Uncertain Reasoning. Environment Description. Configurations. Models. Bayesian Networks

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1 Bayesian Networks A. Objectives 1. Basics on Bayesian probability theory 2. Belief updating using JPD 3. Basics on graphs 4. Bayesian networks 5. Acquisition of Bayesian networks 6. Local computation and message passing B. References a. Xiang (Chapter 2) b. Jensen & Nielsen (Chapter 3) c. Darwiche (2.3) d. Fenton & Neil (Chapter 7) Y. Xiang, CIS 6120, Uncertain Reasoning 1 Environment Description A. An agent can represent the state of its environment through a set V of variables. B. Ex A simple digital circuit C. Each variable x V is associated with a domain D x. D x may be discrete or continuous. Y. Xiang, CIS 6120, Uncertain Reasoning 2 Configurations A. We extend propositional logic to allow variables and AtomicSentence x = x i, where x i D x. B. Write a conjunctive sentence (a = a i ) (z = z k ) over a subset X V of variables as a configuration x = (a = a i,, z = z k ) or x = (a i,, z k ). 1. What does it represent? 2. It is also called a variable assignment or instantiation. C. The space of a subset X V of variables is the set D X of all possible configurations over X. D X = Cartesian product of domains of variables in X Models A. A model (or possible world) is a particular state of the environment. A model m is a function that maps each variable x to a value m(x) D x. B. Ex m 0 : a=hi, b=lo, c=hi, d= hi, e=lo, f=lo, h=hi, and g i =norm for i = 1, 2, 3, 4. C. A model is a complete (over V) configuration. D. Denote the set of all models of an env by. Ex For the circuit env, what is? Y. Xiang, CIS 6120, Uncertain Reasoning 3 Y. Xiang, CIS 6120, Uncertain Reasoning 4 1

2 Sentences and Models A. From a model m, the truth of any sentence is uniquely determined. Ex = (g 4 = norm) (c=hi) (d=lo). What is the truth of in m 0? B. If a sentence is true in a model m, we say that m satisfies or m is a model of. Ex Relation btw sentence and model m 0 C. Denote the set of all models of by M( ). 1. Ex What is M( )? 2. Illustration of M( ) by Venn diagram Y. Xiang, CIS 6120, Uncertain Reasoning 5 Logic and Set Operations on Models A. For sentences and, the following holds. 1. M( ) = M( ) M( ) 2. M( ) = M( ) M( ) 3. M( ) = \M( ) = M(α) 4. and are equivalent iff M( ) = M( ). 5. and are mutually exclusive iff M( ) M( ) =. 6. and are exhaustive iff M( ) M( ) =. 7. iff M( ) M( ). Y. Xiang, CIS 6120, Uncertain Reasoning 6 Possible and Compatible Configurations A. A model or configuration is impossible if it violates dependency among its variables. B. Configurations x and y are compatible iff for every v X Y, its value in x equals its value in y. 1) Can two models of an env be compatible? 2) If x and y are compatible, what is the relation between M(x) and M(y)? 3) What if x and y are incompatible? Encoding Uncertain Knowledge A. To infer the state of an environment, we rely on dependence btw observable and unobservable. How do we represent uncertain dependence? B. Bayesian approach: P(X=x Y=y) 1. Function P: D X D Y [0,1] 2. What does it mean? C. Ex Patient w walks into the office of doctor z. What is p = P z (w has TB B z )? D. Ex Patient w walks in with a positive X-ray film. What is q = P z (w has TB B z )? Y. Xiang, CIS 6120, Uncertain Reasoning 7 Y. Xiang, CIS 6120, Uncertain Reasoning 8 2

3 Probability Distributions 1. Conditional probability P(x y) 2. Conditional probability distribution (CPD) P(X y) 3. Conditional probability table (CPT) P(X Y) 4. Unconditional probability (distribution) P(X) 5. Joint probability distribution (JPD) P(V) 6. Marginal probability distribution P(X) Ex Conditional Probability Table idx h a e g 2 P(h a,e,g 2 ) idx h a e g 2 P(h a,e,g 2 ) norm abnm norm abnm norm abnm norm abnm norm abnm norm abnm norm abnm norm abnm 0.8 Y. Xiang, CIS 6120, Uncertain Reasoning 9 Y. Xiang, CIS 6120, Uncertain Reasoning 10 Axioms of Probability A. Probability assignment must satisfy a set of axioms. 1. X, Y, Z: nonempty, disjoint subsets of V 2. x D X, y D Y, and z D Z B. Range: 0 P(x y) 1 C. Certainty: P(x x) = 1 D. Sum: If x 1 and x 2 are incompatible, P(x 1 x 2 z) = P(x 1 z) + P(x 2 z). E. Product: P(x,y z) = P(x y,z) P(y z) Since it holds for all x, y and z, we often write P(X,Y Z) = P(X Y,Z) P(Y Z). Theorems of Probability A. From axioms, rest of probability theory is derivable. B. Bayes rule: P(X Y,Z) = P(Y X, Z)P(X Z)/P(Y Z). C. Negation: P(X x y) = 1 - P(X=x y). D. Marginalization: For Y = X W and X W =, P(x z) = P( x, w z). w D w Since the above holds for all x, w and z, we write P(X Z) = W P(X,W Z) = W P(Y Z). Y. Xiang, CIS 6120, Uncertain Reasoning 11 Y. Xiang, CIS 6120, Uncertain Reasoning 12 3

4 Probabilistic Inference A. Reasoning task: What is the state of agent env given the percept (observations)? B. The Bayesian theory distinguishes between the prior distribution over X V and the posterior distribution over X after observing Y = y. C. Probabilistic inference is the process to update prior into posterior after observation. D. In practice, probabilistic inference is performed recursively. Ex Dental Hygiene A. Bad habit in tooth cleaning increases chances of cavity which in turn cause toothache. B. V={hygiene, cavity, toothache}, where hygiene {g, b}, and cavity, toothache {y, n}. C. Given P(h, c, t) and percept toothache = yes, what is the posterior P(h t=y)? h c t P(h,c,t) good Yes yes good Yes no good No yes good no no bad yes yes bad yes no bad no yes bad no no Y. Xiang, CIS 6120, Uncertain Reasoning 13 Y. Xiang, CIS 6120, Uncertain Reasoning 14 Inference Using JPD A. Query: What is P(h t=y)? B. Algorithm 1. Constrain P(h,c,t) into P(h,c,t=y). 2. Compute P(t=y) by marginalization. 3. Condition P(h,c,t=y) into P(h,c,t t=y) by product. 4. Marginalize out c and t to get P(h t=y). C. Normalization: Steps 2 and 3 above Normalization constant: = 1/P(t=y) D. Hence, P(h t=y) = c,t P(h,c,t=y). Complexity of Inference Using JPD A. Suppose V =n, D x k for x V, and V =V\{x}. B. How many probability parameters are needed to specify P(V) in O() notation? Acquisition intractability C. How may parameters in P(V) define P(V x=x 0 )? 1. Agent must process these parameters to compute P(y x=x 0 ) for any y V. 2. Updating intractability D. How many summations are needed to get P(y x=x 0 ) from P(V x=x 0 )? Marginalization intractability Y. Xiang, Uncertain Reasoning with PGM 15 Y. Xiang, Uncertain Reasoning with PGM 16 4

5 Explore Independence A. Why is probabilistic inference using JPD intractable? B. Ex To know whether g 3 is normal, is it useful to observe c after observing b, d, and f? C. Observation: Not every env variable is directly dependent on everyone else. D. Ex If d is not observable, is it useful to observe c after observing only b and f? Encode Independence by Graph A. Key to efficient probabilistic inference Encode dependence (independence) relations among variables effectively B. Encoding dependence relations with graph 1) Direct dependence 2) Indirect dependence 3) Asymmetric or causal dependence 4) Unconditional independence 5) Conditional independence Y. Xiang, CIS 6120, Uncertain Reasoning 17 Y. Xiang, CIS 6120, Uncertain Reasoning 18 Basics on Graphs (Quick Review) 1) Graphs G = (V, E) 2) Undirected graphs: E = {<u,v> u,v V, u v} 3) Direct graphs: E = {(u,v) u,v V, u v} Head and tail; parent and child; 4) Degree, in-degree, and out-degree 5) Internal and terminal nodes; root and leaf; 6) Adjacent nodes and adjacency adj(v) 7) Path, simple path, directed path, ancestor and descendant 8) Cycles, directed, undirected, and DAGs 9) Singly connected and multiply connected graphs 10)Subgraphs Y. Xiang, CIS 6120, Uncertain Reasoning 19 Conditional Independence (CI) A. Let X, Y, and Z be disjoint subsets of variables in V. X and Y are conditionally independent given Z, denoted by I(X,Z,Y), iff for every x, and for every y and z such that P(y,z)>0, we have P(x y,z) = P(x z). B. Order of X & Y is unimportant: I(X,Z,Y) I(Y,Z,X). C. Only-if part is key to tackle acquisition intractability. D. Ex Dental hygiene: Acquire P(h,c,t) given I(h,c,t). E. If part yields a numerical test of CI, but cannot be relied on to tackle acquisition intractability. F. Ex Does I(h,c,t) holds for the given P(h,c,t)? Y. Xiang, CIS 6120, Uncertain Reasoning 20 5

6 Establishing Conditional Independence A. Without numerical test, how can we establish CI? B. Causality Given direct causes, a variable is conditionally independent of all others except its own effects. C. Mental test of CI D. Tractable acquisition 1) Identify CI by causality. 2) Acquire local distributions over cause & effect. 3) Define JPD from CI and local distributions. Encoding CI with DAG A. A Bayesian network is a triplet (V,G,P). 1. V is the set of variables of an environment. 2. G is a DAG whose nodes map one-to-one to members of V s.t. each variable is conditionally independent of its non-descendants given its parents. 3. P = { P(v (v)) v V} is a set of CPTs where (v) is the set of parents of v in G. B. Ex BN for a digital circuit Y. Xiang, CIS 6120, Uncertain Reasoning 21 Y. Xiang, CIS 6120, Uncertain Reasoning 22 An Example BN 1. How many parameters are needed to specify JPD directly? 2. How many are needed to specify BN? 3. How do we get JPD from the BN? Y. Xiang, CIS 6120, Uncertain Reasoning 23 JPD of a Bayesian Network A. Theorem [Chain rule] Let (V,G, P) be a BN, then P(V) = v V P(v (v)). B. Proof sketch (induction on V ) 1. Base case for V =1; Inductive assumption for V = n-1; 2. General case for V = n; Pick leaf a; Define BNs S and S ; 3. Apply inductive assumption to P(V ) 4. Apply product rule to P(V) C. A BN uniquely and compactly specifies a JPD. D. Is acquisition intractability resolved? Y. Xiang, CIS 6120, Uncertain Reasoning 24 6

7 How To Specify Bayesian Networks A. Main steps 1. Identify hypothesis variables 2. Define dependence structure 3. Determine variable domains 4. Specify CPTs B. Identify unobservable variables of primary concern a. Diagnosis: diseases of patient b. Web search: desirability of web page for given query c. Tutoring: knowledge state of student C. These variables drive structure specification. Defining Network Structure A. Specify a large network by divide and conquer. 1. Specify a number of network fragments. 2. Connect fragments into a single structure. B. Fragment specification can often be aided by reusing certain natural patterns (idioms). a. An idiom consists of a set of generic variables connected into a DAG. b. An idiom instantiation is an idiom with variables relabeled according to a specific application. Y. Xiang, CIS 6120, Uncertain Reasoning 25 Y. Xiang, CIS 6120, Uncertain Reasoning 26 Cause-Consequence Idiom A. Model a causal relation between a cause and its consequence. cause consequence B. Ex Idiom instantiations 1) Productive 2) Physical 3) Intentional C. An instantiation may have multiple causes. D. Support both predictive and diagnostic inference. Measurement Idiom A. Model uncertainty on an attribute due to sensor inaccuracy. attributevalue sensoraccuracy sensedvalue B. Ex Idiom instantiation on product testing C. Support diagnostic inference. Y. Xiang, CIS 6120, Uncertain Reasoning 27 Y. Xiang, CIS 6120, Uncertain Reasoning 28 7

8 Synthesis Idiom A. Model definitional relations. factor_1 factor_n syntheticfactor B. Ex Idiom instantiation on system reliability C. The relation may be deterministic or uncertain. D. Ex Instantiation on velocity Induction Idiom A. Model learning about a population parameter, in order to make predictions about it in the future, taking into account of contextual differences. populationparameter context obs_1 obs_n forecast B. Ex Idiom instantiation on call center monitoring A manager monitors lost calls for 8 days and predicts lost calls for the 9 th day. Y. Xiang, CIS 6120, Uncertain Reasoning 29 Y. Xiang, CIS 6120, Uncertain Reasoning 30 Specifying Variable Domains A. Associate each variable with a set of values that are mutually exclusive and exhaustive. 1) Testing mutual exclusiveness 2) Testing exhaustiveness B. Clarity criterion a) Values must be defined sufficiently precisely so that which is true can be practically determined. b) Ex Body Temperature Y. Xiang, CIS 6120, Uncertain Reasoning 31 Evaluating Probability Distributions 1. Follow well-founded theory if one exists. Ex One s genotype is aa (property shows), aa (carrier), or AA (non-carrier). A child takes one gene (a or A) from father s genotype (F) and one from mother s (M) with equal chance. What is P(C F,M)? F\M aa aa AA aa aa AA C aa aa AA aa aa AA aa aa AA 2. Use empirical frequency if available. 3. Use subjective estimate when necessary. Y. Xiang, CIS 6120, Uncertain Reasoning 32 8

9 Belief Updating By Message Passing A. Graphical structures help to copy with acquisition intractability. Can they help to copy with updating intractability and marginalization intractability? B. General idea 1. Avoid updating JPD directly and marginalizing it. 2. Update local distributions and propagate impact by passing messages along graphical structures. Ex Message Passing in Chain A. Ex BN for toothache P(h=good)=0.7 habit cavity p(c=yes h=good)=0.1 p(c=yes h=bad)=0.8 B. Compute P(h t=y) by message passing 1. Message P(t=y c) from t to c 2. Message P(t=y h) from c to h 3. Compute result at h p(t=yes c=yes)=0.85 p(t=yes c=no)=0.05 toothache Y. Xiang, CIS 6120, Uncertain Reasoning 33 Y. Xiang, CIS 6120, Uncertain Reasoning 34 Message Passing in Trees A. The - message passing [Pearl 88] computes the posterior for every node in a tree-structured BN with linear time. B. Not all envs admit tree dependence structures. Ex BN for a digital circuit C. When - message passing is applied to multiply connected BNs, there is no theoretical guarantee for exact posteriors. D. Experimental study obtained posteriors highly correlated with correct posteriors for some BNs but oscillated outcomes for others. Y. Xiang, CIS 6120, Uncertain Reasoning 35 Demo: Building Alarm A. Ex A building is typically normal but tampering or fire occurs occasionally. Either may trigger alarm which prompts occupants to evacuate. Verbal reports may reach security guard when gathering occurs outside the building. B. Separate inference engine from knowledge base C. Predictive inference D. Diagnostic inference E. Explaining away Y. Xiang, CIS 6120, Uncertain Reasoning 36 9