Causal and Bayesian Network (Chapter 2)

1. Causal and Bayesian Network(Chapter 2) Book: Bayesian Networks and Decision Graphs Author: Finn V. Jensen, Thomas D. Nielsen CSE 655 Probabilistic Reasoning Faculty of Computer Science, Institute of Business Administration Presented by Quratulain

2. Outline • Reasoning under uncertainty • Causal network and d-separation • Bayesian network • Graphical model Quratulain

3. Reasoning Under Uncertainty Why Reason Probabilistically? • In many problem domains it isn't possible to create complete, consistent models of the world. • If information is given with certainty then Propositional logic (Truth table) can be used. • Want to make rational decisions even when there is not enough information to prove that an action will work. • To deal with uncertain events, we extend truth value of propositional logic to certainties which are number between 0 and 1. Quratulain

4. Example (Type of reasoning that human do daily) “In the morning, my car will not start.” Reasons: • I can here starter tune, so must be power in battery • May be fuel has been stolen overnight • The spark plug are dirty • May be due to the dirt in carburetor • A loose connection in ignition system or any thing serious Quratulain

5. A Causal Perspective – Car Example • Construct a graph to represent causal relationship between events which gives structure to the situation for reasoning. Quratulain

6. Outline • Reasoning under uncertainty • Causal network and d-separation • Bayesian network • Graphical model Quratulain

7. Causal network and d-separation • A causal network consists of a set of variables and a set of directed links between variables. • Mathematically, the structure is called a directed graph. • Causal networks can be used to follow how a change of certainty in one variable may change the certainty for other variables. Quratulain

8. 3-Cases of evidence transmition • Serial Connections • Diverging Connections (common cause) • Converging Connection (common effect) P(C|A^B)=P(C|B) P(C|A^B)=P(C|B) Quratulain

9. Serial Connections • Evidence about A will influence the certainty of B, which then influences the certainty of C. • Similarly, evidence about C will influence the certainty of A through B. • If the state of B is known, then the channel is blocked, A and C become independent. • we say that A and C are d-separated given B. Quratulain

10. Diverging Connections • Influence can pass between all the children of A if A is not known. That is, B,C, . . . , E are d-separated given A. Sex (male, female), length of hair (long, short), and stature (<168 cm, ≥168 cm) Quratulain

11. Converging Connection • If nothing is known about A then the parents are independent evidence about one of them cannot influence the certainties of the others through A. Quratulain

12. D-separation • Two distinct variables A and B in a causal network are d-separated such that either: • The connection is serial or diverging and V is instantiated. • The connection is converging, and neither V nor any of V ’s descendants have received evidence. Quratulain

13. Example • Are B and C independent given A? • Are B and C independent given F Quratulain

14. Markov Blanket • The Markov blanket of a variable A is the set consisting of: • the parents of A, • the children of A, and • the variables sharing a child with A. • The Markov blanket has the property that when instantiated, A is d-separated from the rest of the network. Quratulain

15. Outline • Reasoning under uncertainty • Causal network and d-separation • Bayesian network • Graphical model Quratulain

16. Bayesian Network • A Bayesian network consists of the following • A set of variables and a set of directed edges between variables. • Each variable has a finite set of mutually exclusive states. • The variables together with the directed edges form an acyclic directed graph. • To each variable A with parents B1, . . . , Bn, a conditional probability table P(A|B1, . . . , Bn) is attached. Quratulain

17. Bayesian Network • The probabilities to specify are: • P(A), P(B), • P(C | A,B), • P(E |C), P(D|C), • P(F |E), and P(G| D,E,F) • It has been claimed that prior probabilities are bias to the model • Prior probabilities are necessary because prior certainty assessments are an integral part of human reasoning about certainty • The model should not include conditional independences that do not hold in the real world. • The d-separation properties check’s Conditional independences in model. Quratulain

18. Chain Rule for Bayesian Network • Let BN be a Bayesian network over U= {A1, . ..,An}. Then BN specifies a unique joint probability distribution P(U) given by the product of all conditional probability tables specified in BN: Quratulain

19. Outline • Reasoning under uncertainty • Causal network and d-separation • Bayesian network • Graphical model Quratulain

20. Graphical Model • Graphical specification is easy for humans to read, and helps focus attention. • The basic property of the Bayesian networks is the chain rule for compact representation of joint probability distribution. • Graphical model represents a causal relation in a knowledge domain. Quratulain

21. Questions Quratulain