1Eric Xing 1Machine LearningMachine Learning1010--701/15701/15--781, Fall 2006781, Fall 2006Graphical Models IIGraphical Models IIInferenceInferenceEric XingEric XingLecture 13, October 26, 2006Reading: Chap. 8, C.B bookVisit to AsiaTuberculosisTuberculosisor CancerXRay ResultDyspneaBronchitisLung CancerSmokingX1X2X3X4X5X6X7X8Visit to AsiaTuberculosisTuberculosisor CancerXRay ResultDyspneaBronchitisLung CancerSmokingX1X2X3X4X5X6X7X8X1X2X3X4X5X6X7X8Eric Xing 2),,,,|(),,,|(),,|(),|()|()(=),,,,,( 543216432153214213121654321XXXXXXPXXXXXPXXXXPXXXPXXPXPXXXXXXPX1X2X3X4X5X6p(X6| X2, X5)p(X1)p(X5| X4)p(X4| X1)p(X2| X1)p(X3| X2)P(X1, X2, X3, X4, X5, X6)= P(X1) P(X2| X1) P(X3| X2) P(X4| X1) P(X5| X4) P(X6| X2, X5)Recap of Basic Prob. Conceptsz Joint probability dist. on multiple variables:z If Xi's are independent: (P(Xi|·)= P(Xi))z If Xi's are conditionally independent (as described by a GM), the joint can be factored to simpler products, e.g., ∏==iiXPXPXPXPXPXPXPXXXXXXP)()()()()()()(),,,,,( 6543216543212Eric Xing 3Eric Xing 4Structure: an undirected graph• Meaning: a node is conditionally independent of every other node in the network given its Directed neighbors• Local contingency functions (potentials) and the cliques in the graph completely determine the joint dist. •Give correlations between variables, but no explicit way to generate samplesXY1Y2Markov Random Fields3Eric Xing 5Representationz Defn: an undirected graphical model represents a distribution P(X1 ,…,Xn) defined by an undirected graph H, and a set of positive potential functions ycassociated with cliques of H, s.t.where Z is known as the partition function:z Also known as Markov Random Fields, Markov networks …z The potential function can be understood as an contingency function of its arguments assigning "pre-probabilistic" score of their joint configuration. ∏∈=CcccnZxxP )(),,( xψ11K∑∏∈=nxxCcccZ,,)(K1xψEric Xing 6Density estimationRegressionClassificationParametric and nonparametric methodsLinear, conditional mixture, nonparametricGenerative and discriminative approachQXQXXYm,sXXGMs are your old friends4Eric Xing 7(Picture by ZoubinGhahramani and Sam Roweis)An (incomplete) genealogy of graphical modelsEric Xing 8Probabilistic Inferencez We now have compact representations of probability distributions: Graphical Modelsz A GM Mdescribes a unique probability distribution Pz How do we answer queries about P?z We use inference as a name for the process of computing answers to such queries5Eric Xing 9z Most of the queries one may ask involve evidencez Evidence eis an assignment of values to a set E variables in the domainz Without loss of generality E = { Xk+1, …, Xn}z Simplest query: compute probability of evidencez this is often referred to as computing the likelihood of e∑∑=1)(1xxkk,e,x,xPP(e) KKQuery 1: LikelihoodEric Xing 10z Often we are interested in the conditional probability distribution of a variable given the evidencez this is the a posteriori belief in X, given evidence ez We usually query a subset Yof all domain variables X={Y,Z}and "don't care" about the remaining, Z:z the process of summing out the "don't care" variables zis called marginalization, and the resulting P(y|e) is called a marginal prob.∑===xx,e)P(XP(X,e)P(e)P(X,e)e)P(X |∑==ze)zP(Y,Ze)P(Y ||Query 2: Conditional Probability6Eric Xing 11A CB?A CB?Applications of a posteriori Beliefz Prediction: what is the probability of an outcome given the starting conditionz the query node is a descendent of the evidencez Diagnosis: what is the probability of disease/fault given symptomsz the query node an ancestor of the evidencez Learning under partial observationz fill in the unobserved values under an "EM" setting (more later)z The directionality of information flow between variables is not restricted by the directionality of the edges in a GMz probabilistic inference can combine evidence form all parts of the networkEric Xing 12z In this query we want to find the most probable joint assignment (MPA) for some variables of interestz Such reasoning is usually performed under some given evidence e, and ignoring (the values of) other variables z :z this is the maximum a posteriori configuration of y.∑==zyyezyPeyPeY )|,(maxarg)|(maxarg)|(MPAQuery 3: Most Probable Assignment7Eric Xing 13x y P(x,y)00 0.3501 0.0510 0.311 0.3Applications of MPAz Classification z find most likely label, given the evidencez Explanation z what is the most likely scenario, given the evidenceCautionary note:z The MPA of a variable depends on its "context"---the set of variables been jointly queriedz Example:z MPA of X ?z MPA of (X, Y) ?Eric Xing 14Thm:Computing P(X= x|e) in a GM is NP-hardz Hardness does not mean we cannot solve inferencez It implies that we cannot find a general procedure that works efficiently for arbitrary GMsz For particular families of GMs, we can have provably efficient proceduresComplexity of Inference8Eric Xing 15√√√√Approaches to inferencez Exact inference algorithmsz The elimination algorithmz The junction tree algorithms (but will not cover in detail here)z Approximate inference techniquesz Stochastic simulation / sampling methodsz Markov chain Monte Carlo methodsz Variational algorithms (will be covered in advanced ML courses)Eric Xing 16z A signal transduction pathway:z Query: P(e)z By chain decomposition, we getA B CED∑∑∑∑∑∑∑∑==dcbadcbadePcdPbcPabPaPe)P(a,b,c,d,eP)|()|()|()|()()(a naïve summation needs to enumerate over an exponential number of termsWhat is the likelihood that protein E is active?Marginalization and Elimination9Eric Xing 17A B CED∑∑∑ ∑∑∑∑∑==dcb adcbaabPaPdePcdPbcPdePcdPbcPabPaPeP)|()()|()|()|()|()|()|()|()()(Elimination on Chainsz Rearranging terms ...Eric Xing 18z Now we can perform innermost summationz This summation "eliminates" one variable from our summation argument at a "local cost".A B CEDX∑∑∑∑∑∑∑==dcbdcb abpdePcdPbcPabPaPdePcdPbcPeP)()|()|()|()|()()|()|()|()(Elimination on Chains10Eric Xing 19A B CED∑∑∑∑ ∑∑∑∑===dcdc bdcbcpdePcdPbpbcPdePcdPbpdePcdPbcPeP)()|()|()()|()|()|()()|()|()|()(XXElimination in Chainsz Rearranging and then summing again, we getEric Xing 20z Eliminate nodes one by one all the way to the end, we getz Complexity:z Each step costs O(|Val(Xi)|*|Val(Xi+1)|) operations: O(kn2)z Compare to naïve evaluation that sums over joint values of n-1 variables O(nk)A B CED∑=ddpdePeP )()|()(XXXXElimination in
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