Real-Time Bayesian Networks Inference

1. KDD Group Presentation Real Time Bayesian Networks Inference Wednesday, March 14, 2001 Haipeng Guo KDD Research Group Department of Computing and Information Sciences Kansas State University

2. Presentation Outline • Bayesian Networks Introduction • Bayesian Networks Inference Algorithms Review • Real Time Related Issues • A Distributed Anytime Architecture for Probabilistic reasoning from Santos’ paper[Santos 1995] • Summary

3. Bayesian Networks Introduction • Definition • Why is it important? • Examples • Applications

4. Bayesian Networks • Bayesian Networks, also called Bayesian Belief networks, causal networks, or probabilistic networks, are a network-based framework for representing and analyzing causal models involving uncertainty • A BBN is a directed acyclic graph (DAG) with conditional probabilities for each node. • Nodes represent  random variables in a problem domain • Arcs  represent conditional dependence relationship among these variables.  • Each node contains a CPT(Conditional Probabilistic Table) that contains probabilities of this node being specific values given the values of its parent nodes.

5. Bowel-problem Family-Out Light-On Dog-out Hear-bark Family-Out Example • " Suppose when I go home at night, I want to know if my family is home before I try the doors.(Perhaps the most convenient door to enter is double locked when nobody is home.) Now, often when my wife leaves the houses, she turns on an outdoor light. However, she sometimes turns on the lights if she is expecting a guest. Also, we have a dog. When nobody is home, the dog is put in the back yard. The same is true if the dog has bowel problems. Finally, if the dog is in the back yard, I will probably hear her barking(or what I think is her barking), but sometimes I can be confused by other dogs. "

6. Asia Example from Medical Diagnostics

7. Why is BBN important? • Offers a compact, intuitive, and efficient graphical representation of dependence relations between entities of a problem domain. (model the world in a more natural way than Rule-based systems and neural network) • Handle uncertainty knowledge in mathematically rigorous yet efficient and simple way • Provides a computational architecture for computing the impact of evidence nodes on beliefs(probabilities) of interested query nodes • Growing numbers of creative applications

8. MINVOLSET KINKEDTUBE PULMEMBOLUS INTUBATION VENTMACH DISCONNECT PAP SHUNT VENTLUNG VENITUBE PRESS MINOVL FIO2 VENTALV PVSAT ANAPHYLAXIS ARTCO2 EXPCO2 SAO2 TPR INSUFFANESTH HYPOVOLEMIA LVFAILURE CATECHOL LVEDVOLUME STROEVOLUME ERRCAUTER HR ERRBLOWOUTPUT HISTORY CO CVP PCWP HREKG HRSAT HRBP BP Alarm Example: the power of BBN • The Alarm network • 37 variables, 509 parameters (instead of 237)

9. Applications • Medical diagnostic systems • Real-time weapons scheduling • Jet-engines fault diagnosis • Intel processor fault diagnosis (Intel); • Generator monitoring expert system (General Electric); • Software troubleshooting (Microsoft office assistant, Win98 print troubleshooting) • Space shuttle engines monitoring(Vista project) • Biological sequences analysis and classification • ……

10. Bayesian Networks Inference • Given an observed evidence, do some computation to answer queries • An evidence e is an assignment of values to a set of variables E in the domain, E = { Xk+1, …, Xn } • For example, E = e : { Visit Asia = True, Smoke = True} • Queries: • The posteriori belief: compute the conditional probability of a variable given the evidence, • P(Lung Cancer| Visit Asia = TRUE AND Smoke = TRUE) = ? • This kind of inference tasks is called Belief Updating • MPE: compute the Most Probable Explanation given the evidence • An explanation for the evidence is a complete assignment { X1 = x1, …, Xn= xn } that is consistent with evidence. Computing a MPE is finding an explanation such that no other explanation has higher probability • This kind of inference tasks is called Belief revision

11. Belief Updating • The problem is to compute P(X=x|E=e): the probability of query nodes X, given the observed value of evidence nodes E = e. For example: Suppose that a patient arrives and it is known for certain that he has recently visited Asia and has dyspnea. - What’s the impact that this evidence has on the probabilities of the other variables in the network ? P(Lung Cancer) = ? Smoking Visit to Asia Tuberculosis Lung Cancer tub. or lung cancer Bronchitis Dyspnea X-Ray

12. Belief Revision Let W is the set of all nodes in our given Bayesian network Let the evidence e be the observation that the roses are okay. Our goal is to now determine the assignment to all nodes which maximizes P(w|e). We only need to consider assignments where the node roses is set to okay and maximize P(w), i.e. the most likely “state of the world” given the evidence that rose is ok in “this world”. The best solution then becomes - P(sprinklers = F, rain = T, street = wet, lawn = wet, soil = wet, roses = okay) = 0.2646

13. Complexity of BBN Inference • Probabilistic Inference Using Belief Networks is NP-hard. [Cooper 1990] •  Approximating Probabilistic Inference in Bayesian Belief Networks is NP-hard [Dagum 1993] • Hardness does not mean we cannot solve inference. It implies that • We cannot find a general procedure that works efficiently for all networks • However, for particular families of networks, we can have provably efficient algorithms either exact or approximate • Instead of a general exact algorithm, we seek for special case, average case, approximate algorithms • Various of approximate, heuristic, hybrid and special case algorithms should be taken into consideration

14. BBN Inference Algorithms • Exact algorithms • Pearl’s message propagation algorithm(for single connected networks only) • Variable elimination • Cutset conditioning • Clique tree clustering • SPI(Symbolic Probabilistic Inference) • Approximate algorithms • Partial evaluation methods by performing exact inference partially • Variational approach by exploiting averaging phenomena in dense networks(law of large numbers) • Search based algorithms by converting inference problem to an optimization problem, then using heuristic search to solve it • Stochastic sampling also called Monte Carlo algorithms

15. PolyTree • Singly Connected Networks(or Polytrees) Definition : A directed acyclic graph (DAG) in which at most one undirected path exists between any two nodes. Multiple parents and/or multiple children Polytree structure satisfies definition Do not satisfy definition

16. Propagation Algorithm Objective Data Data • The algorithm’s purpose is “… fusing and propagating the impact of new evidence and beliefs through Bayesian networks so that each proposition eventually will be assigned a certainty measure consistent with the axioms of probability theory.” (Pearl, 1988, p 143)

17. PolyTree Propagation Example “The impact of each new piece of evidence is viewed as a perturbation that propagatesthroughthe network via message-passing betweenneighboring variables . . .” (Pearl, 1988, p 143)  Message to Parent  Message from Parent Data Data • Exact algorithm, for Polytree only, linear in the size of the network

18. conditioning Cutset Conditioning Algorithm • Transfer the network into several simpler polytrees by conditioning the cutset and then call the Polytree propagation algorithm. Each simple network has one or more variable instantiated to a definite value. P(X|E) is computed as a weighted average over the values computed by each polytree. [Pearl 1988] • A cutset is a set of nodes when instantiated will render the network single connected. • First exact algorithm for multiple connected networks, exponential time complexity in the size of the cutset. • There are exponentially many such cutset instantiations

19. A B Clique 1: {A, B, C, D} E C D Clique 2: {B, D, E} Clique Tree Clustering Algorithm • Transform the network into a tree of cliques, then computes probabilities for the cliques during a two-way message passing and the individual node probabilities P(X|E) are calculated from the probabilities of cliques • A clique W of G is a maximal complete subset of G, that is, there is no other complete subset of G which properly contains W • The most common used exact inference algorithm for general networks • Efficient for sparse networks, but could have a very bad performance for more general, dense networks • Exact, for multiple connected networks, exponential time complexity in the size of the network

20. Clique 1 Clique 4 Clq5 Clq3 Clq4 Clq2 Clq6 Clq1 Clique 2 Clique 3 G5 C4 C E B2 C4 B C4 C4 C4 G5 H E3 E F F6 G E3 A B D A1 B2 G5 A D8 H7 F F6 A1 H7 E3 G5 B2 D8 D G H E3 C Clique 5 Clique 6 Clique tree clustering Triangulation Moralization Identify Cliques , Message passing P(Clqi) and P(X|E) Form Clique Tree

21. Variable Elimination Algorithm • General idea: • Write query in the form • Iteratively • Move all irrelevant terms outside of innermost sum • Perform innermost sum, getting a new term • Insert the new term into the product • Computation depends on order of elimination, a “good” elimination orderings can reduce complexity • The size of the largest clique in the induced graph is thus an indicator for the complexity of variable elimination. This quantity is called the induced width of a graph according to the specified ordering • Finding an ordering that minimizes the induced width is NP-Hard • Exact, for all networks, exponential time complexity, inefficient

22. SPI(Symbolic Probabilistic Inference) General idea: • Transform BBN inference problem into a well-defined combinatorial optimization problem - the Optimal Factoring Problem(OFP). Thus the problem becomes to find an optimal factoring given a set of probability distribution. The solution of the OFP is then used to combine the CPT that describe the BBN and extract the desired marginal distribution. • OFP itself is NP-Hard. • Exact, for all networks, exponential time complexity, inefficient Factoring 1 needs 72 multiplications Factoring 2 needs only 28 multiplications

23. Approximate Algorithms • Exact Inference for large-scale networks is apparently infeasible. • Real life network can be up to thousands nodes. For example: QMR(Quick medical Reference) consists of a combination of statistical and expert knowledge for approximately 600 significant diseases and 4000 findings. The median size of the maximal clique of the moralized graph is 151.5 nodes. It’s intractable for all exact inference algorithms. • Approximate algorithms can be categorized into: • Partial evaluation methods by performing exact inference partially • Variational approach by exploiting averaging phenomena in dense networks(law of large numbers) • Search based algorithms by converting inference problem to an optimization problem, then using heuristic search to solve it • Stochastic sampling also called Monte Carlo algorithms

24. Perform Exact Algorithm Partially General idea: reduce the complexity by reducing the solution space Partial sets of nodes instantiation • Partial sets of hypotheses • Partial set of nodes • “Bounded conditioning”[Cooper 1991] • “Localized partial evaluation”[Draper 1994] • “incremental SPI”[D’Ambrosio 1993] • “Probabilistic partial evaluation”[Poole 1997] • “Mini-buckets” algorithm[Dechter 1997] • Approximate, for all networks, complexity not clear

25. Variational Method General idea: exploit averaging phenomena in dense graph • A sum can be avoided if it contains a sufficient number of terms such that a law of large numbers can be invoked • Graphically, the model is transformed into a sub-graph of the original model in which some of the finding nodes are delinked until it’s possible to run an exact algorithm on the resulting graph. [Jaakkola & Jordan 1999] • Approximate, efficient, for dense graph only

26. Search based algorithms • General idea: Convert the problem into an optimization problem then use heuristic search to solve it. • Consider node instantiations across the entire graph • Exploiting characteristics of problem domain to help search • A general hop is that a relatively small fraction of the exponentially many node instantiations contains a majority of the probability mass, and by exploring the high probability instantiations(bounding the unexplored probability mass) one can obtain reasonable bounds on posterior probabilities. • Cooper 1985, Peng & Reggia 1987, Henrion 1991 • Best-first search(A*), linear programming, genetic algorithm • Charniak 1994, Santos 1993, Carlos 1993 • Approximate, heuristic, maybe fail

27. Stochastic Sampling Algorithms • General idea: Run repeated simulations according to the BBN, the probability of an event of interest is estimated using the frequency with which that event occurs in a set of samples. • Logic sampling [henrion 1988] • forward sampling • backward sampling [Fung 1994] • Likelihood weighting [Fung & Chang 1990] • Important sampling [Shachter 1990] • Approximate, performance depends only on the CPTs, can handle very large networks, but has difficulty with extremely unlikely events.

28. Inference Algorithm Conclusions • The general problem of exact inference is NP-Hard. • The general problem of approximate inference is NP-Hard. • Exact inference works for small, sparse networks only. • No single champion either exact or inference algorithms. • The goal of research should be that of identifying effective approximate techniques that work well in large classes of problems. • Another direction is the integration of various kinds of approximate and exact algorithms exploiting the best characteristics of each algorithm.

29. A Distributed Anytime Inference Architecture “On a Distributed Anytime Architecture for Probabilistic Reasoning” Air Force Institute of Technology Eugene Santos Jr. , 1995

30. Anytime algorithms • To meet the demand for real-time inference, an inference algorithm must have two capibilities: • Provide a near optimal solution at any given moment • Improving upon solutions as more time and resources are allocated • Algorithms which have this property of producing a solution at any point in time are called “anytime” algorithms

31. Anywhere Algorithms • To exploit parallelism and distributed processing to reduce the time complexity, the tasks in the distributed environment must be able to exploit intermediate results produced by the other components of the system. • Algorithms with this property are called “anywhere” algorithms. • When different algorithms having both anytime and anywhere properties are harnessed together into a cooperative system, the resultant architecture can exploit the best characteristics of each algorithm.

32. The OVERMIND Architecture • Part of PESKI, an online expert system for engine diagnosis for the Space Shuttle Program • Three components: • IRA(Intelligent Resource Allocator) • Manages and allocates available computing resources • OVERSEER(Overseer task Manager) • Initiates new tasks, directs messages/information • LOTS(Library of Tasks) • A set of BBN inference algorithms suitable for performing various including an A* search algorithm, a genetic algorithm, an integer linear programming algorithm and a hybrid stochastic algorithm(HySS)

33. General Idea • The best algorithm to use is problem-instance dependent. • In a set of anywhere algorithms, if each particular algorithm is good at certain portion of a problem we can then take the partial solution of an algorithm and pass it to another approach which itself works better on the new portion • This leads to an anytime anywhere solution

34. Genetic Algorithms • A heuristic search algorithm modeled after natural genetic evolutions • Has anytime and anywhere property. • No stopping criterion that guarantees an optimal answer. • Its ability to generate solutions early can serve as a starting point if possible for other deterministic algorithm.

35. Best-First Search(A*) • A heuristic algorithm searching for optimal solution from initial state • Provide an approximate answer when interrupted • Allow the algorithm to accept initial guess from another sources • Use Best-first search to find the most probable complete instantiation among those compatible with the guess

36. IRA(Intelligent Resource Allocator) • Serve to maximize processor use by coordinating requests for resources from OVERSEER and the tasks themselves. • Hardware: a network of workstations • Identify resource requirements for different tasks • GA: single CPU • ILP: multi processing

37. The OVERSEER(Task manager) • Currently simple messager role • Advance capabilities involve deliberation scheduling: employing meta-reasoning to consider what computational tasks to execute. • To do this, some estimate of runtime and quality of results should be available foe each algorithm.

38. Implementation and results • The strengths of different methods are combined together: • Gas produce reasonable solution immediately • A* took those solutions near some maximas • HySS fine-tuned those maximas • ILP finished the optimization and generated te optimal solution • Result: • Initial test: multiple instances of GAs • GAs: 20% speed up • HySS: 3%~5% speed up • A* and ILP: 15~25% speed up

39. Summary • Exploited the anytime anywhere properties of several inference algorithms such as Gas, ILP and A* and unified them into a single model of parallel computation. • The architecture can use the best characteristics of each algorithm.

40. Future Research • Consider more algorithms • Study the relationship between the problem domain and the corresponding solutions domain to help deliberation scheduling.

41. The End • Any Questions ?

42. Linear Programming • The problem of finding the most probable explanation has been transformed into an integer linear programming problem with a set of constraints to satisfied. • Efficient algorithms for linear programming can be used to compute the optimal solution