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Berkeley COMPSCI 188 - Bayes nets II (6PP)

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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-ruleBayes 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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Berkeley COMPSCI 188 - Bayes nets II (6PP)

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