# COMP538: Introduction to Bayesian Networks

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1 COMP538: Introduction to Bayesian Networks Lecture 2: Bayesian Networks Nevin L. Zhang Department of Computer Science and Engineering Hong Kong University of Science and Technology Fall 2008 Nevin L. Zhang (HKUST) Bayesian Networks Fall / 55

2 Objective Objective: Explain the concept of Bayesian networks Reading: Zhang & Guo, Chapter 2 References: Russell & Norvig, Chapter 15 Nevin L. Zhang (HKUST) Bayesian Networks Fall / 55

3 Outline Probabilistic Modeling with Joint Distribution 1 Probabilistic Modeling with Joint Distribution 2 Conditional Independence and Factorization 3 Bayesian Networks 4 Manual Construction of Bayesian Networks Building structures Causal Bayesian networks Determining Parameters Local Structures 5 Remarks Nevin L. Zhang (HKUST) Bayesian Networks Fall / 55

4 Probabilistic Modeling with Joint Distribution The probabilistic approach to reasoning under uncertainty A problem domain is modeled by a list of variables X 1, X 2,..., X n, Knowledge about the problem domain is represented by a joint probability P(X 1, X 2,...,X n ). Example: Alarm (Pearl 1988) Story: In LA, burglary and earthquake are not uncommon. They both can cause alarm. In case of alarm, two neighbors John and Mary may call. Problem: Estimate the probability of a burglary based who has or has not called. Variables: Burglary (B), Earthquake (E), Alarm (A), JohnCalls (J), MaryCalls (M). Knowledge required by the probabilistic approach in order to solve this problem: P(B, E, A, J, M) Nevin L. Zhang (HKUST) Bayesian Networks Fall / 55

5 P(B, E, A, J, M) B E A J M Prob B E A J M Prob y y y y y n y y y y.0002 y y y y n n y y y n.0004 y y y n y n y y n y.0004 y y y n n n y y n n.0002 y y n y y n y n y y.0002 y y n y n n y n y n.0002 y y n n y n y n n y.0002 y y n n n.0000 n y n n n.0002 y n y y y n n y y y.0001 y n y y n n n y y n.0002 y n y n y n n y n y.0002 y n y n n.0000 n n y n n.0001 y n n y y n n n y y.0001 y n n y n n n n y n.0001 y n n n y n n n n y.0001 y n n n n n n n n n.996 Nevin L. Zhang (HKUST) Bayesian Networks Fall / 55 Probabilistic Modeling with Joint Distribution Joint probability distribution

6 Probabilistic Modeling with Joint Distribution Inference with joint probability distribution What is the probability of burglary given that Mary called, P(B=y M=y)? Compute marginal probability: P(B, M) = P(B, E, A, J, M) E,A,J B M Prob y y y n n y n n Compute answer (reasoning by conditioning): P(B=y M=y) = = P(B=y, M=y) P(M=y) = 0.61 Nevin L. Zhang (HKUST) Bayesian Networks Fall / 55

7 Probabilistic Modeling with Joint Distribution Advantages Probability theory well-established and well-understood. In theory, can perform arbitrary inference among the variables given a joint probability. This is because the joint probability contains information of all aspects of the relationships among the variables. Diagnostic inference: From effects to causes. Example: P(B=y M=y) Predictive inference: From causes to effects. Example: P(M=y B=y) Combining evidence: P(B=y J=y, M=y, E=n) All inference sanctioned by laws of probability and hence has clear semantics. Nevin L. Zhang (HKUST) Bayesian Networks Fall / 55

8 Probabilistic Modeling with Joint Distribution Difficulty: Complexity in model construction and inference In Alarm example: 31 numbers needed, Quite unnatural to assess: e.g. P(B = y, E = y, A = y, J = y, M = y) In general, Computing P(B=y M=y) takes 29 additions. [Exercise: Verify this.] P(X 1, X 2,..., X n ) needs at least 2 n 1 numbers to specify the joint probability. Exponential model size. Knowledge acquisition difficult (complex, unnatural), Exponential storage and inference. Nevin L. Zhang (HKUST) Bayesian Networks Fall / 55

9 Outline Conditional Independence and Factorization 1 Probabilistic Modeling with Joint Distribution 2 Conditional Independence and Factorization 3 Bayesian Networks 4 Manual Construction of Bayesian Networks Building structures Causal Bayesian networks Determining Parameters Local Structures 5 Remarks Nevin L. Zhang (HKUST) Bayesian Networks Fall / 55

10 Conditional Independence and Factorization Chain Rule and Factorization Overcome the problem of exponential size by exploiting conditional independence The chain rule of probabilities: P(X 1, X 2 ) = P(X 1 )P(X 2 X 1 ) P(X 1, X 2, X 3 ) = P(X 1 )P(X 2 X 1 )P(X 3 X 1, X 2 )... P(X 1, X 2,..., X n ) = P(X 1 )P(X 2 X 1 )...P(X n X 1,..., X n 1 ) n = P(X i X 1,..., X i 1 ). i=1 No gains yet. The number of parameters required by the factors is: 2 n n = 2 n 1. Nevin L. Zhang (HKUST) Bayesian Networks Fall / 55

11 Conditional Independence and Factorization Conditional Independence About P(X i X 1,..., X i 1 ): Then Domain knowledge usually allows one to identify a subset pa(x i ) {X 1,..., X i 1 } such that Given pa(x i), X i is independent of all variables in {X 1,..., X i 1} \ pa(x i), i.e. Joint distribution factorized. P(X i X 1,..., X i 1) = P(X i pa(x i)) P(X 1, X 2,..., X n ) = n P(X i pa(x i )) The number of parameters might have been substantially reduced. i=1 Nevin L. Zhang (HKUST) Bayesian Networks Fall / 55

12 Conditional Independence and Factorization Example continued P(B, E, A, J, M) = P(B)P(E B)P(A B, E)P(J B, E, A)P(M B, E, A, J) = P(B)P(E)P(A B, E)P(J A)P(M A)(Factorization) pa(b) = {}, pa(e) = {},pa(a) = {B, E }, pa(j) = {A},pa(M) = {A}. Conditional probabilities tables (CPT) B Y N P(B) M A P(M A) Y Y.9 N Y.1 Y N.05 N N.95 E Y N P(E) J A P(J A) Y Y.7 N Y.3 Y N.01 N N.99 A B E P(A B, E) Y Y Y.95 N Y Y.05 Y Y N.94 N Y N.06 Y N Y.29 N N Y.71 Y N N.001 N N N.999 Nevin L. Zhang (HKUST) Bayesian Networks Fall / 55

13 Conditional Independence and Factorization Example continued Model size reduced from 31 to =10 Model construction easier Fewer parameters to assess. Parameters more natural to assess:e.g. P(B = Y ), P(E = Y), P(A = Y B = Y, E = Y ), Inference easier.will see this later. P(J = Y A = Y ), P(M = Y A = Y ) Nevin L. Zhang (HKUST) Bayesian Networks Fall / 55

14 Outline Bayesian Networks 1 Probabilistic Modeling with Joint Distribution 2 Conditional Independence and Factorization 3 Bayesian Networks 4 Manual Construction of Bayesian Networks Building structures Causal Bayesian networks Determining Parameters Local Structures 5 Remarks Nevin L. Zhang (HKUST) Bayesian Networks Fall / 55

15 Bayesian Networks From Factorizations to Bayesian Networks Graphically represent the conditional independency relationships: construct a directed graph by drawing an arc from X j to X i iff X j pa(x i ) pa(b) = {}, pa(e) = {}, pa(a) = {B, E }, pa(j) = {A}, pa(m) = {A}. B P(B) E P(E) A P(A B, E) J P(J A) M P(M A) Also attach the conditional probability (table) P(X i pa(x i )) to node X i. What results in is a Bayesian network.also known as belief network, probabilistic network. Nevin L. Zhang (HKUST) Bayesian Networks Fall / 55

16 Formal Definition Bayesian Networks A Bayesian network is: An directed acyclic graph (DAG), where Each node represents a random variable And is associated with the conditional probability of the node given its parents. Recall: In introduction, we said that Bayesian networks are networks of random variables. Nevin L. Zhang (HKUST) Bayesian Networks Fall / 55

17 Bayesian Networks Understanding Bayesian networks Qualitative level: A directed acyclic graph (DAG) where arcs represent direct probabilistic dependence. B E A J M Absence of arc indicates conditional independence: A variable is conditionally independent of all its nondescendants given its parents. (Will prove this later.) The above DAG implies the following conditional independence relationships: B E; J B A; J E A; M B A; M E A; M J A The following are not implied: J B; J E; J M; B E A Nevin L. Zhang (HKUST) Bayesian Networks Fall / 55

18 Bayesian Networks Understanding Bayesian networks Quantitative (numerical) level: Conditional probability tables: B Y N P(B) M A P(M A) Y Y.9 N Y.1 Y N.05 N N.95 E Y N P(E) J A P(J A) Y Y.7 N Y.3 Y N.01 N N.99 A B E P(A B, E) Y Y Y.95 N Y Y.05 Y Y N.94 N Y N.06 Y N Y.29 N N Y.71 Y N N.001 N N N.999 Describe how parents of a variable influence the variable. Nevin L. Zhang (HKUST) Bayesian Networks Fall / 55

19 Bayesian Networks Understanding Bayesian Networks As a whole: A Bayesian network represents a factorization of a joint distribution. n P(X 1, X 2,...,X n ) = P(X i pa(x i )) Multiplying all the CPTs results in a joint distribution over all variables. i=1 Nevin L. Zhang (HKUST) Bayesian Networks Fall / 55

20 Example networks Bayesian Networks Network repository Software Bayesian Network Repository: Genie & Smile Network Repository: Netica Net Library: Hugin Case Studies: Genie & Smile: Free. Netica: Free version for small nets. Nevin L. Zhang (HKUST) Bayesian Networks Fall / 55

21 Outline Manual Construction of Bayesian Networks 1 Probabilistic Modeling with Joint Distribution 2 Conditional Independence and Factorization 3 Bayesian Networks 4 Manual Construction of Bayesian Networks Building structures Causal Bayesian networks Determining Parameters Local Structures 5 Remarks Nevin L. Zhang (HKUST) Bayesian Networks Fall / 55

22 Manual Construction of Bayesian Networks Building structures Procedure for constructing Bayesian network structures 1 Choose a set of variables that describes the application domain. 2 Choose an ordering for the variables. 3 Start with the empty network and add variables to the network one by one according to the ordering. 4 To add the i-th variable X i, 1 Determine a subset pa(x i ) of variables already in the network (X 1,..., X i 1 ) such that P(X i X 1,..., X i 1 ) = P(X i pa(x i )) (Domain knowledge is needed here.) 2 Draw an arc from each variable in pa(x i ) to X i. Nevin L. Zhang (HKUST) Bayesian Networks Fall / 55

23 Examples Manual Construction of Bayesian Networks Building structures Order 1: B, E, A, J, M pa(b) = {}, pa(e) = {}, pa(a) = {B, E }, pa(j) = {A}, pa(m) = {A}. Nevin L. Zhang (HKUST) Bayesian Networks Fall / 55

24 Examples Manual Construction of Bayesian Networks Building structures Order 2: M, J, A, B, E pa(m) = {}, pa(j) = {M}, pa(a) = {M, J}, pa(b) = {A}, pa(e) = {A, B}. Nevin L. Zhang (HKUST) Bayesian Networks Fall / 55

25 Examples Manual Construction of Bayesian Networks Building structures Order 3: M, J, E, B, A pa(m) = {}, pa(j) = {M}, pa(e) = {M, J}, pa(b) = {M, J, E }, pa(a) = {M, J, B, E }. Nevin L. Zhang (HKUST) Bayesian Networks Fall / 55

26 Manual Construction of Bayesian Networks Building structures Building Bayesian network structures Which order? Naturalness of probability assessment (Howard and Matheson). (B, E, A, J, M) is a good ordering because the following distributions natural to assess P(B), P(E): frequency of burglary and earthquake P(A B, E): property of Alarm system. P(M A): knowledge about Mary P(J A): knowledge about John. The order M, J, E, B, A is not good because, for instance, P(B J, M, E) is unnatural and hence difficult to assess directly. Nevin L. Zhang (HKUST) Bayesian Networks Fall / 55

27 Manual Construction of Bayesian Networks Building structures Building Bayesian network structures Which order? Minimize number of arcs (J. Q. Smith). The order (M, J, E, B, A) is bad because too many arcs. In contrast, the order (B, E, A, J, M) is good is because it results in a simple structure. Use causal relationships (Pearl): cause come before their effects. The order (M, J, E, B, A) is not good because, for instance, M and J are effects of A but come before A. In contrast, the order (B, E, A, J, M) is good is because it respects the causal relationships among variables. Nevin L. Zhang (HKUST) Bayesian Networks Fall / 55

28 Manual Construction of Bayesian Networks Exercise in Structure Building Building structures Five variable about what happens to an office building Fire: There is a fire in the building. Smoke: There is smoke in the building. Alarm: Fire alarm goes off. Leave: People leaves the building. Tampering: Someone tamper with the fire system (e.g., open fire exit) Build network structures using the following ordering. Clearly state your assumption. 1 Order 1: tampering, fire, smoke, alarm, leave 2 Order 2: leave, alarm, smoke, fire, tampering Nevin L. Zhang (HKUST) Bayesian Networks Fall / 55

29 Manual Construction of Bayesian Networks Causal Bayesian networks Building structures Build a Bayesian network using casual relationships: Choose a set of variables that describes the domain. Draw an arc to a variable from each of its DIRECT causes. (Domain knowledge needed here.) What results in is a causal Bayesian network, or simply causal networks, Arcs are interpreted as indicating cause-effect relationships. Nevin L. Zhang (HKUST) Bayesian Networks Fall / 55

30 Example: Manual Construction of Bayesian Networks Building structures Travel (Lauritzen and Spiegelhalter) Adventure Smoking Tuberculosis Lung Cancer Bronchitis X ray Dyspnea Nevin L. Zhang (HKUST) Bayesian Networks Fall / 55

31 Manual Construction of Bayesian Networks Use of Causality: Issue 1 Building structures Causality is not a well understood concept. No widely accepted definition. No consensus on Whether it is a property of the world, Or a concept in our minds helping us to organize our perception of the world. Nevin L. Zhang (HKUST) Bayesian Networks Fall / 55

32 Causality Manual Construction of Bayesian Networks Building structures Sometimes causal relations are obvious: Alarm causes people to leave building. Lung Cancer causes mass on chest X-ray. At other times, they are not that clear. Whether gender influences ability in technical sciences. Most of us believe Smoking cause lung cancer,but the tobacco industry has a different story: Surgeon General (1964) s c Tobacco Industry g s c Nevin L. Zhang (HKUST) Bayesian Networks Fall / 55

33 Manual Construction of Bayesian Networks Working Definition of Causality Building structures Imagine an all powerful agent, GOD, who can change the states of variables. Example: X causes Y if knowing that GOD has changed the state of X changes your believe about Y. Smoking and yellow finger are correlated. If we force someone to smoke for sometime, his finger will probably become yellow. So Smoking is a cause of yellow finger. If we paint someone s finger yellow, that will not affect our belief on whether s/he smokes. So yellow finger does not cause smoking. Similar example with Earthquake and Alarm Nevin L. Zhang (HKUST) Bayesian Networks Fall / 55

34 Causality Manual Construction of Bayesian Networks Building structures Coin tossing example revisited: Knowing that GOD somehow made sure the coin drawn from the bag is a fair coin would affect our belief on the results of tossing. Knowing that GOD somehow made sure that the first tossing resulted in a head does not affect our belief on the type of the coin. So arrows go from coin type to results of tossing. Coin Type Toss 1 Result Toss 2 Result... Toss n Result Nevin L. Zhang (HKUST) Bayesian Networks Fall / 55

35 Manual Construction of Bayesian Networks Use of Causality: Issue 2 Building structures Adventure Smoking Tuberculosis Lung Cancer Bronchitis X ray Dyspnea Causality network structure (building process) Network structure conditional independence (Semantics of BN) The causal Markov assumption bridges causality and conditional independence: A variable is independent of all its non-effects (non-descendants) given its direct causes (i.e. parents). We make this assumption if we determine Bayesian network structure using causality. Nevin L. Zhang (HKUST) Bayesian Networks Fall / 55

36 Manual Construction of Bayesian Networks Determining probability parameters Determining Parameters Later in this course, we will discuss in detail how to learn parameters from data. We will not be so much concerned with eliciting probability values from experts. However, people do that some times. In such a case, one would want the number of parameters be as small as possible. The rest of the lecture describe two concepts for reducing the number of parameters: Causal Independence. Context-specific independence. Left to students as reading materials. Nevin L. Zhang (HKUST) Bayesian Networks Fall / 55

37 Manual Construction of Bayesian Networks Determining probability parameters Determining Parameters Sometimes conditional probabilities are given by domain theory Genetic inheritance in Stud (horse) farm (Jensen, F. V. (2001). Bayesian networks and decision graphs. Springer.): P(Child Father, Mother) aa aa AA aa (1, 0, 0) (.5,.5, 0) (0, 1, 0) aa (.5,.5, 0) (.25,.5, 25) (0,.5,.5) AA (0, 1, 0) (0,.5,.5) (0, 0, 1) Genotypes: aa - sick, aa - carrier, AA - pure. Nevin L. Zhang (HKUST) Bayesian Networks Fall / 55

38 Manual Construction of Bayesian Networks Determining probability parameters Determining Parameters Sometimes, we need to get the numbers from the experts. This is a time-consuming and difficult process. Nonetheless, many networks have been built. See Bayesian Network Repository at Combine experts knowledge and data Use assessments by experts as a start point. When data become available, combine data and experts assessments. As more and more data become available, influence of experts is automatically reduced. We will show how this can be done when discussing parameter learning. Note: Much of the course will be about how to learning Bayesian networks (structures and parameters) from data. Nevin L. Zhang (HKUST) Bayesian Networks Fall / 55

39 Manual Construction of Bayesian Networks Reducing the number of parameters Determining Parameters Let E be a variable in a BN and let C 1, C 2,..., C m be its parents. C 1 C 2... C m E The size of the conditional probability P(E C 1, C 2,...,C m ) is exponential in m. This poses a problem for knowledge acquisition, learning, and inference. In application, there usually exist local structures that one can exploit to reduce the size of conditional probabilities Nevin L. Zhang (HKUST) Bayesian Networks Fall / 55

40 Manual Construction of Bayesian Networks Causal independence Determining Parameters Causal independence refers to the situation where the causes C 1, C 2..., and C m influence E independently. In other words, the ways by which the C i s influence e are independent. Burglary Earthquake Burglary Earthquake Alarm Alarm-due-to-Burglary (A b ) Alarm-due-to-Earthquake (A e ) Alarm Example: Burglary and earthquake trigger alarm independently. Precise statement: A b and A e are independent. A = A b A e, hence Noisy-OR gate (Good 1960). Nevin L. Zhang (HKUST) Bayesian Networks Fall / 55

41 Manual Construction of Bayesian Networks Determining Parameters Causal Independence C 1 C 2 C m ξ 1 ξ... 2 ξm * E Formally, C 1, C 2..., and C m are said to be causally independent w.r.t effect E if there exist random variables ξ 1, ξ 2..., and ξ m such that 1 For each i, ξ i probabilistically depends on C i and is conditionally independent of all other C j s and all other ξ j s given C i, and 2 There exists a commutative and associative binary operator over the domain of e such that. E = ξ 1 ξ 2... ξ m Nevin L. Zhang (HKUST) Bayesian Networks Fall / 55

42 Manual Construction of Bayesian Networks Causal Independence Determining Parameters C 1 C 2 C m ξ 1 ξ... 2 ξm * E In words, individual contributions from different causes are independent and the total influence on effect is a combination of the individual contributions. ξ i contribution of C i to E. * base combination operator. E independent cause (IC) variable. Known as convergent variable in Zhang & Poole (1996). Nevin L. Zhang (HKUST) Bayesian Networks Fall / 55

43 Manual Construction of Bayesian Networks Causal Independence Determining Parameters Example: Lottery C i : money spent on buying lottery of type i. E: change of wealth. ξ i : change in wealth due to buying the ith type lottery. Base combination operator: +. (Noisy-adder) Other causal independence models: 1 Noisy MAX-gate max 2 Noisy AND-gate Nevin L. Zhang (HKUST) Bayesian Networks Fall / 55

44 Manual Construction of Bayesian Networks Determining Parameters Causal Independence Theorem (2.1) If C 1, C 2,..., C m are causally independent w.r.t E,then the conditional probability P(E C 1,..., C m ) can be obtained from the conditional probabilities P(ξ i C i ) through P(E=e C 1,..., C m ) = α 1... α k =e P(ξ 1 =α 1 C 1 )...P(ξ m =α m C m ), (1) for each value e of E. Here is the base combination operator of E. See Zhang and Poole (1996) for the proof. Nevin L. Zhang (HKUST) Bayesian Networks Fall / 55

45 Manual Construction of Bayesian Networks Causal Independence Determining Parameters Notes: Causal independence reduces model size: In the case of binary variable, it reduces model sizes from 2 m+1 to 4m. Examples: CPSC, Carpo It can also be used to speed up inference (Zhang and Poole 1996). Relationship with logistic regression? (Potential term project) Nevin L. Zhang (HKUST) Bayesian Networks Fall / 55

46 Parent divorcing Manual Construction of Bayesian Networks Determining Parameters Another technique to reduce the number of parameters Top figure: A more natural model for the Adventure Smoking Travel example. But it requires =20 parameters. Tuberculosis X ray Adventure Tuberculosis X-ray Tuberculosis or Lung Cancer Lung Cancer Lung Cancer Dyspnea Smoking Dyspnea Bronchitis Bronchitis Low figure: requires only =18 parameters. The difference would be bigger if, for example, D have other parents. The trick is to introduce a new node (TB-or-LC). It divorces T and L from the other parent B of D. Note that the trick would not help if the new node TB-or-LC has 4 or more states. Nevin L. Zhang (HKUST) Bayesian Networks Fall / 55

47 Manual Construction of Bayesian Networks Determining Parameters Context specific independence (CSI) Let C be a set of variables. A context on C is an assignment of one value to each variable in C. We denote a context by C=c, where c is a set of values of variables in C. Two contexts are incompatible if there exists a variable that is assigned different values in the contexts. They are compatible otherwise. Nevin L. Zhang (HKUST) Bayesian Networks Fall / 55

48 Manual Construction of Bayesian Networks Context-specific independence Determining Parameters Let X, Y, Z, and C be four disjoint sets of variables. X and Y are independent given Z in context C=c if whenever P(Y, Z, C=c)>0. P(X Z,Y,C=c) = P(X Z,C=c) When Z is empty, one simply says that X and Y are independent in context C=c. Nevin L. Zhang (HKUST) Bayesian Networks Fall / 55

49 Manual Construction of Bayesian Networks Context-specific independence Determining Parameters Age Gender Number of Pregnancies Shafer s Example: Number of pregnancies (N) is independent of Age (A) in the context Gender=Male (G =m). P(N A, G=m) = P(N G=m) Number of parameters reduced by ( A 1)( N 1). Nevin L. Zhang (HKUST) Bayesian Networks Fall / 55

50 Manual Construction of Bayesian Networks Context-specific independence Determining Parameters Income independent of Weather in context Profession=Programmer. P(I W, P=Prog, Q) = P(I P=Prog, Q) Weather Profession Qualification (W) (P) (Q) Income independent of Qualification in context Profession=Farmer. Income (I) P(I W, P, Q) P(I W, P=Farmer, Q) = P(I W, P=Farmer) Number of parameters reduced by: ( W 1) Q ( I 1) + ( Q 1) W ( I 1) CSI can also be exploited to speed up inference (Zhang and Poole 1999). Nevin L. Zhang (HKUST) Bayesian Networks Fall / 55

51 Outline Remarks 1 Probabilistic Modeling with Joint Distribution 2 Conditional Independence and Factorization 3 Bayesian Networks 4 Manual Construction of Bayesian Networks Building structures Causal Bayesian networks Determining Parameters Local Structures 5 Remarks Nevin L. Zhang (HKUST) Bayesian Networks Fall / 55

52 Remarks Reasons for the popularity of Bayesian networks It s graphical language is intuitive and easy to understand because it captures what might be called intuitive causality. Pearl (1986) claims that it is a model for human s inferential reasoning: Notations of dependence and conditional dependence are basic to human reasoning. The fundamental structure of human knowledge can be represented by dependence graphs. Nevin L. Zhang (HKUST) Bayesian Networks Fall / 55

53 Remarks Reasons for the popularity of Bayesian networks In practice, the graphical language Functions as a convenient language to organizes one s knowledge about a domain. Facilitates interpersonal communication. On the other hand, the language is well-defined enough to allow computer processing. Correctness of results guaranteed by probability theory. For probability theory, Bayesian networks provide a whole new perspective: Probability is not really about numbers; It is about the structure of reasoning. (Glenn Shafer) Nevin L. Zhang (HKUST) Bayesian Networks Fall / 55

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