Bayes Nets. CS 188: Artificial Intelligence Fall Example: Alarm Network. Bayes Net Semantics. Building the (Entire) Joint. Size of a Bayes Net

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1 CS 188: Artificial Intelligence Fall 2010 Lecture 15: ayes Nets II Independence 10/14/2010 an Klein UC erkeley A ayes net is an efficient encoding of a probabilistic model of a domain ayes Nets Questions we can ask: Inference: given a fixed N, what is P( e)? epresentation: given a N graph, what kinds of distributions can it encode? Modeling: what N is most appropriate for a given domain? 3 ayes Net Semantics Example: Network Let s formalize the semantics of a ayes net A set of nodes, one per variable A directed, acyclic graph A conditional distribution for each node A collection of distributions over, one for each combination of parents values CP: conditional probability table escription of a noisy causal process A ayes net = opology (graph) + Local Conditional Probabilities A 1 A n 4 P() +b b urglary John calls A J P(J A) +a +j 0.9 +a j 0.1 a +j 0.05 a j 0.95 Earthqk Mary calls A M P(M A) +a +m 0.7 +a m 0.3 a +m 0.01 a m 0.99 E P(E) +e e E A P(A,E) +b +e +a b +e a b e +a b e a 0.06 b +e +a 0.29 b +e a 0.71 b e +a b e a uilding the (Entire) Joint We can take a ayes net and build any entry from the full joint distribution it encodes ypically, there s no reason to build ALL of it We build what we need on the fly o emphasize: every N over a domain implicitly defines a joint distribution over that domain, specified by local probabilities and graph structure 6 Size of a ayes Net How big is a joint distribution over N oolean variables? 2 N How big is an N-node net if nodes have up to k parents? O(N * 2 k+1 ) oth give you the power to calculate Ns: Huge space savings! Also easier to elicit local CPs Also turns out to be faster to answer queries (coming) 7 1

2 ayes Nets So Far We now know: What is a ayes net? What joint distribution does a ayes net encode? Now: properties of that joint distribution (independence) Key idea: conditional independence Last class: assembled Ns using an intuitive notion of conditional independence as causality oday: formalize these ideas Main goal: answer queries about conditional independence and influence Next: how to compute posteriors quickly (inference) 8 Conditional Independence eminder: independence and are independent if and are conditionally independent given (Conditional) independence is a property of a distribution 9 Example: Independence opology Limits istributions For this graph, you can fiddle with θ (the CPs) all you want, but you won t be able to represent any distribution in which the flips are dependent! 1 2 All distributions 10 Given some graph topology G, only certain joint distributions can be encoded he graph structure guarantees certain (conditional) independences (here might be more independence) Adding arcs increases the set of distributions, but has several costs Full conditioning can encode any distribution 11 Independence in a N Important question about a N: 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: Causal Chains his configuration is a causal chain Is independent of given? : Low pressure : ain : raffic Question: are and necessarily independent? Answer: no. Example: low pressure causes rain, which causes traffic. can influence, can influence (via ) Addendum: they could be independent: how?! Evidence along the chain blocks the influence 13 2

3 Common Cause Common Effect Another basic configuration: two effects of the same cause Are and independent? Are and independent given? Observing the cause blocks influence between effects.! : Project due : Newsgroup busy : Lab full 14 Last configuration: two causes of one effect (v-structures) Are and independent? : the ballgame and the rain cause traffic, but they are not correlated Still need to prove they must be (try it!) Are and independent given? No: seeing traffic puts the rain and the ballgame in competition as explanation? his is backwards from the other cases Observing an effect activates influence between possible causes. : aining : allgame : raffic 15 he General Case eachability Any complex example can be analyzed using these three canonical cases General question: in a given N, are two variables independent (given evidence)? ecipe: shade evidence nodes Attempt 1: if two nodes are connected by an undirected path not blocked by a shaded node, they are conditionally independent L Solution: analyze the graph Almost works, but not quite Where does it break? Answer: the v-structure at doesn t count as a link in a path unless active eachability (-Separation) Example Question: Are and conditionally independent given evidence vars {}?, if and separated by Look for active paths from to No active paths = independence! Active riples Inactive riples A path is active if each triple is active: Causal chain A C where is unobserved (either direction) Common cause A C where is unobserved Common effect (aka v-structure) A C where or one of its descendents is observed All it takes to block a path is a single inactive segment [emo] 19 3

4 Example Example L Variables: : aining : raffic : oof drips S: I m sad Questions: S Causality? When ayes nets reflect the true causal patterns: Often simpler (nodes have fewer parents) Often easier to think about Often easier to elicit from experts Example: raffic asic traffic net Let s multiply out the joint Ns need not actually be causal Sometimes no causal net exists over the domain E.g. consider the variables raffic and rips End up with arrows that reflect correlation, not causation What do the arrows really mean? opology may happen to encode causal structure opology only guaranteed to encode conditional independence r 1/4 r 3/4 r t 3/4 t 1/4 r t 3/16 r t 1/16 r t 6/16 r t 6/16 r t 1/2 22 t 1/2 23 Example: everse raffic everse causality? Example: Coins Extra arcs don t prevent representing independence, just allow non-independence t 9/16 t 7/16 t r 1/3 r 2/3 t r 1/7 r 6/7 r t 3/16 r t 1/16 r t 6/16 r t 6/ Adding unneeded arcs isn t wrong, it s just inefficient 1 2 h t h t 25 4

5 Changing ayes Net Structure Example: Alternate he same joint distribution can be encoded in many different ayes nets Causal structure tends to be the simplest urglary Earthquake If we reverse the edges, we make different conditional independence assumptions Analysis question: given some edges, what other edges do you need to add? One answer: fully connect the graph etter answer: don t make any false conditional independence assumptions John calls Mary calls John calls Mary calls 26 o capture the same joint distribution, we have to add more edges to the graph urglary Earthquake 27 Summary ayes nets compactly encode joint distributions Guaranteed independencies of distributions can be deduced from N graph structure -separation gives precise conditional independence guarantees from graph alone A ayes net s joint distribution may have further (conditional) independence that is not detectable until you inspect its specific distribution 28 5