Icy Roads. Bayesian Networks. Icy Roads. Icy Roads. Icy Roads. Holmes and Watson in LA

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1 6.825 Techniques in rtificial ntelligence ayesian Networks To do probabilistic reasoning, you need to know the joint probability distribution ut, in a domain with N propositional variables, one needs 2 N numbers to specify the joint probability distribution We want to exploit independences in the domain Two components: structure and numerical parameters cy Roads nspector Smith is waiting for and, who are driving (separately) to meet him. t is winter. is secretary tells him that has had an accident. e says, t must be that the roads are icy. bet that will have an accident too. should go to lunch. ut, his secretary says, guess better wait for. ausal omponent cy ecture 15 1 ecture 15 2 cy Roads nspector Smith is waiting for and, who are driving (separately) to meet him. t is winter. is secretary tells him that has had an accident. e says, t must be that the roads are icy. bet that will have an accident too. should go to lunch. ut, his secretary says, guess better wait for. ausal omponent cy Roads nspector Smith is waiting for and, who are driving (separately) to meet him. t is winter. is secretary tells him that has had an accident. e says, t must be that the roads are icy. bet that will have an accident too. should go to lunch. ut, his secretary says, guess better wait for. ausal omponent cy cy and W are dependent, ecture 15 3 ecture 15 4 cy Roads nspector Smith is waiting for and, who are driving (separately) to meet him. t is winter. is secretary tells him that has had an accident. e says, t must be that the roads are icy. bet that will have an accident too. should go to lunch. ut, his secretary says, guess better wait for. and in and have moved to. e wakes up to find his lawn wet. e wonders if it has rained or if he left his sprinkler on. e looks at his neighbor s lawn and he sees it is wet too. So, he concludes it must have rained. ausal omponent cy and W are dependent, but conditionally independent given Sprinkler awn Rain awn ecture 15 5 ecture

2 and in Forward Serial onnection and have moved to. e wakes up to find his lawn wet. e wonders if it has rained or if he left his sprinkler on. e looks at his neighbor s lawn and he sees it is wet too. So, he concludes it must have rained. Sprinkler awn Rain awn Given W, P(R) goes up and P(S) goes down explaining away Transmit evidence from to through unless is instantiated (its truth value is known) = battery dead = car won t start = car won t move nowing about will tell us something about ut, if we know, then knowing about will not tell us anything new about. ecture 15 7 ecture 15 8 ackward Serial onnection iverging onnection Transmit evidence from to through unless is instantiated (its truth value is known) = battery dead = car won t start = car won t move nowing about will tell us something about ut, if we know, then knowing about will not tell us anything new about Transmit evidence through unless it is instantiated = crash = cy = crash nowing about will tell us something about nowing about will tell us something about ut, if we know, then knowing about will not tell us anything new about, or vice versa ecture 15 9 ecture onverging onnection Transmit evidence from to only if or a descendant of is instantiated = acterial infection = Sore throat = Viral nfection Without knowing, finding does not tell us anything about f we see evidence for, then and become dependent (potential for explaining away ). f we find bacteria in patient with a sore throat, then viral infection is less likely. -separation Two variables and are d-separated iff for every path between them, there is an intermediate variable V such that either The connection is or diverging and V is known The connection is converging and neither V nor any descendant is instantiated Two variables are d-connected iff they are not d-separated diverging --:, blocked when is --:, blocked when is --: diverging, blocked when is --: converging, blocked when has no evidence, connected otherwise ecture converging ecture

3 -Separation etail -Separation Example diverging converging --:, blocked when is --:, blocked when is --: diverging, blocked when is --: converging, blocked when has no evidence, connected otherwise No instantiation, are d-connected (-- connected, -- connected), are d-connected (-- connected, -- blocked) instantiated, are d-separated (-- blocked, -- blocked) and instantiated, are d-connected (-- blocked, -- connected) instantiated, are d-connected (-- blocked, -- connected) and instantiated, are d-separated (-- blocked, -- blocked) ecture E F G Given is known, is ecture Separation Example -Separation Example Given is known, is Given is known, is E F G E F G ecture ecture Separation Example -Separation Example Given is known, is Given is known, is E F G E F G Since at least one path is not blocked, is not d-separated from E Since at least one path is not blocked, is not d-separated from E ecture ecture

4 Recitation Problems Use the ayesian network from the previous slides to answer the following questions: re and F d-separated if is instantiated? re and F d-separated if nothing is instantiated? re and E d-separated if is instantiated? re and E d-separated if and are instantiated? escribe a situation in which and G are d- separated. escribe a situation in which and G are d- connected. ayesian (elief) Networks Set of variables, each has a finite set of values Set of directed arcs between them forming acyclic graph Every node, with parents 1,, n, has P( 1,, n ) specified Theorem: f and are d-separated given evidence e, then P( e) = P(, e) ecture ecture hain Rule Variables: V 1,, V n Values: v 1,, v n P(V 1 =v 1, V 2 =v 2,, V n =v n ) = i P(V i =v i parents(v i )) hain Rule Variables: V 1,, V n Values: v 1,, v n P(V 1 =v 1, V 2 =v 2,, V n =v n ) = i P(V i =v i parents(v i )) P() = P(=true, =true, =true, =true) P() P() P() = P(=true, =true, =true, =true) P() P() P(,) P() = P(,) P( )P() = P( ) P( ) P() = P( ) d-separated from given d-separated from given ecture ecture hain Rule Variables: V 1,, V n Values: v 1,, v n P(V 1 =v 1, V 2 =v 2,, V n =v n ) = i P(V i =v i parents(v i )) ey dvantage The conditional independencies (missing arrows) mean that we can store and compute the joint probability distribution more efficiently P() = P(=true, =true, =true, =true) P() P() P() = P(,) P( )P() = P( ) P() = P( ) P( ) P( ) P() = P( ) P( ) P()P() d-separated from given d-separated from given d-separated from ecture ecture

5 cy Roads with Numbers Numerical Example: Shorthand t = true cy P(=t) 0.7 P(=f) 0.3 cy P()=0.7 f = false P( ) P(W ) P(=t ) P(=f ) P(W=t ) P(W=f ) =t 0.2 =t 0.2 =f 0.9 =f 0.9 The right-hand column in these tables is redundant, since we know the entries in each row must add to 1. N: the columns need NOT add to 1. ecture ecture Probability that es Probability of cy given cy P()=0.7 cy P()=0.7 P( ) P(W ) P( ) P(W ) P(W) = P(W ) P() + P(W ) P( ) = = = 0.59 P( W) = P(W ) P() / P(W) = 0.7 / 0.59 = 0.95 We started with P() = 0.7; knowing that crashed raised the probability to 0.95 ecture ecture Probability of given Prob of given cy and cy P()=0.7 cy P()=0.7 P( ) P(W ) P( ) P(W ) P( W) = P( W,)P( W) + P( W, ) P( W) = P( )P( W) + P( ) P( W) = = P( W, ) = P( ) = and W are d-separated given, so and W are conditionally independent given We started with P() = 0.59; knowing that crashed raised the probability to ecture ecture

6 Recitation Problems n the and visit network, use the following conditional probability tables. P(R) = 0.2 P( R,S) P(S) = R,S 1.0 P(W R) R, S 1.0 R 1.0 R, S R R, S alculate: P(), P(R ), P(S ), P(W ), P(R W,), P(S W,) ecture