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Multi-Entity Bayesian Networks Without Multi-Tears

Multi-Entity Bayesian Networks Without Multi-Tears. Bayesian Networks Seminar Jan 3-4, 2007. Limitations of Bayesian networks. Not expressive enough for many real world applications. fixed number of attributes varying numbers of related entities of different types.

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Multi-Entity Bayesian Networks Without Multi-Tears

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  1. Multi-Entity Bayesian Networks Without Multi-Tears Bayesian Networks Seminar Jan 3-4, 2007

  2. Limitations of Bayesian networks Not expressive enough formany real world applications. • fixed number ofattributes • varying numbers of relatedentities of different types.

  3. Systems based on first-orderlogic (FOL) • the ability to represent entities ofdifferent types interacting with each other in variedways. • “has enoughexpressive power to define all of mathematics .. and thesemantics of every version of logic, including itself” • lack a theoretically principled, widely accepted, logically coherent methodology for reasoning under uncertainty.

  4. Multi-entity Bayesiannetworks (MEBN) • a knowledge representationformalism that combines the expressive power offirst-order logic with a sound and logically consistenttreatment of uncertainty. • not a computer language or anapplication.

  5. Multi-entity Bayesiannetworks (MEBN) (cont..) A formal system that instantiates first-order Bayesianlogic. • MEBN provides syntax, a set of modelconstruction and inference processes, and semanticsthat together provide a means of defining probabilitydistributions over unbounded and possibly infinitenumbers of interrelated hypotheses. • As such, MEBNprovides a logical foundation for the many emerginglanguages that extend theexpressiveness of Bayesiannetworks.

  6. An example: Star Trek • illustrate thelimitations of standard BNs for situations thatdemand a more powerful representation formalism.

  7. Decision Support Systems in the 24th Century

  8. The Basic Starship Bayesian Network

  9. Problems • little use in a real life starship environment

  10. The BN for Four Starships

  11. Limitations of Bayesian networks • BNs lack the expressive power to represententity types (e.g., starships) that can be instantiated as many times as required for the situation at hand.

  12. The Premise • the likelihood ratio for ahigh MDR is 7/5 = 1.4 in favor of a starship in cloakmode. • Although this favors a cloaked starship in thevicinity, the evidence is not overwhelming.

  13. Repetition • Repetition is a powerful way to boost thediscriminatory power of weak signals. • an example: • from airport terminal radars, a single pulse reflectedfrom an aircraft usually arrives back to the radarreceiver very weakened, making it hard to set apartfrom background noise. • However, a steady sequenceof reflected radar pulses is easily distinguishable frombackground noise.

  14. Repetition (cont..) • Following the same logic, it isreasonable to assume that an abnormal backgrounddisturbance will show random fluctuation, whereas adisturbance caused by a starship in cloak mode wouldshow a characteristic temporal pattern. • Thus, whenthere is a cloaked starship nearby, the MDR state atany time depends on its previous state.

  15. The BN for One Starship with Recursion

  16. Temporal Recursion • DBNs • PDBN • a more generalrecursion capability is needed, as well as aparsimonious syntax for expressing recursiverelationships.

  17. Using MEBN Logic • a more “realistic” sci-fi scenario. • different alien species: • Friends, Cardassians,Romulans, and Klingons while addressing encounterswith other possible races using the general labelUnknown. • consider each starship’s type, offensive power, the ability of inflict harm to the Enterprise given its range, and numerous other features pertinent to the model’s purpose.

  18. Using MEBN Logic (cont..) • MEBN logic represents the world as comprised ofentities that have attributes and are related to otherentities. • Random Variables.

  19. Using MEBN Logic (cont..) • Knowledge about attributes and relationships is expressed as a collection of MFrags organized into MTheories. • MEBN Fragments (MFrags). • represents a conditional probability distribution forinstances of its resident RVs given their parents in thefragment graph and the context nodes. • MEBN Theories (MTheories). • a set of MFrags that collectively satisfiesconsistency constraints ensuring the existence of aunique joint probability distribution over instances ofthe RVs represented in each of the MFrags within theset.

  20. The DangerToSelf MFrag

  21. Using MEBN Logic (cont..) • Nodes: • Contex nodes. • Input nodes. • Resdent nodes. • Arguments. • Unique identifier • Exclamation point.

  22. Using MEBN Logic (cont..) • Instances of the RV. • HarmPotential(!ST1, !T1), HarmPotential(!ST2,!T1) • Home MFrag. • Local distribution • Boolean context nodes. • True, False, or Absurd. • Relevant only for deciding whether to use a resident random variable’s local distribution or its default distribution.

  23. Using MEBN Logic (cont..) • No probability values are shown for the states of the nodes. • a node in an MFrag represents a generic class of random variables. • Identify all the instances • none • pseudo code.

  24. Using MEBN Logic (cont..) • Local distributions in standard BNs are typically represented by static tables, which limits each node to a fixed number of parents. • An instance of a node in an MTheory might have any number of parents. • MEBN implementations(i.e. languages based on MEBN logic) must provide an expressive language for defining local distributions.

  25. An Instance of the DangerToSelf MFrag

  26. An Instance of the DangerToSelf MFrag (cont..) • the belief for state Unacceptable is .975 (.90 + .025*3) and the beliefs for states High, Medium, and Low are .02 ((1-.975)*.8), .005 ((1-.975)*.2), and zero respectively.

  27. Using MEBN Logic (cont..) • more complex knowledge patterns could be accommodated as needed to suit the requirements of the application. • MEBN logic has built-in logical MFrags that provide the ability to express anything that can be expressed in first-order logic.

  28. Recursive MFrags • One of the main limitations of BNs is their lack of support for recursion. • MEBN provides theoretically grounded support for very general recursive definitions of local distributions.

  29. Recursive MFrags (cont..) • MEBN logic allows influences between instances of the same random variable. • The recursion is grounded by specifying an initial distribution at time !T0 that does not depend on a previous magnetic disturbance.

  30. Recursive MFrags (cont..) • How recursive definitions can be applied to construct a situation-specific Bayesian Network (SSBN) to answer a query. • Example: • !Z0, !T3, !ST0 and !ST1 • ZoneMD(!Z0,!T3)

  31. The Zone MFrag

  32. SSBN Constructed from Zone MFrag

  33. Recursive MFrags (cont..) • Steps: • begin by creating an instance of the home MFrag of the query node ZoneMD(!Z0,!T3). • build any CPTs we can already build. • recursively creating instances of the home MFrags until we have added all the nodes.

  34. Building MEBN models with MTheories • MFrags provide a flexible means to represent knowledge about specific subjects within the domain of discourse. • but the true gain in expressive power is revealed when we aggregate these “knowledge patterns” to form a coherent model of the domain of discourse that can be instantiated to reason about specific situations and refined through learning. • just collecting a set MFrags that represent specific parts of a domain is not enough to ensure a coherent representation of that domain. • a set of MFrags with cyclic influences

  35. Building MEBN models with MTheories (cont..) • In order to build a coherent model we have to make sure that our set of MFrags collectively satisfies consistency constraints ensuring the existence of a unique joint probability distribution over instances of the random variables mentioned in the MFrags. Such a coherent collection of MFrags is called an MTheory. • An MTheory represents a joint probability distribution for an unbounded, possibly infinite number of instances of its random variables.

  36. Building MEBN models with MTheories (cont..) • A generative MTheory • summarizes statistical regularities that characterize a domain. These regularities are captured and encoded in a knowledge base using some combination of expert judgment and learning from observation. • To apply a generative MTheory to reason about particular scenarios, we need to provide the system with specific information about the individual entity instances involved in the scenario. • Bayesian inference • answer specific questions of interest • refine the MTheory

  37. Building MEBN models with MTheories (cont..) • Findings • the basic mechanism for incorporating observations into MTheories. • a finding is represented as a special 2-node MFrag containing a node from the generative MTheory and a node declaring one of its states to have a given value.

  38. Building MEBN models with MTheories (cont..) • Inserting a finding into an MTheory corresponds to asserting a new axiom in a first-order theory. • In other words, MEBN logic is inherently open, having the ability to incorporate new axioms as evidence and update the probabilities of all random variables in a logically consistent way.

  39. Building MEBN models with MTheories (cont..) • A valid MTheory • Each random variable must have a unique home MFrag. • It must ensure that all recursive definitions terminate in finitely many steps and contain no circular influences.

  40. The Star Trek Generative MTheory

  41. Building MEBN models with MTheories (cont..) • It is important to understand the power and flexibility that MEBN logic gives to knowledge base designers by allowing multiple, equivalent ways of portraying the same knowledge.

  42. Equivalent MFrag Representations of Knowledge

  43. Inference in MEBN Logic • BN • Assessing the impact of new evidence involves conditioning on the values of evidence nodes and applying a belief propagation algorithm. • MEBN • have an initial generative MTheory, a Finding set and Target set. • construct SSBN. • Creating instances of Finding and Target random variables. • standard BN inference is applied. • Inspecting the posterior probabilities of the target nodes.

  44. SSBN for the Star Trek MTheory with Four Starships within Enterprise’s Range

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