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Entropy2016,18, 442 30. Aprausheva,N.N.; Sorokin,S.V. Exactequationof theboundaryofunimodalandbimodaldomainsof a two-componentGaussianmixture. PatternRecognit. ImageAnal. 2013,23, 341–347. 31. Learned-Miller,E.;DeStefano, J.Aprobabilisticupperboundondifferentialentropy. IEEETrans. Inf. Theory 2008,54, 5223–5230. 32. Amari,S.-I. α-Divergence IsUnique,BelongingtoBoth f-DivergenceandBregmanDivergenceClasses. IEEETrans. Inf. Theory2009,55, 4925–4931. 33. Cichocki,A.;Amari, S.I. FamiliesofAlpha-Beta-andGamma-Divergences: FlexibleandRobustMeasures ofSimilarities. Entropy2010,12, 1532–1568. 34. Póczos,B.; Schneider, J. On theEstimationofα-Divergences. InProceedingsof the14th International ConferenceonArtificial IntelligenceandStatistics,Ft. Lauderdale,FL,USA,11–13April 2011;pp.609–617. 35. Nielsen,F.;Nock,R. OnRényiandTsallisentropiesanddivergences forexponential families. arXiv2011, arXiv:1105.3259. 36. Minka,T.DivergenceMeasuresandMessagePassing; TechnicalReportMSR-TR-2005-173;MicrosoftResearch: Cambridge,UK,2005. 37. Améndola, C.; Drton, M.; Sturmfels, B. MaximumLikelihood Estimates forGaussianMixturesAre Transcendental. arXiv2015, arXiv:1508.06958. 38. Hellinger,E.NeueBegründungderTheoriequadratischerFormenvonunendlichvielenVeränderlichen. J.ReineAngew.Math. 1909,136, 210–271. (InGerman) 39. VanErven,T.;Harremos,P. RényidivergenceandKullback-Leiblerdivergence. IEEETrans. Inf. Theory 2014,60, 3797–3820. 40. Nielsen, F.;Nock,R. Aclosed-formexpression for theSharma-Mittal entropyof exponential families. J.Phys.AMath. Theor. 2012,45, 032003. 41. Nielsen,F.;Nock,R.OntheChiSquareandHigher-OrderChiDistances forApproximating f-Divergences. IEEESignalProcess. Lett. 2014,21, 10–13. 42. Nielsen, F.; Boltz, S. TheBurbea-Rao andBhattacharyya centroids. IEEETrans. Inf. Theory 2011, 57, 5455–5466. 43. Jarosz,W. EfficientMonteCarloMethodsforLightTransport inScatteringMedia. Ph.D.Thesis,University ofCalifornia,SanDiego,CA,USA,2008. 44. Fujisawa,H.; Eguchi, S. Robust parameter estimationwith a small bias against heavy contamination. J.Multivar.Anal.2008,99, 2053–2081. 45. Havrda, J.;Charvát,F. Quantificationmethodofclassificationprocesses.Conceptofstructuralα-entropy. Kybernetika1967,3, 30–35. 46. Liang,X. ANoteonDivergences.NeuralComput. 2016,28, 2045–2062. 47. Lin, J. DivergencemeasuresbasedontheShannonentropy. IEEETrans. Inf. Theory1991,37, 145–151. 48. Endres,D.M.;Schindelin, J.E. Anewmetric forprobabilitydistributions. IEEETrans. Inf. Theory2003,49, 1858–1860. 49. Nielsen,F.;Boissonnat, J.D.;Nock,R. OnBregmanVoronoidiagrams. InProceedingsof theEighteenth AnnualACM-SIAMSymposiumonDiscreteAlgorithms, NewOrleans, LA,USA, 7–9 January 2007; Society for IndustrialandAppliedMathematics: Philadelphia,PA,USA,2007;pp. 746–755. 50. Boissonnat, J.D.;Nielsen,F.;Nock,R.BregmanVoronoidiagrams.Discret. Comput.Geom.2010,44,281–307. 51. Foster,D.V.;Grassberger,P. Lowerboundsonmutual information. Phys. Rev. E2011,83, 010101. 52. Nielsen, F.; Sun, K. PyKLGMM: Python Software for Computing Bounds on the Kullback-Leibler DivergencebetweenMixtureModels. 2016. Availableonline: https://www.lix.polytechnique.fr/~nielsen/ KLGMM/(accessedon6December2016). 53. Cobb, L.; Koppstein, P.; Chen, N.H. Estimation and moment recursion relations for multimodal distributionsof theexponential family. J.Am. Stat.Assoc. 1983,78, 124–130. 54. Nielsen,F.;Nock,R. Patchmatchingwithpolynomialexponential familiesandprojectivedivergences. InProceedingsof the9th InternationalConferenceSimilaritySearchandApplications (SISAP),Tokyo, Japan,24–26October2016. c©2016bytheauthors. LicenseeMDPI,Basel,Switzerland. Thisarticle isanopenaccess articledistributedunder the termsandconditionsof theCreativeCommonsAttribution (CCBY) license (http://creativecommons.org/licenses/by/4.0/). 311
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Differential Geometrical Theory of Statistics
Title
Differential Geometrical Theory of Statistics
Authors
Frédéric Barbaresco
Frank Nielsen
Editor
MDPI
Location
Basel
Date
2017
Language
English
License
CC BY-NC-ND 4.0
ISBN
978-3-03842-425-3
Size
17.0 x 24.4 cm
Pages
476
Keywords
Entropy, Coding Theory, Maximum entropy, Information geometry, Computational Information Geometry, Hessian Geometry, Divergence Geometry, Information topology, Cohomology, Shape Space, Statistical physics, Thermodynamics
Categories
Naturwissenschaften Physik
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Differential Geometrical Theory of Statistics