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Intelligent Science Platforms

Intelligent Science Platforms. Kevin H. Knuth Departments of Physics and Informatics University at Albany. Encoding Knowledge With Lattices. apple. banana. cherry. State Space. States describe Systems Antichain. exp. a b c. log. Exp and Log. exp.

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Intelligent Science Platforms

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  1. Intelligent SciencePlatforms Kevin H. Knuth Departments of Physics and Informatics University at Albany

  2. Encoding Knowledge • With Lattices Kevin H KnuthCIDU 2008

  3. apple banana cherry State Space States describe SystemsAntichain Kevin H KnuthCIDU 2008

  4. exp a b c log Exp and Log Kevin H KnuthCIDU 2008

  5. exp a b c log Exp and Log Kevin H KnuthCIDU 2008

  6. exp a b c log Exp and Log States Statements(sets of states) (potential states) Kevin H KnuthCIDU 2008

  7. Three Spaces exp exp a b c log log Kevin H KnuthCIDU 2008

  8. Three Spaces exp exp a b c log log States Statements(sets of states) (potential states) Questions(sets of statements) (potential statements) Kevin H KnuthCIDU 2008

  9. apple banana cherry State Space States describe SystemsAntichain Kevin H KnuthCIDU 2008

  10. implies Hypothesis Space Statements are sets of StatesBoolean Lattice Kevin H KnuthCIDU 2008

  11. Inquiry Space answers Questions are sets of StatementsFree Distributive Lattice Kevin H KnuthCIDU 2008

  12. Relevance “Is it an Apple or Cherry, or is it a Banana or Cherry?” “Is it an Apple?” Relevance Decreases answers Central Issue“Is it an Apple, Banana, or Cherry?” Kevin H KnuthCIDU 2008

  13. The Central Issue I = “Is it an Apple, Banana, or Cherry?” This question is answered by the following set of statements: I = { a = “It is an Apple!”, b = “It is a Banana!”, c = “It is a Cherry!” } Kevin H KnuthCIDU 2008

  14. Some Questions Answer Others Now consider the binary question B = “Is it an Apple?” B = {a = “It is an Apple!”, ~a = “It is not an Apple!”} As the defining set of I is exhaustive, Kevin H KnuthCIDU 2008

  15. Ordering Questions I answers B B includes I I = “Is it an Apple, Banana, or Cherry?” B = “Is it an Apple?” Kevin H KnuthCIDU 2008

  16. ValuationsonLattices

  17. Valuations Valuations are functions that take lattice elements to real numbers Valuation: ℝ Bi-Valuation:ℝ Kevin H KnuthCIDU 2008

  18. Valuations Valuations are functions that take lattice elements to real numbers Valuation: ℝ Bi-Valuation:ℝ By assigning valuations in accordance to the lattice structure, this generalizes lattice inclusion to degrees of inclusion. The valuation inherits meaning from the ordering relation! Kevin H KnuthCIDU 2008

  19. Context The context of a bi-valuation can be made implicit Valuation Bi-Valuation Context y is explicit Measure of x with respect to Context y Context y is implicit Kevin H KnuthCIDU 2008

  20. Consistency Valuation assignments must be consistent with lattice structure Kevin H KnuthCIDU 2008

  21. Consistency Valuation assignments must be consistent with lattice structure In general Kevin H KnuthCIDU 2008

  22. Associativity of Join Associativity leads to a Sum Rule… or more specifically Kevin H KnuthCIDU 2008

  23. Distributivity Distributivity leads to a Product Rule… or more specifically Kevin H KnuthCIDU 2008

  24. Commutativity Commutativity leads to a Bayes Theorem… Note that Bayes Theorem involves a change of context. Valuations are not sufficient… need bi-valuations. Kevin H KnuthCIDU 2008

  25. Inclusion-Exclusion (The Sum Rule) The Sum Rule for Lattices Kevin H KnuthCIDU 2008

  26. Inclusion-Exclusion (The Sum Rule) The Sum Rule for Probability Kevin H KnuthCIDU 2008

  27. Inclusion-Exclusion (The Sum Rule) Definition of Mutual Information Kevin H KnuthCIDU 2008

  28. Inclusion-Exclusion (The Sum Rule) Polya’s Min-Max Rule for Integers Kevin H KnuthCIDU 2008

  29. Inclusion-Exclusion (The Sum Rule) “Measuring Integers”, Knuth 2008 The Sum Rule derives from the Möbius function of the lattice, And is related to its Zeta function Kevin H KnuthCIDU 2008

  30. Probability Probabilities are degrees of implication! Constraint Equations! Kevin H KnuthCIDU 2008

  31. Relevance Relevance quantifies the degree to which one question answers another Constraint Equations Kevin H KnuthCIDU 2008

  32. Probability and Relevance Relevance is a function of probability The degree to which one question answers another must depend on the probabilities of the possible answers. Kevin H KnuthCIDU 2008

  33. Relevance Kevin H KnuthCIDU 2008

  34. Normalization Conditions Relevance is minimized when Relevance is maximized when We may choose to normalize relevance between zero and one. Kevin H KnuthCIDU 2008

  35. Relevance and Entropy Kevin H KnuthCIDU 2008

  36. Higher-Order Informations This relevance is related to the mutual information. In this way one can obtain higher-order informations. Kevin H KnuthCIDU 2008

  37. Higher-Order Informations The Sum Rule can be applied in larger lattices to obtain even higher-order informations as introduced by McGill (1954) and as co-informations by Bell (2003). However, it has been noted that these informations suffer from the strange properties that they can become negative. Problem Solved! Normalize properly! Kevin H KnuthCIDU 2008

  38. EXAMPLE Kevin H KnuthCIDU 2008

  39. Guessing Game apple banana cherry Can only ask binary (YES or NO) questions! Kevin H KnuthCIDU 2008

  40. Which Question to Ask? Is it or is it not an Apple?Is it or is it not a Banana?Is it or is it not a Cherry? AVM VAM AVAM AVVM If you believe that there is a 75% chance that it is an Apple, and a 10% chance that it is a Banana,which question do you ask? Kevin H KnuthCIDU 2008

  41. Relevance Depends on Probability Is it an Apple? Is it a Banana? Is it a Cherry? ABC BAC CAB c c c a a a b b b AVAM AVVM If you believe that there is a 75% chance that it is an Apple, and a 10% chance that it is a Banana,which question do you ask? Kevin H KnuthCIDU 2008

  42. Relevance Depends on Probability Is it an Apple? Is it a Banana? Is it a Cherry? ABC BAC CAB c c c a a a b b b AVAM AVVM AMVM If you believe that there is a 75% chance that it is an Apple, and a 10% chance that it is a Banana,which question do you ask? Kevin H KnuthCIDU 2008

  43. Results ABC BAC CAB c c c a a a b b b ACAB ABBC ACBC c c c a a a b b b Kevin H KnuthCIDU 2008

  44. EXPERIMENTAL DESIGN Kevin H KnuthCIDU 2008

  45. Doppler Shift • PROBLEM:Determine the relative radial velocity relative to a Sodium lamp. We can measure light intensities near the doublet at 589 nm and 589.6 nm • We can take ONE MEASUREMENT • Which wavelength shall we examine? • Recall, we don’t know the Doppler shift! Kevin H KnuthCIDU 2008

  46. What Can We Ask? • The question that can be asked is: • “What is the intensity at wavelength λ ?” • There are many questions to choose from, each corresponding to a different wavelength λ Kevin H KnuthCIDU 2008

  47. What are the Possible Answers? • Say that the intensity can be anywhere between 0 and 1. Kevin H KnuthCIDU 2008

  48. Given Possible Doppler Shifts… • Say we have information about the velocity.The Doppler shift is such that the shift in wavelength has zero mean with a standard deviation of 0.1 nm. Kevin H KnuthCIDU 2008

  49. Probable Answers for Each Question • We now look at the set of probable answers for each question Kevin H KnuthCIDU 2008

  50. Entropy of Distribution of Probable Results • Red shows the entropy of the distribution of probable results. Kevin H KnuthCIDU 2008

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