Inference in Bayes Nets

Transcription

1 Inference in Bayes Nets Bruce D Ambrosio Oregon State University and Robert Fung Institute for Decision Systems Research July 23, 1994 Summer Institute on Probablility in AI 1994 Inference 1 1

2 Introduction l Inference is the computation of results to queries with respect to a given network in the presence of given evidence. Network E Evidence p(a E) Inference a0 a1 p(a E )? Query Summer Institute on Probablility in AI 1994 Inference 1 2

3 Introduction l Exact inference is NP-hard Topology is the main cause for complexity in exact inference Current exact algorithms are efficient and can solve real-world problems There is a steep cliff for inference (and also for knowledge acquisition) l Approximate inference is also NP-hard The effect of this categorization is not clear for real-world problems Low likelihood evidence and functional relationships are the main source of complexity for approximate inference Summer Institute on Probablility in AI 1994 Inference 1 3

4 I. Applying Inference l A. Characterization of Inference Problem features l B. Inference Algorithms B. 1. Selected Algorithms B. 2. Algorithm Utilities B. 3. Algorithm Review l C. Practical Considerations C. 1. Choosing an Algorithm C. 2. Tools l F. Summary and Recommendations Summer Institute on Probablility in AI 1994 Inference 1 4

5 I. A. Inference Problem Characterization l Network features Topology of network Size of network Type of variables in the network Entropy of the variable distributions l Query-related features Task Type of queries Available computational resources Evidence-related features Summer Institute on Probablility in AI 1994 Inference 1 5

6 Network Features l Network Types Probabilistic Networks Influence Diagrams l Network Topology singly-connected sparsely-connected two-level asymmetric BN2O l Network Dynamics static formulation interspersed with inference l Types of variables discrete continuous linear-gaussian mixed discretecontinuous discrete preceding linear-gaussian noisy-or functional skewness of the distribution temporal extension Summer Institute on Probablility in AI 1994 Inference 1 6

7 Query Features l Query task prediction posteriors given evidence all marginal posteriors specific marginal posteriors specific joint-conditional queries most likely hypotheses the most likely the n most likely decision policies the first decision policies for all decisions l Type of queries batch or asynchronous known in advance response time requirements real-time issued by user l Available computational resources consumer application embedded system parallel processing l Evidence-related features batch or asynchronous interspersed with queries Summer Institute on Probablility in AI 1994 Inference 1 7

8 Query Task l Prediction given a model, what might happen? l Posterior computation given evidence, what is happening? given evidence, what is the model of what will happen? l Most likely composite hypothesis given evidence, what is the most likely explanation? l Decision analysis what should I do about it? Summer Institute on Probablility in AI 1994 Inference 1 8

9 Prediction l Expected values given a model l P(Output)? I1 Output N1 N2 I2 N1 N2 I1 P1 O Carry A1 Inv I2 P2 C A1 Inv Summer Institute on Probablility in AI 1994 Inference 1 9

10 Posterior Computation l Beliefs given evidence Given I1=0, I2=0,O=1 What is probability A1 is ok? C is correct? N1 N2 I1 P1 O I2 P2 C A1 Inv Summer Institute on Probablility in AI 1994 Inference 1 10

11 Most Likely Composite Hypothesis Given I1=0, I2=0,O=1 What is most likely state of system? N1 N2 I1 P1 O I2 P2 C A1 Inv Summer Institute on Probablility in AI 1994 Inference 1 11

12 Policy Determination l Given a decision situation, what actions should be taken under what conditions? N1 N2 N1 N2 I1 P1 O I1 P1 O I2 P2 C I2 P2 C A1 Inv A1 Inv Value Action Summer Institute on Probablility in AI 1994 Inference 1 12

13 I. B. Algorithms l Exact Simple Bayes Graph Reduction Polytree Algorithm Clustering Symbolic Probabilistic Inference QuickScore Recursive Decomposition Potential ID Vegas l Approximate Forward Sampling Stochastic Simulation BNRAS Top N Term computation Poole Backward Sampling l Qualitative l QPNs l kappa-calculus Summer Institute on Probablility in AI 1994 Inference 1 13

14 I. B. Algorithms l B. 1. Selected Algorithms Exact Algorithms Joint Computation Simple Bayes Graph Reduction Polytree Algorithm Clustering Symbolic Probabilistic Inference Approximate Algorithms Forward Simulation Stochastic Simulation l B. 2. Algorithm Utilities l B. 3. Algorithm Review Summer Institute on Probablility in AI 1994 Inference 1 14

15 Exact Inference l Conditional independence relations make inference tractable l The Basic Problem: dealing with loops l Historical development marginalization Bayes rule development of singly-connected structures factoring (Smart Marginalization) Shachter, Andersen, Szolovits 1991; D Ambrosio 90 A B C D l Duality of probabilistic inference and decision analysis Summer Institute on Probablility in AI 1994 Inference 1 15

16 A Bayesian Network represents a joint probability distribution l each node distribution is a conditional probability of the node s variable given the parent variables. l the joint distribution is the product of the individual node distributions A B C D P(A,B,C,D) = P(D B,C)P(B A)P(C A)P(A) Summer Institute on Probablility in AI 1994 Inference 1 16

17 Method 1: Probabilistic Inference by Computing the Joint Distribution A B a P(a) t.4 f.6 a P(b) t f t.9.1 f.2.8 l Any query can be answered by computing the full joint P(AB) = P(B A)*P(A) P(AB) = {.36,.04,.12,.48} l Marginalize out the variables not in the query P(B) = ΣAP(AB) P(B) = {.48,.52} l However, very inefficient Summer Institute on Probablility in AI 1994 Inference 1 17

18 Method 2: Simple Bayes l Two-level Network l Inference is linear in size of network l Has been used extensively D F1 F2... Fn-1 Fn P(D F 1...Fn) = P(D)*Πi P(F i D) Summer Institute on Probablility in AI 1994 Inference 1 18

19 Method 3: Graph Reduction (Influence Diagram) l Development Olmstead 1982 Shachter 1986 Shachter 1988 Others (Optimal Operation Sequencing, Linear Gaussian formulation, Dynamic Programming, Asymmetries) l Basic Idea Any probabilistic or decision-theoretic query can be represented as a (sub) network of the network. Reduce the network to that (sub) network. l Three basic operations arc reversal (Bayes rule) barren node removal (marginalization) merge with value node (expectation/maximization) Summer Institute on Probablility in AI 1994 Inference 1 19

20 Arc Reversal A B X Y Z X = Pa(A)\Pa(b) Y = Pa(A)vPa(b) X = Pa(B)\Pa(A) A B X Y Z P(A B,X,Y,Z) = P(B A,Y,Z)*P(A X,Y)/P(B X,Y,Z) Summer Institute on Probablility in AI 1994 Inference 1 20

21 Other Operations l Barren Node Removal A A B C D B C ΣDP(A,B,C,D) = P(A)P(B A)P(C B,A)Σ D P(D C) l Removal into Value Node (by Expectation) X D V D V V(D) = Σ X V(X,D)*P(X) Summer Institute on Probablility in AI 1994 Inference 1 21

22 Probabilistic Inference Example l Reduce graph to sub-network of interest Barren node removal (1) Arc reversal (3) Chance node removal (2, 4, 5) l Query: P(C) A B C D A 1 2 B C A B C E E A 3 4 B C B C 5 C Summer Institute on Probablility in AI 1994 Inference 1 22

23 Decision Example l Reverse X-Y Arc l Removal of X by Expectation into V l Removal of D by maximization into V l gives you the optimal policy for D given Y Y X Y X D V D V Y Y D V V Summer Institute on Probablility in AI 1994 Inference 1 23

24 Method 4: Polytree Algorithm l History Pearl 1982 Kim and Pearl 1983 Other (Conditioning, Node Aggregation, Linear Gaussian Continuous) l Basic Idea Message-passing architecture for computing marginal posteriors. lambda: message up to your parents pi: message down to your children Summer Institute on Probablility in AI 1994 Inference 1 24

25 Singly Connected Nets Example l Initialization from roots to leafs A-G, A-C, B-D, C-D, C-F, D-F l Collection of evidence messages at central node (C) G-A, A-C, E-C, F-D, B-D, D-C l Broadcast of messages out from central node C-A,A-G, C-E, C-D, D-B, D-F A B * G C D E * * F Summer Institute on Probablility in AI 1994 Inference 1 25

26 Node Aggregration l Combine variables to make net singly connected P(A1/Inc) = P(Inv A1)*P(A1) P(P1/P2/C N1,N2,I1,I2,A1/Inv) = P(P1 N1,I1,I2)*P(P2 I1,I2,A1)*P(C P2,Inv) N1 N2 N1 N2 I1 P1 O I1 P1/P2/C I2 P2 C I2 A1 Inv A1/Inv Summer Institute on Probablility in AI 1994 Inference 1 26 O

27 Method 5: Clustering (Hugin) l History Lauritzen and Spiegelhalter (1988) Jensen (1990) Others (mixed discrete and linear gaussian variables, most likely hypotheses, updating of probabilities with data, optimized junction tree finding) Currently the dominant exact algorithm l Basic idea Cluster nodes together to make a singly-connected structure. Message passing along that structure. Clustering need not be mutually exclusive as in node aggregation for the polytree algorithm. Naturally computes posterior of cluster joints --> all node marginal posteriors Summer Institute on Probablility in AI 1994 Inference 1 27

28 Moralizing and Triangularization l Moralize: marry parents l Triangulate: no cycles more than three nodes H E G A B C A F H D B E C F G D Summer Institute on Probablility in AI 1994 Inference 1 28

29 Junction Tree Formation and Message Passing l Create singly-connected structure from triangulated graph H E G A B C F D BEF ABC BCF FCD EF FG Summer Institute on Probablility in AI 1994 Inference 1 29

30 Method 6: SPI l History D Ambrosio 88 Shachter, D Ambrosio, del Favero 90 Li and D Ambrosio 93 Other (Mixed discrete and continuous, most likely composite hypothesis, parallel, pure continuous, SPI with join trees) l Basic Idea: Smart marginalization. Symbolic manipulation of the marginalization expression. Query Driven use d-separation to find expressions to be combined Finds an efficient form for evaluation Handles multiple queries Interspersing evidence and queries Conjunctive queries Summer Institute on Probablility in AI 1994 Inference 1 30

31 Set Factoring l Construct a factor set A containing all node expressions. l Construct a candidate set B containing all pairs from A. l Choose best candidate from B. l Combine factors in candidate. l Update A: remove factors in candidate from A. add result to A. l Update B: remove all candidates using one of factors just combined. form new candidates with new factor. l Iterate until only one factor. Summer Institute on Probablility in AI 1994 Inference 1 31

32 SPI Example l Back to the chain rule l P(C) = Σ A,B,D,E (P(D C)(P(C A,B,E)(P(E)(P(B A)(P(A))))) l = Σ D P(D C)(Σ E (Σ A,B (P(C A,B,E)P(B A)P(A)))P(E)) Σ D A B C D E P(C A,B,E) Σ A,B Σ E P(D C) P(E) P(A) P(B A) Summer Institute on Probablility in AI 1994 Inference 1 32

33 Choosing Best Candidate l Choose candidates with minimum result size. l If more than one, choose one with maximum vars. l Loop 1: A: { } B:{ {1 2} {1 3} {1 4} {2 3} {2 4} {3 4} } Result Vars: {1 2} {1 3} { } {1 2 3} { } { } Result Size: vars : Choose: {1,2} Summer Institute on Probablility in AI 1994 Inference 1 33

34 More than one query l Suppose we now query marginal for node 3? l Use partial results from previous queries 3 Σ Σ 2,3 1 Σ Summer Institute on Probablility in AI 1994 Inference 1 34

35 Evidence l Incrementally adjust network and eval polytree re-write node expression selecting only observed value P( 4 2,3 ) -> P( 4* 2,3 ) eliminate all results incorporating previous expression Σ 2 Σ 2 Σ Σ 1 Σ 2 4* Summer Institute on Probablility in AI 1994 Inference 1 35

36 Conjunctive Queries l Use d-separation to find all relevant expressions l use set factoring to find efficient evaluation form l don t marginalize out query vars 2,3 Σ 1 4* Summer Institute on Probablility in AI 1994 Inference 1 36

37 Approximate Inference l All exact methods have problems with highly connected graphs l Can we avoid full computation? Simulation Bounded Conditioning Search Summer Institute on Probablility in AI 1994 Inference 1 37

38 Method 7: Forward Simulation o History General formulation (Shachter and Peot 1989, 1992) Likelihood weighting (Fung and Chang 1989) Logic sampling (Henrion 1986) o Importance sampling procedure Choose a sampling distribution Ps For each trial:»sample forward from root nodes to evidence nodes»compute weight of each trial to be P(X)/Ps(X)» For likelihood weighting this is P P(E Pa(E))»Accumulate weight for each node Normalize weights for each node Summer Institute on Probablility in AI 1994 Inference 1 38

39 Forward Sampling Example D1 E1 D2 o Possible Order D1, D2, D3, D4 D3 o Weight P(E1 d1)*p(e2 d3) E2 D4 Summer Institute on Probablility in AI 1994 Inference 1 39

40 Method 8: Stochastic Simulation and RAS methods l History Pearl (1987) Chavez (1990) l Basic Idea: Clamp the evidence variables to the observed values and then sample from the remaining variables based on neighboring variable values. Sampling each trial starts from the node configurations of the previous trial. l Trials are not independent as in forward sampling. Can be trapped by functional dependencies to a subset of the scenarios. l RAS methods provide for a number of transitions before taking the the trial values in order to provide more independence of trials. Summer Institute on Probablility in AI 1994 Inference 1 40

41 Stochastic Simulation Example l Local Computations are based on Markov Blanket l Markov Blanket is union of the parents, children, and mates D1 1. Choose sampling order 2. Randomly set initial values 3. For each node (X) sample from: Ps(X) = k*p(x Pa(X))Π y in Ch(x) P(Y X,Pa(Y)\X) E1 D2 Example: D3 Ps(D3) = k*p(d3 d1)*p(e2 D3) *P(d4 D3,d2) E2 D4 Summer Institute on Probablility in AI 1994 Inference 1 41

42 Algorithm Review l Exact Simple Bayes Graph Reduction Polytree Algorithm Clustering Symbolic Probabilistic Inference QuickScore Recursive Decomposition Potential ID Valuation networks l Approximate Monte Carlo Forward Sampling Stochastic Simulation BNRAS Top N Term computation Search (Poole) Backward Sampling Summer Institute on Probablility in AI 1994 Inference 1 42

43 I. C. Practical Considerations l Review Inference task l Tradeoff between representation and inference l Approximate inference if exact inference not feasible l Factoring is the primary role in efficiency of exact inference l For exact inference, most algorithms could be developed to have the same capabilities flexible factoring, mixed discrete, linear gaussian variables, decisions, asymmetries but currently, algorithms still reflect different computational architectures Summer Institute on Probablility in AI 1994 Inference 1 43

44 Practical Considerations l What current architecture/tool meets your application needs? l With what architecture do you feel most comfortable? l Roll your own or use an existing tool? Most people have rolled their own (Rockwell, Boeing, ADS, Microsoft, NRL) Availability of tools is increasing with closer to the ideal capabilities Summer Institute on Probablility in AI 1994 Inference 1 44

45 Available Systems System Organization Type Description Algorithm Language Platforms GUI Interface IDEAL Rockwell Science Center (Jim White) CL-SPI Oregon State U (Bruce D Ambrosio) HUGIN Hugin (Bob Fung Demos Lumina Decision Systems (Max Henrion) SPI++ Prevision (Bruce D Ambrosio) DPL Applied Decision Analysis PC-DX Knowledge Industries (Mark Peot) Research Research Testbed Research Research Testbed Commercial Probabilistic Inference Commercial Modeling Environment (Risk Analysis) Commercial Probabilistic Inference / Decision Analysis many Lisp CLIM SPI Lisp TCL/TK Hugin C Unix/PC X/Windows Monte Carlo C++ Mac/Unix/ PC SPI C++ Unix/PC/ Mac Mac/none/ Windows Commercial Decision?? PC Windows Analysis Commercial Diagnosis?? C++ PC Windows?? Summer Institute on Probablility in AI 1994 Inference 1 45

46 Example Applications l Intellipath large single-fault network --> Hugin type exact inference l QMR-DT large BN2O network ---> approximate inference is necessary: search or simulation l IES/BTI very large, dynamic singly-connected network --> polytree algorithm l Schedule Risk Analysis large mixed discrete and continuous network --> Monte Carlo l HP Sequential Diagnosis and Repair discrete, dynamic influence diagram --> term-computation Summer Institute on Probablility in AI 1994 Inference 1 46