Belief networks Chapter 15.1í2. AIMA Slides cæstuart Russell and Peter Norvig, 1998 Chapter 15.1í2 1
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1 Belief networks Chapter 15.1í2 AIMA Slides cæstuart Russell and Peter Norvig, 1998 Chapter 15.1í2 1
2 í Conditional independence Outline í Bayesian networks: syntax and semantics í Exact inference í Approximate inference AIMA Slides cæstuart Russell and Peter Norvig, 1998 Chapter 15.1í2 2
3 Independence random variables ABare èabsolutelyè independent iæ Two èajbè =P èaè P P èa; Bè =P èajbèp èbè =P èaèp èbè or A and B are two coin tosses e.g., n Boolean variables are independent, the full joint is If 1 ;:::;X n è=æ i PèX i è PèX hence can be speciæed by just n numbers Absolute independence is a very strong requirement, seldom met AIMA Slides cæstuart Russell and Peter Norvig, 1998 Chapter 15.1í2 3
4 Conditional independence the dentist problem with three random variables: Consider oothache, Cavity, Catch èsteel probe catches in my toothè T The full joint distribution has 2 3, 1 = 7 independent entries I have a cavity, the probability that the probe catches in it doesn't If on whether I have a toothache: depend P ècatchjt oothache; Cavityè =P ècatchjcavityè è1è Catch is conditionally independent of T oothache given Cavity i.e., same independence holds if I haven't got a cavity: The P ècatchjt oothache; :Cavityè=P ècatchj:cavityè è2è AIMA Slides cæstuart Russell and Peter Norvig, 1998 Chapter 15.1í2 4
5 Conditional independence contd. Equivalent statements to è1è è1aè P èt oothachejcatch; Cavityè = P èt oothachejcavityè Why?? P èt oothache; CatchjCavityè=P èt oothachejcavityèp ècatchjcavityè è1bè Why?? joint distribution can now be written as Full oothache; Catch; Cavityè =PèT oothache; CatchjCavityèPèCavityè PèT PèT oothachejcavityèpècatchjcavityèpècavityè = = 5 independent numbers èequations 1 and 2 remove 2è i.e., AIMA Slides cæstuart Russell and Peter Norvig, 1998 Chapter 15.1í2 5
6 Conditional independence contd. Equivalent statements to è1è è1aè P èt oothachejcatch; Cavityè = P èt oothachejcavityè Why?? P èt oothachejcatch; Cavityè = P ècatchjt oothache; CavityèP èt oothachejcavityè=p ècatchjcavityè = P ècatchjcavityèp èt oothachejcavityè=p ècatchjcavityèèfrom1è = P èt oothachejcavityè P èt oothache; CatchjCavityè=P èt oothachejcavityèp ècatchjcavityè è1bè Why?? P èt oothache; CatchjCavityè = P èt oothachejcatch; CavityèP ècatchjcavityè èproduct ruleè = P èt oothachejcavityèp ècatchjcavityè èfrom 1aè AIMA Slides cæstuart Russell and Peter Norvig, 1998 Chapter 15.1í2 6
7 Belief networks simple, graphical notation for conditional independence assertions A hence for compact speciæcation of full joint distributions and Syntax: set of nodes, one per variable a directed, acyclic graph èlink ç ëdirectly inæuences"è a conditional distribution for each node given its parents: a PèX i jp arentsèx i èè the simplest case, conditional distribution represented as In conditional probability table ècptè a AIMA Slides cæstuart Russell and Peter Norvig, 1998 Chapter 15.1í2 7
8 è Oèd k nè numbers vs. Oèd n è Example at work, neighbor John calls to say myalarm is ringing, but neighbor I'm doesn't call. Sometimes it's set oæ by minorearthquakes. Is there Mary a burglar? Burglar, Earthquake, Alarm, JohnCalls, MaryCalls Variables: topology reæects ëcausal" knowledge: Network Burglary P(B).001 Earthquake P(E).002 Alarm B T T F F E T F T F P(A) JohnCalls A T F P(J) MaryCalls A T F P(M) Note: ç k parents AIMA Slides cæstuart Russell and Peter Norvig, 1998 Chapter 15.1í2 8
9 Semantics semantics deænes the full joint distribution as ëglobal" product of the local conditional distributions: the PèX 1 ;:::;X n è=æ n i =1PèX i jparentsèx i èè P èj ^ M ^ A ^:B ^:Eè is given by?? e.g., = AIMA Slides cæstuart Russell and Peter Norvig, 1998 Chapter 15.1í2 9
10 Semantics semantics deænes the full joint distribution as ëglobal" product of the local conditional distributions: the PèX 1 ;:::;X n è=æ n i =1PèX i jparentsèx i èè P èj ^ M ^ A ^:B ^:Eè is given by?? e.g., P è:bèp è:eèp èaj:b ^:EèP èjjaèp èmjaè = semantics: each node is conditionally independent ëlocal" its nondescendants given its parents of Theorem: Local semantics, global semantics AIMA Slides cæstuart Russell and Peter Norvig, 1998 Chapter 15.1í2 10
11 Markov blanket node is conditionally independent of all others given its Each blanket: parents + children + children's parents Markov U 1... U m Z 1j X Z nj Y 1... Y n AIMA Slides cæstuart Russell and Peter Norvig, 1998 Chapter 15.1í2 11
12 Constructing belief networks a method such that a series of locally testable assertions of Need independence guarantees the required global semantics conditional Choose an ordering of variables X 1 ;:::;X n 1. For i =1ton 2. X i to the network add parents from X 1 ;:::;X i,1 such that select PèX i jp arentsèx i èè = PèX i jx 1 ;:::;X i,1 è choice parents guarantees the global semantics: This n of ;:::;X n è=æ i =1PèX i jx 1 ; :::; X i,1 è èchain ruleè 1 PèX = æ n i =1PèX i jparentsèx i èè by construction AIMA Slides cæstuart Russell and Peter Norvig, 1998 Chapter 15.1í2 12
13 Example Suppose we choose the ordering M, J, A, B, E MaryCalls JohnCalls P èjjm è=p èjè? AIMA Slides cæstuart Russell and Peter Norvig, 1998 Chapter 15.1í2 13
14 . Alarm No. èajj;mè=pèajjè? P èajj;mè=pèaè? P AIMA Slides cæstuart Russell and Peter Norvig, 1998 Chapter 15.1í2 14
15 . Burglary. No. èbja; J;Mè=PèBjAè? P èbja; J;Mè=PèBè? P AIMA Slides cæstuart Russell and Peter Norvig, 1998 Chapter 15.1í2 15
16 . Earthquake.. Yes. No. èejb; A;J;Mè=PèEjAè? P èejb; A;J;Mè=PèEjA; Bè? P AIMA Slides cæstuart Russell and Peter Norvig, 1998 Chapter 15.1í2 16
17 ..... No. Yes. AIMA Slides cæstuart Russell and Peter Norvig, 1998 Chapter 15.1í2 17
18 Example: Car diagnosis evidence: engine won't start Initial variables èthin ovalsè, diagnosis variables èthick ovalsè Testable battery age alternator broken fanbelt broken Hidden variables èshadedè ensure sparse structure, reduce parameters battery dead no charging battery flat no oil no gas fuel line blocked starter broken lights oil light gas gauge engine won t start AIMA Slides cæstuart Russell and Peter Norvig, 1998 Chapter 15.1í2 18
19 Example: Car insurance claim costs èmedical, liability, propertyè Predict data on application form èother unshaded nodesè given Age GoodStudent RiskAversion SeniorTrain SocioEcon Mileage VehicleYear ExtraCar DrivingSkill MakeModel DrivingHist Antilock DrivQuality Airbag CarValue HomeBase AntiTheft Ruggedness Accident OwnDamage Theft Cushioning OtherCost OwnCost MedicalCost LiabilityCost PropertyCost AIMA Slides cæstuart Russell and Peter Norvig, 1998 Chapter 15.1í2 19
20 Compact conditional distributions grows exponentially with no. of parents CPT becomes inænite with continuous-valued parent or child CPT Solution: canonical distributions that are deæned compactly nodes are the simplest case: Deterministic = f èp arentsèxèè for some function f X Boolean functions E.g., orthamerican, Canadian _ U S _ M exican N E.g., numerical relationships among = inæow + precipation - outæow - evaporation AIMA Slides cæstuart Russell and Peter Norvig, 1998 Chapter 15.1í2 20
21 Compact conditional distributions contd. distributions model multiple noninteracting causes Noisy-OR Parents U 1 :::U k include all causes ècan add leak nodeè 1è Independent probability q i for each cause alone 2è j failure èxju 1 :::U j ; :U j+1 ::::U k è=1, æ i =1q i P è Flu Malaria P èfeverè P è:feverè Cold F F 0.0 1:0 F F T 0:9 0.1 F T F 0:8 0.2 F T T 0:98 0:02 = 0:2 æ 0:1 F F F 0:4 0.6 T F T 0:94 0:06 = 0:6 æ 0:1 T T F 0:88 0:12 = 0:6 æ 0:2 T T T T 0:988 0:012 = 0:6 æ 0:2 æ 0:1 Number of parameters linear in number of parents AIMA Slides cæstuart Russell and Peter Norvig, 1998 Chapter 15.1í2 21
22 Hybrid èdiscrete+continuousè networks Subsidy? Harvest Discrete èsubsidy? and Buys?è; continuous èharvest and Costè Cost Buys? Option 1: discretization possibly large errors, large CPTs Option 2: ænitely parameterized canonical families Continuous variable, discrete+continuous parents èe.g., Costè 1è Discrete variable, continuous parents èe.g., Buys?è 2è AIMA Slides cæstuart Russell and Peter Norvig, 1998 Chapter 15.1í2 22
23 Continuous child variables one conditional density function for child variable given continuous Need for each possible assignment to discrete parents parents, common is the linear Gaussian model, e.g.,: Most ècost = cjharvest = h; Subsidy?= trueè P = N èa t h + b t ;ç t èècè = 1 çt p2ç exp 0 B B B@ c, èa t h + b t è çt 1 C CA 2 1 C CCA Cost varies linearly with Harvest, variance is æxed Mean variation is unreasonable over the full range Linear but works OK if the likely range of Harvest is narrow AIMA Slides cæstuart Russell and Peter Norvig, 1998 Chapter 15.1í2 23
24 Continuous child variables network with LG distributions All-continuous full joint is a multivariate Gaussian è LG network is a conditional Gaussian network i.e., Discrete+continuous multivariate Gaussian over all continuous variables for each combina- a tion of discrete variable values AIMA Slides cæstuart Russell and Peter Norvig, 1998 Chapter 15.1í2 24 P(Cost Harvest,Subsidy?=true) Cost P(Cost Harvest,Subsidy?=false) Harvest 5 Cost Harvest P(Cost Harvest) Cost Harvest
25 èbuys?=true j Cost = cè = æèè,c + çè=çè P view as hard threshold whose location is subject to noise Can Discrete variable wè continuous parents ofbuys? given Cost should be a ësoft" threshold: Probability P(Buys?=false Cost=c) Cost c distribution integral of Gaussian: Probit R uses x,1 N è0; 1èèxèdx = æèxè AIMA Slides cæstuart Russell and Peter Norvig, 1998 Chapter 15.1í2 25
26 Discrete variable contd. Sigmoid èor logitè distribution also used in neural networks: 1 P èbuys?=true j Cost= cè =,c+ç ç è 1+expè,2 P(Buys?=false Cost=c) Cost c Sigmoid has similar shape to probit but much longer tails: AIMA Slides cæstuart Russell and Peter Norvig, 1998 Chapter 15.1í2 26
Outline } Conditional independence } Bayesian networks: syntax and semantics } Exact inference } Approximate inference AIMA Slides cstuart Russell and
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