地球的臭氧层

nullLocal structures; Causal Independence, Context-sepcific independanceLocal structures; Causal Independence, Context-sepcific independanceCOMPSCI 276 Fall 2007Reducing parameters of familiesReducing parameters of familiesDeterminizm Causal independence Context-specific independanc Continunous variablesnullCausal IndependenceCausal IndependenceEvent X has two possible causes: A,B. It is hard to elicit P(X|A,B) but it is easy to determine P(X|A) and P(X|B). Example: several diseases causes a symptom. Effect of A on X is independent from the effect of B on X Causal Independence, using canonical models: Noisy-O, Noisy AND, noisy-maxABXBinary ORBinary ORABXABP(X=0|A,B)001P(X=1|A,B)0010110011101Noisy-ORNoisy-OR“noise” is associated with each edge described by noise parameter   [0,1] : Let q b=0.2, qa =0.1 P(x=0|a,b)= (1-a) (1-b) P(x=1|a,b)=1-(1-a) (1-b) ABXABP(X=0|A,B)001P(X=1|A,B)0ab010.10.9100.20.8110.020.98qi=P(X=0|A_i=1,…else =0)Noisy-OR with LeakNoisy-OR with LeakUse leak probability 0  [0,1] when both parents are false: Let a =0.2, b =0.1, 0 = 0.0001 P(x=0|a,b)= (1-0)(1-a)a(1-b)b P(x=0|a,b)=1-(1-0)(1-a)a(1-b)b ABXABP(X=0|A,B)000.9999P(X=1|A,B)0.0001ab010.10.9100.20.8110.020.98Formal Definition for Noisy-Or Formal Definition for Noisy-Or Definition 1 Let Y be a binary-valued random variable with k binary-valued parents X1,…,Xk. The CPT P(Y|X1,…Xk) is a noisy-or if there are k+1 noise parameters 0, 1,… k such that P(y=0| X1,…,Xk) = (1- 0)  i,Xi=1 (1- i)Closed Form Bel(X) - 1Closed Form Bel(X) - 1Given: noisy-or CPT P(x|u) noise parameters i Tu = {i: Ui = 1} Define: qi = 1 - I, Then:q_i is the probability that the inhibitor for u_i is active while theClosed Form Bel(X) - 2Closed Form Bel(X) - 2Using Iterative Belief Propagation:Set piix = pix (uk=1). Then we can show that:Closed Form Bel(X) - 2Closed Form Bel(X) - 2Using Iterative Belief Propagation:Set piix = pix (uk=1). Then we can show that:Causal Influence DefinedCausal Influence DefinedDefinition 2 Let Y be a random variable with k parents X1,…,Xk. The CPT P(Y|X1,…Xk) exhibits independence of causal influence (ICI) if it is described via a network fragment of the structure shown in on the left where CPT of Z is a deterministic functions f. ZYX1X1X1Z0Z1Z2ZknullnullnullnullnullContext Specific IndependenceContext Specific IndependenceWhen there is conditional independence in some specific variable assignmentnullnullnullnullThe impact during inferenceThe impact during inferenceCausal independence in polytrees is linear during inference Causal independence in general can sometime be exploited but not always CSI can be exploited by using operation (product and summation) over trees.Representing CSIRepresenting CSIUsing decision trees Using decision graphsA student’s exampleA student’s exampleIntelligenceDifficultyGradeLetterSATJobApplyTree CPDTree CPDIf the student does not apply, SAT and L are irrelevant Tree-CPD for jobASL(0.8,0.2)(0.9,0.1)(0.4,0.6)(0.1,0.9)s1a0a1s0l1l0Definition of CPD-treeDefinition of CPD-treeA CPD-tree of a CPD P(Z|pa_Z) is a tree whose leaves are labeled by P(Z) and internal nodes correspond to parents branching over their values.Captures irrelevant variables Captures irrelevant variables CL2(0.1,0.9)l21c1c2l20L1(0.8,0.2)(0.3,0.7)l11l10(0.9,0.1)Multiplexer CPDMultiplexer CPDA CPD P(Y|A,Z1,Z2,…,Zk) is a multiplexer iff Val(A)=1,2,…k, and P(Y|A,Z1,…Zk)=Z_aRule-based representationRule-based representationA CPD-tree that correponds to rules.ABC(0.3,0.7)(0.4,0.6)(0.1,0.9)b1a0a1b0c1c0CB(0.3,0.7)(0.5,0.5)(0.2,0.8)c1c0b1b0Continuous VariablesContinuous VariablesICS 275b 2002Gaussian DistributionGaussian DistributionnullnullMultivariate GaussianMultivariate GaussianDefinition: Let X1,…,Xn. Be a set of random variables. A multivariate Gaussian distribution over X1,…,Xn is a parameterized by an n-dimensional mean vector  and an n x n positive definitive covariance matrix . It defines a joint density via:Linear Gaussian DistributionLinear Gaussian DistributionDefinition: Let Y be a continuous node with continuous parents X1,…,Xk. We say that Y has a linear Gaussian model if it can be described using parameters 0, …,k and 2 such that: P(y| x1,…,xk)=N (0 + 1x1 +…,kxk ; 2 )nullnullLinear Gaussian NetworkLinear Gaussian NetworkDefinition Linear Gaussian Bayesian network is a Bayesian network all of whose variables are continuous and where all of the CPTs are linear Gaussians. Linear Gaussian BN  Multivariate Gaussian =>Linear Gaussian BN has a compact representationHybrid ModelsHybrid ModelsContinuous Node, Discrete Parents (CLG) Define density function for each instantiation of parents Discrete Node, Continuous Parents Treshold SigmoidContinuous Node, Discrete ParentsContinuous Node, Discrete ParentsDefinition: Let X be a continuous node, and let U={U1,U2,…,Un} be its discrete parents and Y={Y1,Y2,…,Yk} be its continuous parents. We say that X has a conditional linear Gaussian (CLG) CPT if, for every value uD(U), we have a a set of (k+1) coefficients au,0, au,1, …, au,k+1 and a variance u2 such that:CLG NetworkCLG NetworkDefinition: A Bayesian network is called a CLG network if every discrete node has only discrete parents, and every continuous node has a CLG CPT.Discrete Node, Continuous Parents Threshold ModelDiscrete Node, Continuous Parents Threshold ModelDiscrete Node, Continuous Parents Sigmoid Binomial LogitDiscrete Node, Continuous Parents Sigmoid Binomial LogitDefinition: Let Y be a binary-valued random variable with k continuous-valued parents X1,…Xk. The CPT P(Y|X1…Xk) is a linear sigmoid (also called binomial logit) if there are (k+1) weights w0,w1,…,wk such that:nullReferencesReferencesJudea Pearl “Probabilistic Reasoning in Inteeligent Systems”, section 4.3 Nir Friedman, Daphne Koller “Bayesian Network and Beyond”