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Chronology of Development of Hyperbolic Dirac Net (HDN) Inference.
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Bayes’s solution to a problem of inverse probability was presented in “An Essay towards solving a Problem in the Doctrine of Chances” which was read to the Royal Society in 1763 after Bayes’ death
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When applied, the probabilities involved in Bayes’ theorem may have different probability interpretations. In one of these interpretations, the theorem is used directly as part of a particular approach to statistical inference. With the Bayesian probability interpretation the theorem expresses how a subjective degree of belief should rationally change to account for evidence: this is Bayesian inference, which is fundamental to Bayesian statistics. However, Bayes’ theorem has applications in a wide range of calculations involving probabilities, not just in Bayesian inference.
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Bayesian inference is a method of statistical inference in which Bayes’ theorem is used to update the probability for a hypothesis as more evidence or information becomes available. Bayesian inference is an important technique in statistics, and especially in mathematical statistics. Bayesian updating is particularly important in the dynamic analysis of a sequence of data. Bayesian inference has found application in a wide range of activities, including science, engineering, philosophy, medicine, sport, and law. In the philosophy of decision theory, Bayesian inference is closely related to subjective probability, often called “Bayesian probability“.
2. 1859, Georg Friedrich Bernhard Riemann proposed Riemann zeta function,function useful in number theory for investigating properties of prime numbers. Written as ζ(x), it was originally defined as the infinite series
ζ(x) = 1 + 2−x + 3−x + 4−x + ⋯.
The theory should perhaps be distinguished from an existing purely number-theoretic area sometimes also known as Zeta Theory, which focuses on the Riemann Zeta Function and the ways in which it governs the distribution of prime numbers
The Riemann zeta function is an extremely important special function of mathematics and physics that arises in definite integration and is intimately related with very deep results surrounding the prime number theorem. While many of the properties of this function have been investigated, there remain important fundamental conjectures (most notably the Riemann hypothesis) that remain unproved to this day. The Riemann zeta function is defined over the complex plane for one complex variable, which is conventionally denoted (instead of the usual ) in deference to the notation used by Riemann in his 1859 paper that founded the study of this function (Riemann 1859). It is implemented in the Wolfram Language as Zeta[s].
3. 1900. Ramanujan’s mathematical work was primarily in the areas of number theory and classical analysis. In particular, he worked extensively with infinite series, integrals, continued fractions, modular forms, q-series, theta functions, elliptic functions, the Riemann Zeta-Function, and other special functions.
Hardy wrote in Ramanujan’s obituary :
There is always more i n one of Ramanujan’s formulae than meets the eye, as anyone who sets to work to verify those which look the easiest will soon discover. In some the interest lies very deep, in others comparatively near the surface; but there is not one, which is not curious and entertaining.
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Initially Hardy was inclined to think that he was dealing with a fraud, but on further reflection realized that Ramanujan was in fact describing the Riemann Zeta Function (for s = – 1). He could then appreciate his brilliance as one, who though considerably isolated and without any formal training, had independently covered much of the same ground as Riemann.
However it still begs the question as to what the actual meaning of such a result can be, for in the standard conventional manner of mathematical interpretation, the sum of the series of natural numbers clearly diverges.
The startling fact is that this result – though indirectly expressed in a quantitative manner – actually expresses a qualitative type relationship (pertaining to holistic mathematical interpretation).
Uncovering Ramanujan’s “Lost” Notebook: An Oral History
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Whereas Ramanujan’s earlier work dealt largely with classical number-theoretic objects such as q-series, theta functions, partitions and prime numbers—exotic, startling, breathtaking identities built up from infinite series, integrals and continued fractions—in these newfound papers, Andrews found never-before-seen work on the mysterious “mock theta functions” hinted at in a letter written to Hardy in Ramanujan’s final months, pointing to realms at the very edge of the mathematical landscape. The content of Ramanujan’s lost notebook is too rich, too ornate, too strange to be developed within the scope of the present article. We provide a handful of stunning examples below, intended only to tantalize—perhaps mystify—the reader, who is encouraged to let his or her eyes wander across the page, picking patterns like spring flowers from the wild field of symbols.
In the words of mathematician Ken Ono, a contemporary trailblazer in the field of mock theta functions, “Obviously Ramanujan knew much more than he revealed .” Indeed, Ramanujan then “miraculously claimed” that the coefficients of this mock theta function obey the asymptotic relation
The new realms pointed to by the work of Ramanujan’s final year are now understood to be ruled by bizarre mathematical structures known as harmonic Maass forms. This broader perspective was only achieved in the last ten years, and has led to cutting-edge science, ranging from cancer research to the physics of black holes to the completion of group theory.
Yet details of George Andrews’s unearthing of Ramanujan’s notes are only sparsely sketched in the literature; one can detect but an outline of the tale surrounding one of the most fruitful mathematical discoveries of our era. In hopes of contributing to a more complete picture of this momentous event and its significance, here we weave together excerpts from interviews we conducted with Andrews documenting the memories of his trip to Trinity College, as well as from separate interviews with mathematicians Bruce Berndt and Ken Ono, who have both collaborated with Andrews in proving and extending the contents of Ramanujan’s famous lost notebook.
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In geometry and physics, spinors are elements of a (complex) vector space that can be associated with Euclidean space. Like geometric vectors and more general tensors, spinors transform linearly when the Euclidean space is subjected to a slight (infinitesimal) rotation. When a sequence of such small rotations is composed (integrated) to form an overall final rotation, however, the resulting spinor transformation depends on which sequence of small rotations was used, unlike for vectors and tensors. A spinor transforms to its negative when the space is rotated through a complete turn from 0° to 360° (see picture), and it is this property that characterizes spinors. It is also possible to associate a substantially similar notion of spinor to Minkowski space in which case the Lorentz transformations of special relativity play the role of rotations. Spinors were introduced in geometry by Élie Cartan in 1913. In the 1920s physicists discovered that spinors are essential to describe the intrinsic angular momentum, or “spin”, of the electron and other subatomic particles.
The quantum electrodynamical law which applies to spin-1/2 particles and is the relativistic generalization of the Schrödinger equation. In dimensions (three space dimensions and one time dimension), it is given by
6. 1930. Dirac publishes his book on his pivotal view of quantum mechanics, including his earliest mentions of an operator with the properties of the hyperbolic h number such that hh = +1. It extends the theory of wave mechanics to particle mechanics. P. A. M. Dirac, The Principles of Quantum Mechanics, First Edition, Oxford University Press, Oxford (1930).
7. 1933. In his Nobel Prize Dinner speech, Dirac states that mechanical methods are applicable to all forms of human thought where numbers are involved. http://www.nobelprize.org/nobel_prizes/physics/laureates/1933/dirac-speech.html
8. 1939. DIRAC PUBLISHES HIS BRAKET NOTATION. It is incorporated into the third edition of his book.
P.A.M. Dirac (1939). A new notation for quantum mechanics, Mathematical Proceedings of the Cambridge Philosophical Society 35 (3): 416–418
9. 1974. Robson develops his Expected Information approach that preempts the Bayes Net method.
B. Robson, Analysis of the Code Relating Sequence to Conformation in Globular Proteins: Theory and Application of Expected Information, Biochem. J141, 853-867 (1974).
10. 1978. The Expected Information approach crystallizes as the GOR method widely used in bioinformatics.
Garnier, D. J. Osguthorpe, and B. Robson, Analysis of the Accuracy and Implications of Simple Methods for Predicting the Secondary Structure of Globular Proteins”, J. Mol. Biol. 120, 97-120 (1978).
11. 1982 . Buchannan and Shortliffe describe the first medical Expert System. It is based on probabilistic statements, but sets a tradition of innovation and diverse controversial methods in automated medical inference.
Buchanan, E.H. Shortliffe, (1982) Rule Based Expert Systems. The Mycin Experiments of the Stanford Heuristic Programming Project, Addison-Wesley: Reading, Massachusetts.
12. 1985. Pearl Gives Full Accound the Bayes Net method.
Pearl, Probabilistic Reasoning in Intelligent Systems. San Francisco CA: Morgan Kaufmann (1985).
13. March 1989, Sir Tim Berners-less invented WWW: – Introduced non-linear linking of information across systems.
Tim laid out his vision for what would become the Web in a document called “Information Management: A Proposal”.Believe it or not, Tim’s initial proposal was not immediately accepted. In fact, his boss at the time, Mike Sendall, noted the words “Vague but exciting” on the cover. The Web was never an official CERN project, but Mike managed to give Tim time to work on it in September 1990. He began work using a NeXT computer, one of Steve Jobs’ early products.
14. 1997. Clifford Algebra using h becomes more widely recognized as a tool for engineers as well as scientists and physicists.
Gürlebeck, W. Sprössig, Quaternionic and Clifford Calculus for Physicists and Engineers, Wiley, Chichester (1997)
15. 1999. Tim Berners-Lee described the Semantic Web vision in the following terms
I have a dream for the Web [in which computers] become capable of analyzing all the data on the Web, the content, links, and transactions between people and computers. A Semantic Web, which should make this possible, has yet to emerge, but when it does, the day-to-day mechanisms of trade, bureaucracy and our daily lives will be handled by machines talking to machines. The intelligent agents people have touted for ages will finally materialize. (1999)
16. 2000. Khrennikov gives description of a primarily h-complex quantum mechanics.
Khrenikov, Hyperbolic quantum mechanics, Cornell University Library, arXiv:quant-ph/0101002v1 (2000).
17. 2000. Bucholz and Sommer refine work showing that neural networks as inference systems modeled on the brain can usefully use the hypercomplex imaginary number h.
S. Buchholz, and G. Sommer, A hyperbolic multilayer perceptron International Joint Conference on Neural Networks, IJCNN 2000, Como,Italy, Vol. 2 of pp. 129-133. Amari, S-I and. Giles, C.L M. Gori. M. and Piuri, V. Eds. IEEE Computer Society Press, (2000).
18. 2003. Robson Points out that the Expected Information method in bioinformatics is really the use of the partially summated Riemann Zeta function, and a best choice for treatment of sparse data in data mining in general.
B Robson (2003) “Clinical and Pharmacogenomic Data Mining. 1. The generalized theory of expected information and application to the development of tools” J. Proteome Res. (Am. Chem. Soc.) 283-301, 2
19. 2003. Nitta Shows that the power of the h-complex approach in neural nets is primarily due to its ability to solver the notorious exclusive-or logical problem in a single neuron.
Nitta, Solving the XOR problem and the detection of symmetry using a single complex-valued neuron, Neural Networks 16:8, 1101-1105, T. (2003).
20. 2003. Khrennikov consolidates the notion of an extensively h-complex quantum mechanics, but feels that i-complex, h-complex, and real world mechanics are three spate systems.
A.Khrennikov, A. Hyperbolic quantum mechanics, Adv. in Applied Clifford Algebras, Vol.13, 1 (2003).
21.2004. Khrennikov notes possible relation between h-complex quantum mechanics and mental function.
Khrennikov, On Quantum-Like Probabilistic Structure of Mental Information, Open Systems Information Dynamics, Vol. 11, 3, 267-275 (2004).
22. 2004 Rochon shows that the full Riemann Zeta function is both i-complex and h-complex.
Rochon, A Bicomplex Riemann Zeta Function, Tokyo J. of Math.
23. 2004. Robson argues that zeta theory is a solution to high dimensionality problems in data mining.
Robson, The Dragon on the Gold: Myths and Realities for Data Mining in Biotechnology using Digital and Molecular Libraries, J. Proteome Res. (Am. Chem. Soc.) 3 (6), 1113 – 9 (2004).
24. 2005. Robson argues that all statements in zeta theory and in prime number theory are really statements relevant to data and data mining, and describes first link to Dirac’s quantum mechanics and Dirac’s braket notation.
Robson, Clinical and Pharmacogenomic Data Mining: 3. Zeta Theory As a General Tactic for Clinical Bioinformatics, J. Proteome Res. (Am. Chem. Soc.) 4(2); 445-455 (2005)
25. 2005. Code CliniMiner/Fano based on Zeta Theory and prime number theory is used in first pioneering effort in data mining large number of patient records.
Mullins, I. M., M.S. Siadaty, J. Lyman, K. Scully, G.T. Garrett, G. Miller, R. Muller, B. Robson, C. Apte, C., S. Weiss, I. Rigoutsos, D. Platt, and S. Cohen, Data mining and clinical data repositories: Insights from a 667,000 patient data set, Computers in Biology and Medicine, 36(12) 1351 (2006).
26. 2007. Robson recognizes that the imaginary number required to reconcile zeta theory with quantum mechanics and to allow Dirac notation to be used in inference is the hyperbolic imaginary number h, not the imaginary number i. Unaware of the work of Khrennikov, he makes no Khrennikov-like distinction between h-complex quantum mechanics and the everyday world.
Mullins, I. M., M.S. Siadaty, J. Lyman, K. Scully,G.T. Garrett, G.Miller, R. Muller, B.Robson, C. Apte, C., S. Weiss, I. Rigoutsos, D. Platt, and S. Cohen, Data mining and clinical data repositories: Insights from a 667,000 patient data set, Computers*in*Biology* and*Medicine, 36(12) 1351 (2006)
27. 2007. Robson recognizes that the imaginary number required to reconcile zeta theory with quantum mechanics and to allow Dirac notation to be used in inference is the hyperbolic imaginary number h, not the imaginary number i. Unaware of the work of Khrennikov, he makes no Khrennikov like distinction between h complex quantum mechanics and the every day world.
Robson, The New Physician as Unwitting Quantum Mechanic: Is Adapting Dirac’s Inference System Best Practice for Personalized Medicine, Genomics and Proteomics, J. Proteome Res. (A. Chem. Soc.), Vol. 6, No. 8: 3114 – 3126, (2007).
Robson, B. (2007) “Data Mining and Inference Systems for Physician Decision Support in Personalized Medicine” Lecture and Circulated Report at the 1st Annual Total Cancer Care Summit, Bahamas 2007.
28. 2008. Data Mining techniques using the full i-complex and h-complex zeta function are developed.
Robson, Clinical and Pharmacogenomic Data Mining: 4. The FANO Program and Command Set as an Example of Tools for Biomedical Discovery and Evidence Based Medicine” J. Proteome Res., 7 (9), pp 3922–3947 (2008).
29. 2008. Nitta and Bucholtz explore decision process boundaries of h-complex neural nets.
Nitta, and S. Bucholtz, On the Decision Boundaries of Hyperbolic Neurons. In 2008 International Joint Conference on Neural Networks (IJCNN).
30. 2009. Semantic Web starts to emerge but runs into bottleneck regarding the best approach for probabilistic treatment.
Prediou and H. Stuckenschmidt, H. Probabilistic Models for the SW – A Survey. http://ki.informatik.unimannheim.de/fileadmin/ publication/ Predoiu08Survey.pdf (last accessed 4/29/2010)
31. 2009. Baek and Robson propose that, for reasons of bandwidth limitations and security, the Internet should consist of data-centric computing by smart software robots. Robson indicates that they could be based on h-complex inference systems and link to semantic theory.
Robson B.. and Baek OK. The Engines of Hippocrates. From the Dawn of Medicine to Medical and Phrmaceuteutical Infomatics, Wiley, 2009.
Robson B. (2009) “Towards Intelligent Internet-Roaming Agents for Mining and Inference from Medical Data”, Future of Health Technology Congress, Technology and Informatics, Vol. 149, 157-177 IOS Press
Robson, B. (2009) “Links Between Quantum Physics and Thought” (A. I. Applications in Medicine) , Future of Health Technology Congress, Technology and Informatics, Vol. 149, 157-177 IOS Press.
32. 2009. Nivitha et al. develop new learning algorithms for complex-valued networks.
S. Savitha, S. Suresh, S. Sundararajan, and P, Saratchandran, A new learning algorithm with logarithmic performance index for complex-valued neural networks, Neurocomputing 72 (16-18), 3771-3781 (2009).
33. 2009. Khrennikov argues for the h-complex Hilbert space as providing the “contextual” (underlying rationale, hidden variables etc.) for all quantum mechanics.
Khrennikov, Contextual Approach to Quantum Formalism, Springer (2009)
34. 2010. Robson and Vaithiligam describe how zeta theory and h-complex probabilistic algebra can resolves challenges in data mining by the pharmaceutical industry.
Robson and A. Vaithiligam, Drug Gold and Data Dragons: Myths and Realities of Data Mining in the Pharmaceutical Industry pp25-85 in Pharmaceutical Data Mining, Ed Balakin, K. V. , John Wiley Sons (2010).
35. 2010. PCAST. December Report by the US President’s Council of Advisors on science and Technology calls for an XML-like Universal Exchange Langue for medicine including disaggregation for the patient record on the Internet for patient access, security, and privacy.
36. 2011. First description of Q-UEL in response to PCAST 2010.
Robson, B., Balis, U. G. J. and Caruso, T. P. (2011)“Considerations for a Universal Exchange Language for Healthcare.” In Proceedings of 2011 IEEE 13th International Conference on e-Health Networking, Applications and Services (Healthcom 2011), 173– 176. Columbus, MO: IEEE, 2011.
37. 2011. Robson and Colleagues develop the method of match-and-edit instructions for extracting
Robson, B., Li, J., Dettinger, R., Peters, A., and Boyer, S.K. (2011), Drug discovery using very large numbers of patents. General strategy with extensive use of match and edit operations. Journal of Computer-Aided Molecular Design 25(5): 427-441
38. 2011. Kuroe et al. consolidate the theory of h– complex neural nets.
Kuroe, T. Shinpei, and H. Iima, Models of Hopfield-Type Clifford Neural Networks and Their Energy Functions – Hyperbolic and Dual Valued Networks, Lecture Notes in Computer Science, 7062, 560 (2011).
39. 2012. Robson argues that h-complex algebra is an appropriate basis for Artificial Intelligence in the Pharmaceutical Industry.
Robson, B. (2012) “Towards Automated Reasoning for Drug Discovery and Pharmaceutical Business Intelligence”, Pharmaceutical Technology and Drug Research, 2012 1: 3 ( 27 March 2012 )
40. 2013. Goodman and Lassiter attempt to reconcile and restore interest in probabilistic semantics after a long period of domination by classical logic. N. D. Goodman and D. Lassiter, Probabilistic Semantics and Pragmatics: Uncertainty in Language and Thought,
41. 2013. Robson argues for importance of h-complex approach for measures in epidemiology. Robson, B. (2013)
“Towards New Tools for Pharmacoepidemiology”, Advances in Pharmacoepidemiology and Drug Safety, 1:6,
42. 2013 Robson promotes Q-UEL from a public health perspective. B. Robson, Rethinking Global Interoperability in Healthcare. Reflections and Experiments of an e-Epidemiologist from Clinical Record to Smart Medical Semantic Web Johns Hopkins Grand Rounds Lectures (last accessed 3/14/2013).
43. 2013 Robson and Caruso describe first version of Q-UEL in greater Detail.
Robson, B, and TP Caruso (2013) “A Universal Exchange Language for Healthcare” MedInfo ’13: Proceedings of the 14th World Congress on Medical and Health Informatics, Copenhagen, Denmark, Edited by CU Lehmann, E Ammenwerth, and C Nohr. IOS Press, Washington, DC, USA. http://quantalsemantics.com/documents/MedInfo13-RobsonCaruso_V6.pdf; http://ebooks.iospress.nl/publication/34165
44. 2014. Robson et al. release formal description of consolidated second version of Q-UEL.
Robson, T. P. Caruso and U. G. J. Balis, Suggestions for a Web Based Universal Exchange and Inference Language for Medicine, Computers in Biology and Medicine, 43(12) 2297 (2013).
45. 2013. Moldoveneua expresses view that hyperbolic quantum mechanics can’t also include wave mechanics. Possible attack on Khrennikov’s idea that hyperbolic quantum mechanics can show interference as for waves. Signs of growing sense that hyperbolic quantum mechanics is simply the everyday world described in terms of the machinery of traditional quantum mechanics.
Moldoveanu, Non viability of hyperbolic quantum mechanics as a theory of Nature, Cornell University Library, arXiv:1311.6461v2 [quant-ph] (2013).
46. 2013. First full description of the Hyperbolic Dirac Net and its relation to Q-UEL and to Bayes Nets.
Robson, Hyperbolic Dirac Nets for Medical Decision Support. Theory, Methods, and Comparison with Bayes Nets, Computers in Biology and Medicine, 51, 183 (2013).
47. 2014. Kunegis et al.c develop h-complex algorithms for dating recommender systems.
Kunegis, G. Gröner, and T, Gottrron, On-Line Dating Recommender Systems, the Split Complex Number Approach, (Like/Dislike, Similar/Disimilar) http://userpages.uni- koblenz.de/~kunegis/paper/kunegis-online-dating-recommender-systems-the-split- complex-number-approach.pdf (last accessed 6/1/2014).
48. 2015. Robson describes extension of Hyperbolic Dirac Net to semantic reasoning and probabilistic lingusitics.
Robson, B. “POPPER, a Simple Programming Language for Probabilistic Semantic Inference in Medicine. Computers in Biology and Medicine ” Computers in biology and Medicine”, (in press), DOI: 10.1016/j.compbiomed.2014.10.011 (2015).
49. 2014. Yosemite Manifesto – a response to PCAST 2010 that the Semantic Web should provide healthcare IT, al though preempted by Q-UEL
http://yosemitemanifesto.org/ (last accessed 7/5/2014).
50. 2015. Robson et al. describe medical records in Q-UEL format and PCAST disaggregation for patient security and privacy.
Robson, B., Caruso, T, and Balis, U. G. J. (2015) “Suggestions for a Web Based Universal Exchange and Inference Language for Medicine. Continuity of Patient Care with PCAST Disaggregation.” Computers in Biology and Medicine (in press) 01/2015; 56:51. DOI: 10.1016/j.compbiomed.2014.10.022
51. 2015. Mathematician Steve Deckelman of U. Wisconsin-Stout and Berkeley validates the theoretical principles Hyperbolic Dirac Net.
Deckelman and Robson, B. (2015)“Split-Complex Numbers and Dirac Bra-Kets” Communications in Information andSystems (CIS), in press.
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The inference net on which this dualization is performed is defined as an estimate of a probability as an expression comprising simpler probabilities and or association measures, i.e. each with fewer attributes (i.e. arguments, events, states, observations or measurements) that the joint probability estimated, where each attribute corresponds to nodes of a general graph and the probabilities or association measures represent their interdependencies as edges. It is not required that the inference net be an acyclic directed graph, but the widely used BN that satisfies that description by definition is a useful starting point for making use of the given probabilities to address the same or similar problems. Specifically for the estimation of a joint probability, and HDN properly constructed with prior probabilities, and whether or not it contains cyclic paths, is purely real valued and its construction principles represent a generalization of Bayes Theorem. Any imaginary part indicates the degree of departure from Bayes Theorem over the net as a whole, and the direction of conditionality in which the degree of departure occurs, and thus the HDN provides an excellent book-keeping tool that Bayes Theorem is satisfied overall. Specially for the estimation of a conditional probability, it follows conversely from the above that any expression for a joint probability validated by the above means can serve as the generator of an HDN for the estimation of a conditional probability simply by dividing it by the HDN counterparts of prior probabilities, whence the resulting net is not purely real save by coincidence of probability values.
52. 2015. Implementation of a web based universal exchange and inference language for medicine: Sparse data, probabilities and inference in data mining of clinical data repositories
Barry Robson and Srinidhi Boray
52. 2015. Robson, B., and S. Boray, The Structure of Reasoning in Answering Multiple Choice Medical Licensing Examination Questions. Computer Studies towards Formal Theories of Clinical Decision Support and Setting and Answering Medical Licensing Examinations, Workshop Lecture presentation, Proceedings of the IEEE International conference of Bioinformatics and Biomedicine, 9th-11th November, Washington DC (2015)