CS 188: Artificial Intelligence Spring 2007Example Bayes’ NetBayes’ Net SemanticsBuilding the (Entire) JointExample: Alarm NetworkBayes’ NetsReview: Useful RulesConditional IndependenceExample: IndependenceTopology Limits DistributionsExample: CoinsIndependence in a BNAnalyzing IndependenceCausal ChainsCommon CauseCommon EffectThe General CaseReachabilityReachability (the Bayes’ Ball)ExampleSlide 21Deriving Useful PropertiesSlide 23Slide 24Slide 25Slide 26Slide 27Causality?Example: TrafficExample: Reverse TrafficAlternate BNsSummaryCS 188: Artificial IntelligenceSpring 2007Lecture 13: Graphical models II2/27/2007Srini Narayanan – ICSI and UC BerkeleyExample Bayes’ NetBayes’ Net Semantics•A Bayes’ net:•A set of nodes, one per variable X•A directed, acyclic graph•A conditional distribution of each variable conditioned on its parents (the parameters )•Semantics:•A BN defines a joint probability distribution over its variables:A1XAnBuilding the (Entire) Joint•We can take a Bayes’ net and build any entry from the full joint distribution it encodes•Typically, there’s no reason to build ALL of it•We build what we need on the fly•To emphasize: every BN over a domain implicitly represents some joint distribution over that domain, but is specified by local probabilitiesExample: Alarm Network.001 * .002 * .05 * .05 * .01 = 5x10-11Bayes’ Nets•So far: how a Bayes’ net encodes a joint distribution•Next: how to answer qualitative queries about that distribution•Key idea: conditional independence•Last class: assembled BNs using an intuitive notion of conditional independence as causality•Today: formalize these ideas•Main goal: answer queries about conditional independence and influence•After that: how to answer numerical queries (inference)Review: Useful Rules•Conditional Probability (definition)•Chain Rule•Bayes RuleConditional Independence•Reminder: independence•X and Y are independent ( ) iff or equivalently, •X and Y are conditionally independent given Z ( ) iff or equivalently, •(Conditional) independence is a property of a distributionExample: Independence•For this graph, you can fiddle with (the CPTs) all you want, but you won’t be able to represent any distribution in which the flips are dependent!h 0.5t 0.5h 0.5t 0.5X1X2All distributionsTopology Limits Distributions•Given some graph topology G, only certain joint distributions can be encoded•The graph structure guarantees certain (conditional) independences•(There might be more independence)•Adding arcs increases the set of distributions, but has several costsXYZXYZXYZExample: Coins•Extra arcs don’t prevent representing independence, just allow non-independenceh 0.5t 0.5h 0.5t 0.5X1X2h 0.5t 0.5h | h 0.5t | h 0.5X1X2h | t 0.5t | t 0.5Independence in a BN•Important question about a BN:•Are two nodes independent given certain evidence?•If yes, can calculate using algebra (really tedious)•If no, can prove with a counter example•Example:•Question: are X and Z independent?•Answer: not necessarily, we’ve seen examples otherwise: low pressure causes rain which causes traffic.•X can influence Z, Z can influence X (via Y)•Addendum: they could be independent: how?X Y ZAnalyzing Independence•Arc between nodes ==> (poss) dependence•What if there is no direct arc?•To answer this question in general, we only need to understand 3-node graphs with 2 arcs•Cast of characters: X Y Z“Causal Chain”XYZ“Common Effect”XYZ“Common Cause”Causal Chains•This configuration is a “causal chain”•Is X independent of Z given Y?•Evidence along the chain “blocks” the influenceX Y ZYes!X: Low pressureY: RainZ: TrafficCommon Cause•Another basic configuration: two effects of the same cause•Are X and Z independent?•Consider the project due example•Are X and Z independent given Y?•Observing the cause blocks influence between effects.XYZYes!Y: Project dueX: Newsgroup busyZ: Lab fullCommon Effect•Last configuration: two causes of one effect (v-structures)•Are X and Z independent?•Yes: remember the ballgame and the rain causing traffic, no correlation?•Still need to prove they must be (homework)•Are X and Z independent given Y?•No: remember that seeing traffic put the rain and the ballgame in competition?•This is backwards from the other cases•Observing the effect enables influence between causes.XYZX: RainingZ: BallgameY: TrafficThe General Case•Any complex example can be analyzed using these three canonical cases•General question: in a given BN, are two (sets of) variables independent given some evidence?•Solution: graph search!Reachability•Recipe: shade evidence nodes•Attempt 1: if two nodes are connected by an undirected path not blocked by a shaded node, they are conditionally independent•Almost works, but not quite•Where does it break?•Answer: the v-structure at T doesn’t count as a link in a path unless shadedRTBDLT’Reachability (the Bayes’ Ball)•Correct algorithm:•Shade in evidence•Start at source node•Try to reach target by search•States: pair of (node X, previous state S)•Successor function:•X unobserved:•To any child•To any parent if coming from a child•X observed:•From parent to parent•If you can’t reach a node, it’s conditionally independent of the start node given evidenceSX XSSX XSExampleRTBDLT’YesYesYesExample•Variables:•R: Raining•T: Traffic•D: Roof drips•S: I’m sad•Questions:TSDRYesDeriving Useful Properties•Which nodes are (unconditionally) independent of S?Deriving Useful Properties•Which nodes are (unconditionally) independent of S?•Nodes w/o common ancestorsDeriving Useful Properties•Which nodes are (unconditionally) independent of S?•Nodes w/o common ancestors•Which nodes are conditionally indep. of S given S’s parents?Deriving Useful Properties•Which nodes are (unconditionally) independent of S?•Nodes w/o common ancestors•Which nodes are conditionally indep. of S given S’s parents?•All nondescendantsDeriving Useful Properties•Which nodes are (unconditionally) independent of S?•Nodes w/o common ancestors•Which nodes are conditionally indep. of S given S’s parents?•All nondescendants•Given S’s Markov Blanket?(parents, kids, kids’ parents)Deriving Useful Properties•Which nodes are (unconditionally) independent of S?•Nodes w/o common ancestors•Which nodes
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