1CS 188: Artificial IntelligenceFall 2009Lecture 15: Bayes’ Nets II – Independence10/15/2009Dan Klein – UC BerkeleyAnnouncements Midterm 10/22: see prep page on web One page note sheet, non-programmable calculators Review sessions (times TBD, probably Sunday and Tuesday) Topics go through today, not next class Next reading needs login/password22Bayes’ Nets A Bayes’ net is anefficient encodingof a probabilisticmodel of a domain Questions we can ask: Inference: given a fixed BN, what is P(X | e)? Representation: given a BN graph, what kinds of distributions can it encode? Modeling: what BN is most appropriate for a given domain?3Bayes’ Net Semantics Let’s formalize the semantics of a Bayes’ net A set of nodes, one per variable X A directed, acyclic graph A conditional distribution for each node A collection of distributions over X, one for each combination of parents’ values CPT: conditional probability table Description of a noisy “causal” processA1XAnA Bayes net = Topology (graph) + Local Conditional Probabilities43Example: Alarm NetworkBurglaryEarthqkAlarmJohn callsMary callsB P(B)+b 0.001¬b 0.999E P(E)+e 0.002¬e 0.998B E A P(A|B,E)+b +e +a 0.95+b +e ¬a 0.05+b ¬e +a 0.94+b ¬e ¬a 0.06¬b +e +a 0.29¬b +e ¬a 0.71¬b ¬e +a 0.001¬b ¬e ¬a 0.999A J P(J|A)+a +j 0.9+a ¬j 0.1¬a +j 0.05¬a ¬j 0.95A M P(M|A)+a +m 0.7+a ¬m 0.3¬a +m 0.01¬a ¬m 0.99Building 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 defines a joint distribution over that domain, specified by local probabilities and graph structure64Size of a Bayes’ Net How big is a joint distribution over N Boolean variables?2N How big is an N-node net if nodes have up to k parents?O(N * 2k+1) Both give you the power to calculate BNs: Huge space savings! Also easier to elicit local CPTs Also turns out to be faster to answer queries (coming)7Bayes’ Nets So Far We now know: What is a Bayes’ net? What joint distribution does a Bayes’ net encode? Now: properties of that joint distribution (independence) 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 Next: how to compute posteriors quickly (inference)85Conditional Independence Reminder: independence X and Y are independent if X and Y are conditionally independent given Z (Conditional) independence is a property of a distribution9Example: Independence For this graph, you can fiddle with θ (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 distributions106Topology 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 costs Full conditioning can encode any distributionXYZXYZXYZ11Independence in a BN Important question about a BN: Are two nodes independent given certain evidence? If yes, can prove using algebra (tedious in general) If no, can prove with a counter example Example: Question: are X and Z necessarily independent? Answer: no. Example: 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 Z7Causal 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: Traffic13Common Cause Another basic configuration: two effects of the same cause Are X and Z independent? Are X and Z independent given Y? Observing the cause blocks influence between effects.XYZYes!Y: Project dueX: Newsgroup busyZ: Lab full148Common Effect Last configuration: two causes of one effect (v-structures) Are X and Z independent? Yes: the ballgame and the rain cause traffic, but they are not correlated Still need to prove they must be (try it!) Are X and Z independent given Y? No: seeing traffic puts the rain and the ballgame in competition as explanation? This is backwards from the other cases Observing an effect activates influence between possible causes.XYZX: RainingZ: BallgameY: Traffic15The General Case Any complex example can be analyzed using these three canonical cases General question: in a given BN, are two variables independent (given evidence)? Solution: analyze the graph169Reachability 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 “active”RTBDL17Reachability (D-Separation) Question: Are X and Y conditionally independent given evidence vars {Z}? Yes, if X and Y “separated” by Z Look for active paths from X to Y No active paths = independence! A path is active if each triple is active: Causal chain A → B → C where B is unobserved (either direction) Common cause A ← B → C where B is unobserved Common effect (aka v-structure)A → B ← C where B or one of its descendents is observed All it takes to block a path is a single inactive segmentActive Triples Inactive Triples[Demo]10ExampleYes19RTBT’ExampleRTBDLT’YesYesYes2011Example Variables: R: Raining T: Traffic D: Roof drips S: I’m sad Questions:TSDRYes21Causality? When Bayes’ nets reflect the true causal patterns: Often simpler (nodes have fewer parents) Often easier to think about Often easier to elicit from experts BNs need not actually be causal Sometimes no causal net exists over the domain E.g. consider the variables Traffic and Drips End up with arrows that reflect correlation, not causation What do the arrows really mean? Topology may happen to encode causal structure Topology only guaranteed to encode
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