is linear in the size of network • Size is defined by the number of CPT entries • If the number of parents of each node is bounded by a constant, then the complexity will be also linear in the number of nodes

  27. For multiply connected networks variable elimination can have exponentional time and space complexity

  28. Constructing Bayesian Networks • A Bayesian network is a correct representation of the domain only if each node is conditionally independent of its predecessors in the ordering, given its parents P(MarryCalls|JohnCalls,Alarm,Eathquake,Bulgary)=P(MaryCalls|Alarm)

  29. Conditional Independence relations in Bayesian networks • The toopological semantics is given either of the spqcifications of DESCENDANTS or MARKOV BLANKET

  30. Local semantics

  31. Example • JohnCalls is indipendent of Burglary and Earthquake given the value of Alarm

  32. Example • Burglary is indipendent of JohnCalls and MaryCalls given Alarm and Earthquake

  33. Constructing Bayesian networks • 1. Choose an ordering of variables X1, … ,Xn • 2. For i = 1 to n • add Xi to the network • select parents from X1, … ,Xi-1 such that P (Xi | Parents(Xi)) = P (Xi | X1, ... Xi-1) This choice of parents guarantees: P (X1, … ,Xn) = πni =1P (Xi | X1, … , Xi-1) (chain rule) = πni =1P (Xi | Parents(Xi)) (by construction)

  34. The compactness of Bayesian networks is an example of locally structured systems • Each subcomponent interacts directly with only bounded number of other components • Constructing Bayesian networks is difficult • Each variable should be directly influenced by only a few others • The network topology reflects thes direct influences

  35. Example • Suppose we choose the ordering M, J, A, B, E P(J | M) = P(J)?

  36. 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)? No P(B | A, J, M) = P(B | A)? P(B | A, J, M) = P(B)?

  37. 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)? No P(B | A, J, M) = P(B | A)? Yes P(B | A, J, M) = P(B)? No P(E | B, A ,J, M) = P(E | A)? P(E | B, A, J, M) = P(E | A, B)?

  38. 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)? No P(B | A, J, M) = P(B | A)? Yes P(B | A, J, M) = P(B)? No P(E | B, A ,J, M) = P(E | A)? No P(E | B, A, J, M) = P(E | A, B)? Yes

  39. 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 • Some links represent tenous relationship that require difficult and unnatural probability judgment, such the probability of Earthquake given Burglary and Alarm

  40. Aprendizagem Redes Bayesianas • Como preencher as entradas numa Tabela de Probabilidade Condicional 1º Caso: Se a estrutura da rede bayesiana fôr conhecida, e todas as variavéis podem ser observadas do conjunto de treino. Então: Entrada (i,j) = utilizando os valores observados no conjunto de treino 2º Caso: Se a estrutura da rede bayesiana fôr conhecida, e algumas das variavéis não podem ser observadas no conjunto de treino. Então utiliza-se método do algoritmo do gradiente ascendente

  41. Exemplo 1º caso Family History Smoker • Person FH S E LC PXRay D • P1 Sim Sim Não Sim + Sim • P2 Sim Não Não Sim - Sim • P3 Sim Não Sim Não + Não • P4 Não Sim Sim Sim - Sim • P5 Não Sim Não Não + Não • P6 Sim Sim ? ? ? ? LungCancer Emphysema (FH, S) (FH, ~S) (~FH, S) (~FH, ~S) P(LC = Sim \ FH=Sim, S=Sim) =0.5 LC … 0.5 … … ~LC … … … …

  42. Exemplo 2º caso • Person FH S E LC PXRay D • P1 --- Sim --- Sim + Sim • P2 --- Não --- Sim - Sim • P3 --- Não --- Não + Não • P4 --- Sim --- Sim - Sim • P5 --- Sim --- Não + Não • P6 Sim Sim ? ? ? ? • Suppose structure known, variables partially observable • Similar to training neural network with hidden units • In fact, can learn network conditional probability tables using gradient ascent

  43. Summary • Bayesian networks provide a natural representation for (causally induced) conditional independence • Topology + CPTs = compact representation of joint distribution • Generally easy for domain experts to construct