Product rule. Chain rule
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1 Probability Recap CS 188: Artificial Intelligence ayes Nets: Independence Conditional probability Product rule Chain rule, independent if and only if: and are conditionally independent given if and only if: Instructors: Pieter Abbeel & Dan Klein --- University of California, erkeley [These slides were created by Dan Klein and Pieter Abbeel for CS188 Intro to AI at UC erkeley. All CS188 materials are available at ayes Nets A ayes net is an efficient encoding of a probabilistic model of a domain Questions we can ask: Inference: given a fixed N, what is P( e)? Representation: given a N graph, what kinds of distributions can it encode? Modeling: what N is most appropriate for a given domain? ayes Net Semantics A directed, acyclic graph, one node per random variable A conditional probability table (CPT) for each node A collection of distributions over, one for each combination of parents values ayes nets implicitly encode joint distributions As a product of local conditional distributions To see what probability a N gives to a full assignment, multiply all the relevant conditionals together:
2 Example: Alarm Network Example: Alarm Network P() +b E E P(E) +e P() +b E E P(E) +e b A J P(J A) +a +j 0.9 +a -j 0.1 -a +j a -j 0.95 J A M -e A M P(M A) +a +m 0.7 +a -m 0.3 -a +m a -m 0.99 E A P(A,E) +b +e +a b +e -a b -e +a b -e -a b +e +a b +e -a b -e +a b -e -a b A J P(J A) +a +j 0.9 +a -j 0.1 -a +j a -j 0.95 J A M -e A M P(M A) +a +m 0.7 +a -m 0.3 -a +m a -m 0.99 E A P(A,E) +b +e +a b +e -a b -e +a b -e -a b +e +a b +e -a b -e +a b -e -a Size of a ayes Net ayes Nets 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 CPTs Also faster to answer queries (coming) Representation Conditional Independences Probabilistic Inference Learning ayes Nets from Data
3 Conditional Independence and are independent if and are conditionally independent given (Conditional) independence is a property of a distribution Example: ayes Nets: Assumptions Assumptions we are required to make to define the ayes net when given the graph: P (x i x 1 x i 1 )=P (x i parents( i )) eyond above chain rule à ayes net conditional independence assumptions Often additional conditional independences They can be read off the graph Important for modeling: understand assumptions made when choosing a ayes net graph Example Independence in a N W Conditional independence assumptions directly from simplifications in chain rule: Additional implied conditional independence assumptions? 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: 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?
4 D-separation: Outline D-separation: Outline Study independence properties for triples Analyze complex cases in terms of member triples D-separation: a condition / algorithm for answering such queries Causal Chains This configuration is a causal chain Guaranteed independent of? No! Causal Chains This configuration is a causal chain Guaranteed independent of given? One example set of CPTs for which is not independent of is sufficient to show this independence is not guaranteed. Example: Low pressure causes rain causes traffic, high pressure causes no rain causes no traffic : Low pressure : Rain : Traffic In numbers: P( +y +x ) = 1, P( -y - x ) = 1, P( +z +y ) = 1, P( -z -y ) = 1 : Low pressure : Rain : Traffic es! Evidence along the chain blocks the influence
5 Common Cause This configuration is a common cause Guaranteed independent of? No! One example set of CPTs for which is not independent of is sufficient to show this independence is not guaranteed. : Project due Common Cause This configuration is a common cause Guaranteed and independent given? : Project due Example: Project due causes both forums busy and lab full In numbers: : Forums busy : Lab full P( +x +y ) = 1, P( -x -y ) = 1, P( +z +y ) = 1, P( -z -y ) = 1 : Forums busy : Lab full es! Observing the cause blocks influence between effects. Common Effect Last configuration: two causes of one effect (v-structures) : Raining : allgame Are and independent? es: 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. This is backwards from the other cases : Traffic Observing an effect activates influence between possible causes. The General Case
6 The General Case Reachability General question: in a given N, are two variables independent (given evidence)? Solution: analyze the graph Any complex example can be broken into repetitions of the three canonical cases Recipe: shade evidence nodes, look for paths in the resulting graph L Attempt 1: if two nodes are connected by an undirected path not blocked by a shaded node, they are conditionally independent R 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 Active / Inactive Paths Question: Are and conditionally independent given evidence variables {}? es, if and d-separated by Consider all (undirected) paths from to No active paths = independence! 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 Active Triples D D-Separation Inactive Triples Query: i j {k1,..., kn } Check all (undirected!) paths between? and If one or more active, then independence not guaranteed i j {k1,..., kn } Otherwise (i.e. if all paths are inactive), then independence is guaranteed i j {k1,..., kn } T
7 Example Example L es R es es R T D T T es T Example Structure Implications Variables: R: Raining T: Traffic D: Roof drips S: I m sad Questions: es T R S D Given a ayes net structure, can run d- separation algorithm to build a complete list of conditional independences that are necessarily true of the form i j { k1,..., kn } This list determines the set of probability distributions that can be represented
8 Computing All Independences Topology 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 distribution {} ayes Nets Representation Summary ayes Nets ayes nets compactly encode joint distributions Guaranteed independencies of distributions can be deduced from N graph structure D-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 Representation Conditional Independences Probabilistic Inference Enumeration (exact, exponential complexity) Variable elimination (exact, worst-case exponential complexity, often better) Probabilistic inference is NP-complete Sampling (approximate) Learning ayes Nets from Data
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