Cognitive Systems II Contact: david@elsweiler.co.uk

Last Week's 1 minute paper●We have 2 boxes (red and blue) each containing anumber of oranges and apples.●Without looking they pick a box at random and get anapple.●What is the probability that it was picked out of the redbox?

BurglaryP(B)0.001EarthquakeP(E)0.002B EP(A| B,E)AlarmT T 0.95T F 0.94F T 0.29F F 0.001John CallsAP(J|A)T 0.9F 0.05Mary CallsAP(M|A)T 0.7F 0.01

Inferences with Belief NetworksDiagnostic Inferences (from effects to causes)– What is the probability of a burglary given weknow that John calls → P (B | JC) = 0.016Causal Inferences (from causes to effects)– Given a burglary what is :P (JC | B) = 0.86 andP (MC |B) = 0.67

Inferences SummaryQEQEEQEQEDiagnosticCausalInter-CausalMixed

Algorithm for Belief Networks thatare Poly-treesAlgorithm for making inferences where the beliefnetwork is a poly-tree– There is at most one undirected path betweennodesWill work with all four kinds of inferences outlinedabove

U 1U mx

U 1U mxY 1Y n

U 1U mxZ 1jZ njY 1Y n

+EU x1U mxZ 1j Z njY 1Y n

Answering Queries+EU x1U mxZ 1j Z njY 1Y nE x-

Answering Queries●X is the query variable● U = U 1… U mare parents of node X● Y = Y 1… Y nare children of node X●E is a set of evidence variables●The aim is to compute P (X | E)

Answering Queries● E X + is the causal support for X●The evidence variables “above” X that areconnected to X through its parents● E X - is the evidential support for X●The evidence variables “below” X that areconnected to X through its children● E ui \ Xrefers to all the evidence connected to nodeU iexcept via the path from X● E yi \ X+refers to all the evidence connected tonode Yi through its parents except X

E ui \ XXXXXU 1U mXXXXx

E yi \ X+XXxXXXZ 1jXY 1XY nXZ njX

Computing P(X|E)Since X d-separates from we canuse conditional independence to simplify thefirst term in the numerator

Answering Queries+EU x1U mxZ 1j Z njY 1Y nE x-

Computing P(X|E)Since X d-separates from we canuse conditional independence to simplify thefirst term in the numeratorWe can treat the denominator as a constant

ComputingWe want to consider all the possible combinationsof parents of X and examine how likely they aregivenLet be the vector of parents andlet u be the assignment of values to them

Computingd-separates X fromsimplifies toso the first term

XXXU 1 U mE x+XXXXXx

ComputingWe can simplify the second term by noting●Given X D-separates from the otherparents●Joint probability of independent variables issimply their product

U 1 U mE x+x

Computingd-separates X fromsimplifies toso the first termWe can simplify the second term by noting●Given X D-separates from the otherparents●Joint probability of independent variables issimply their product

ComputingThe last term can be simplified by splittingintoand noting thatis D-separated by X from the rest of

XXXU 1 U mE x+XXXXXx

Computing●is a simple lookup in the conditionalprobability table for X●is a recursive (smaller) subproblem

Computing P(X|E)Since X d-separates from we canuse conditional independence to simplify thefirst term in the numeratorWe can treat the denominator as a constant

ComputingLet Z ibe the parents of Y iother than X, and z ibethe assignment of values to the parentsThe evidence of each Y iis conditionallyindependent of the others given X

XXxXXXZ 1jXY 1XY nXZ njXX X X X

ComputingAveraging over and gives

ComputingBreakingcomponents ofinto the two independentand

XXxXZ 1jXZ 1jXY 1X

ComputingBreakingcomponents ofinto the two independentand

Computingis independent of X and given andis independent of X and

XXxZ 1jY 1Z 1jXXXY 1X

Computingis independent of X and given andis independent of X and

ComputingApply Bayes' rule to :

ComputingRewriting the conjunction of and gives:

Computing= because Z and X are d-separated. Alsois a constant

xZ 1jY 1Y 1

Computing= because Z and X are d-separated. Alsois a constant

ComputingThe Zijs are all independent of each otherWe can combine thes into a single

xZ 11Y 1Z 1jZ 12Z 1jY 1

Computing●is a recursive instance of●is a cond prob table entry for Y i●is a recursive instance of

Multiply-connected Belief NetworksCloudyP(C) = 0.5CP(S)CP(R)T 0.1F 0.5SprinklerRainT 0.8F 0.2Wet GrassS R P(W)T T 0.99T F 0.9F T 0.9F F 0.00

Inference with Multiply-connectedBelief NetworksClustering methods– Transform the net into a probabilisticallyequivalent (but topologically different) Poly-Treeby merging problem nodesConditioning methods– Instantiate variables to definite values and thenevaluate a poly-tree for each possibleinstantiation

Inference with Multiply-connectedBelief NetworksStochastic Simulation Methods– Use the network to generate a large number ofconcrete models of the domain that areconsistent with the network distribution– Gives an approximation of the exact evaluation