1CS 188: Artificial IntelligenceFall 2011Lecture 14: Bayes’ Nets II – Independence10/11/2011Dan Klein – UC BerkeleyBayes’ 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?2Bayes’ 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 Probabilities3Example: Alarm NetworkBurglaryEarthqkAlarmJohn callsMary callsB P(B)+b 0.001¬b0.999E P(E)+e 0.002¬e0.998B E A P(A|B,E)+b +e +a 0.95+b +e¬a0.05+b¬e+a 0.94+b¬e ¬a0.06¬b+e +a 0.29¬b+e¬a0.71¬b ¬e+a 0.001¬b ¬e ¬a0.999A J P(J|A)+a +j 0.9+a¬j0.1¬a+j 0.05¬a ¬j0.95A M P(M|A)+a +m 0.7+a¬m0.3¬a+m 0.01¬a ¬m0.99Size 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)5Building 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 structure62Bayes’ 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)7Bayes Nets: Assumptions Assumptions we are required to make to define the Bayesnet when given the graph: Probability distributions that satisfy the above (“chain-ruleBayes net”) conditional independence assumptions Often guaranteed to have many more conditional independences Additional conditional independences can be read off the graph Important for modeling: understand assumptions made when choosing a Bayes net graph8Example Conditional independence assumptions directly from simplifications in chain rule: Additional implied conditional independence assumptions?9X Y Z WConditional Independence Reminder: independence X and Y are independent if X and Y are conditionally independent given Z (Conditional) independence is a property of a distribution10D-separation: Outline Study independence properties for triples Any complex example can be analyzed using these three canonical cases11Independence 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 Z3Causal 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 full14Common 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 graph16Reachability 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 Triples4D-Separation Given query Shade all evidence nodes For all (undirected!) paths between and Check whether path is active If active return (If reaching this point all paths have been checked and shown inactive) Return 19?ExampleYes20RTBT’ExampleRTBDLT’YesYesYes21Example Variables: R: Raining T: Traffic D: Roof drips S: I’m sad Questions:TSDRYes22All Conditional Independences Given a Bayes net structure, can run d-separation to build a complete list of conditional independences that are necessarily true of the form This list determines the set of probability distributions that can be represented 23Example: Independence For this graph, you can fiddle with θ (the CPTs) all you want, but you won’t
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