1. Introduction to graphical models

2. Outline Course goals and policies Overview of graphical models, examples Constraint networks Probabilistic networks and reasoning under uncertainty Brief review of probability basics Intro into Belief networks Slides courtesy Rina Dechter, Adnan Darwish, Alex Ihler and Andrew Moore

3. Outline Course goals and policies Overview of graphical models Constraint networks Probabilistic networks and reasoning under uncertainty Brief review of probability basics Intro into Belief networks

4. Course goals Focus on graphical models: - constraint networks - Bayesian (probabilistic, belief) networks - Markov networks - learning of models

5. 5 Recommended reading Rina Dechter, Constraint Processing, Morgan Kaufmann

6. 6 Recommended reading Adnan Darwiche, Modeling and Reasoning with Bayesian Networks MIT Press

7. Course requirements Homeworks : There will be 4-5 problem sets A term project: a programming project and/or a paper presentation Grading: Homeworks: 65% Project: 35%

8. Course outline The constraint network and its graphical representations and queries Constraint propagation algorithms; arc-consistency and unit propagation Inference algorithm for constraint networks Backtracking search algorithms; Bayesian Networks, Markov networks and their queries: probability of evidence, belief updating and most probable explanation The Belief Propagation algorithm and the basic of graph-based conditional independence Inference algorithms for probabilistic networks: Bucket elimination, join-tree algorithms Cutset-conditioning algorithms and approximation schemes (sampling and belief propagation) Learning Graphical models: Learning parameters Learning Graphical models: Learning structure

9. Outline Course goals and policies Overview of graphical models Constraint networks Probabilistic networks and reasoning under uncertainty Brief review of probability basics Intro into Belief networks

10. Graphical models

11. Constraint networks

12. Combinatorial optimization

15. Probabilistic (Bayesian) networks

19. Genetic linkage analysis

20. Sample domains for graphical models

21. Outline Course goals and policies Overview of graphical models Constraint networks Probabilistic networks and reasoning under uncertainty Brief review of probability basics Intro into Belief networks

22. Sudoku Each row, column and major block must be alldifferent “Well posed” if it has unique solution: 27 constraints 2 3 4 6 2 Constraint propagation Variables: 81 slots Domains = {1,2,3,4,5,6,7,8,9} Constraints: 27 not-equal

23. Sudoku Each row, column and major block must be alldifferent “Well posed” if it has unique solution

24. Constraint satisfaction problems (CSPs) Standard search problem: state is a "black box“ – any data structure that supports successor function, heuristic function, and goal test CSP: state is defined by variables X i with values from domain D i goal test is a set of constraints specifying allowable combinations of values for subsets of variables Simple example of a formal representation language Allows useful general-purpose algorithms with more power than standard search algorithms

25. AB redgreen redblack greenred greenblack blackgreen black red Constraint Satisfaction Example: map coloring Variables - countries (A,B,C,etc.) Values - colors (e.g., red, green, black) Constraints: C A B D E F G

26. Example: Map-Coloring Variables WA, NT, Q, NSW, V, SA, T Domains D i = {red,green,blue} Constraints: adjacent regions must have different colors e.g., WA ≠ NT, or (WA,NT) in {(red,green),(red,blue),(green,red), (green,blue),(blue,red),(blue,green)}

27. Example: Map-Coloring Solutions are complete and consistent assignments, e.g., WA = red, NT = green,Q = red,NSW = green,V = red,SA = blue,T = green

28. Constraint graph Binary CSP: each constraint relates two variables Constraint graph: nodes are variables, arcs are constraints

29. Varieties of CSPs Discrete variables n variables, domain size d  O(d n ) complete assignments e.g., Boolean CSPs, incl.~Boolean satisfiability (NP-complete) infinite domains: integers, strings, etc. e.g., job scheduling, variables are start/end days for each job need a constraint language, e.g., StartJob 1 + 5 ≤ StartJob 3 Continuous variables linear constraints solvable in polynomial time by linear programming e.g., start/end times for Hubble Space Telescope observations finite domains:

30. Varieties of constraints Unary constraints involve a single variable, e.g., SA ≠ green Binary constraints involve pairs of variables, e.g., SA ≠ WA Higher-order constraints involve 3 or more variables, e.g., cryptarithmetic column constraints

31. Example: Cryptarithmetic Variables: F T U W R O X 1 X 2 X 3 Domains: {0,1,2,3,4,5,6,7,8,9} Constraints: Alldiff (F,T,U,W,R,O) O + O = R + 10 · X 1 X 1 + W + W = U + 10 · X 2 X 2 + T + T = O + 10 · X 3 X 3 = F, T ≠ 0, F ≠ 0

32. Real-world CSPs Assignment problems e.g., who teaches what class Timetabling problems e.g., which class is offered when and where? Transportation scheduling Factory scheduling Notice that many real-world problems involve real- valued variables

33. Standard search formulation (incremental) Let's start with the straightforward approach, then fix it Successor function: assign a value to an unassigned variable that does not conflict with current assignment  fail if no legal assignments Goal test: the current assignment is complete 1.This is the same for all CSPs 2.Every solution appears at depth n with n variables  use depth-first search 3.Path is irrelevant, so can also use complete-state formulation 4.b = (n - l )d at depth l, hence n! · d n leaves States are defined by the values assigned so far Initial state: the empty assignment { }

34. Example The 4-queen problem Q Q Q Q Q Q Q Q Place 4 Queens on a chess board of 4x4 such that no two queens reside in the same row, column or diagonal. Standard CSP formulation of the problem: Variables: each row is a variable. Q Q Q Q 1 2 3 4 Constraint Graph :

35. Spring 201035 Example: The N-queens problem The network has four variables, all with domains Di = {1, 2, 3, 4}. (a) The labeled chess board. (b) The constraints between variables. Constraints: There are = 6 constraints involved ( ) 4 2 Domains:

36. Spring 201036 Example: configuration and design

37. Spring 201037 Configuration and design Want to build: recreation area, apartments, houses, cemetery, dump Recreation area near lake Steep slopes avoided except for recreation area Poor soil avoided for developments Highway far from apartments, houses and recreation Dump not visible from apartments, houses and lake Lots 3 and 4 have poor soil Lots 3, 4, 7, 8 are on steep slopes Lots 2, 3, 4 are near lake Lots 1, 2 are near highway

38. 38 Constraint Graphs: Primal, Dual and Hypergraphs A (primal) constraint graph: a node per variable arcs connect constrained variables. A dual constraint graph: a node per constraint’s scope, an arc connect nodes sharing variables =hypergraph

39. Outline Course goals and policies Overview of graphical models Constraint networks Probabilistic networks and reasoning under uncertainty Brief review of probability basics Intro into Belief networks

40. Example of common sense reasoning Explosive noise at UCI The missing garage door Years to finish an undergrad degree in college 40

41. 41 Why uncertainty Summary of exceptions Birds fly, smoke means fire (cannot enumerate all exceptions. Why is it difficult? Exception combines in intricate ways e.g., we cannot tell from formulas how exceptions to rules interact: A  C B  C --------- A and B -  C

42. 42 The problem All men are mortalT All penguins are birdsT … Socrates is a man Men are kindp1 Birds flyp2 T looks like a penguin Turn key –> car startsP_n Q: Does T fly? P(Q)? True propositions Uncertain propositions Logic?....but how we handle exceptions Probability: astronomical

43. 43 Managing Uncertainty Knowledge obtained from people is almost always loaded with uncertainty Most rules have exceptions which one cannot afford to enumerate Antecedent conditions are ambiguously defined or hard to satisfy precisely First-generation expert systems combined uncertainties according to simple and uniform principle Lead to unpredictable and counterintuitive results Early days: logicist, new-calculist, neo-probabilist

44. 44 Burglery Example Alarm Earthquake Burglery Radio Phone call A  B A more credible ------------------ B more credible IF Alarm  Burglery A more credible (after radio) But B is less credible Issue: Rule from effect to causes

45. Probabilistic networks Monitoring patients in intensive care