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Probability Density Functions from the Fisher Information Metric [article]

T. Clingman, Jeff Murugan, Jonathan P. Shock
2015 arXiv   pre-print
We show a general relation between the spatially disjoint product of probability density functions and the sum of their Fisher information metric tensors.  ...  We then utilise this result to give a method for constructing the probability density functions for an arbitrary Riemannian Fisher information metric tensor.  ...  Suppose we desire a family of probability density functions whose Fisher information metric is the metric tensor of S 2 .  ... 
arXiv:1504.03184v1 fatcat:6zximcrbnfe4zpzzrkp3cpbh24

Classical and quantum Fisher information in the geometrical formulation of quantum mechanics

Paolo Facchi, Ravi Kulkarni, V.I. Man'ko, Giuseppe Marmo, E.C.G. Sudarshan, Franco Ventriglia
2010 Physics Letters A  
By putting these two aspects together, we show that the Fisher information metric, both classical and quantum, can be described by means of the Hermitian tensor on the manifold of pure states.  ...  The tomographic picture of quantum mechanics has brought the description of quantum states closer to that of classical probability and statistics.  ...  In classical probability theory the Fisher information metric can be used to characterize the distance between probability distributions.  ... 
doi:10.1016/j.physleta.2010.10.005 fatcat:efih6blxwzhjformbo3srk2drm

A study on some geometrical and statistical properties of the continuous power-law distribution with a comparison of similar properties of a particular exponential distribution

N. Yapage
2021 Journal of the National Science Foundation of Sri Lanka  
A continuous real random variable x with a power-law distribution has a probability p(x) of taking a value in the interval from x to x + dx, where, 2 by 〈 〉 .  ...  Finally, an example of an exponential distribution having the same information divergence and Fisher information as that of power-law distributions is given, thus having the same lower bound in the Cramer-Rao  ...  We also hope that the same Fisher information for power-law and exponential distributions we have obtained will definitely be helpful to understand the geometry of such networks.  ... 
doi:10.4038/jnsfsr.v49i1.9710 fatcat:7z65gzt7yfcz5ilotzg3uoxypa

Quantum and Fisher information from the Husimi and related distributions

Paul B. Slater
2006 Journal of Mathematical Physics  
In the Petz-Sudar work (completing a program of Chentsov), the quantum analogue of the (classically unique) Fisher information (montone) metric of a probability simplex was extended to define an uncountable  ...  We also investigate the Fisher information metrics corresponding to the escort Husimi, positive-P and certain Gaussian probability distributions, as well as, in some sense, the discrete Wigner pseudoprobability  ...  ACKNOWLEDGMENTS The author expresses gratitude to the Kavli Institute for Theoretical Physics ͑KITP͒ for computational support in this research and to C.  ... 
doi:10.1063/1.2168125 fatcat:cgrmknrtanhq3clsw27mh5zpui

The Brody-Hughston Fisher Information Metric [article]

Paul B. Slater
2003 arXiv   pre-print
We study the interrelationships between the Fisher information metric recently introduced, on the basis of maximum entropy considerations, by Brody and Hughston (quant-ph/9906085) and the monotone metrics  ...  is less noninformative than those obtained from any of the monotone metrics, even the minimal monotone (Bures) metric.  ...  BH indicated how to associate a Fisher information metric to their family of density functions (i. e., probability distributions), each distribution, of course, corresponding to a density matrix.  ... 
arXiv:quant-ph/0304029v2 fatcat:wdwcoujxbnhznfyp5zah4fxmay

Information metric from a linear sigma model

U. Miyamoto, S. Yahikozawa
2012 Physical Review E  
The idea that a spacetime metric emerges as a Fisher-Rao 'information metric' of instanton moduli space has been examined in several field theories such as the Yang-Mills theories and nonlinear sigma models  ...  In this brief paper, we report that the flat Euclidean or Minkowskian metric, rather than an anti-de Sitter metric that generically emerges from instanton moduli spaces, can be obtained as the Fisher-Rao  ...  Fund of the Ministry of Education, Culture, Sports, Science and Technology, Japan (Young Scientists (B) 22740176).  ... 
doi:10.1103/physreve.85.051133 pmid:23004729 fatcat:ojeyo7sslraxpogtje75wzh6u4

Extending the Fisher metric to density matrices [article]

D. Petz, Cs. Sudar
2001 arXiv   pre-print
Chentsov studied Riemannian metrics on the set of probability measures from the point of view of decision theory.  ...  He proved that up to a constant factor the Fisher information is the only metric which is monotone under stochastic transformations.  ...  Without some restrictions this would be pointless, the emphasis is put on statisticaly relevant metrics which on the submanifold of probability distributions recover the Fisher information metric. 2 Chentsov's  ... 
arXiv:quant-ph/0102132v1 fatcat:kqhcluvtsnbhheoewl65ko73iq

Hydrodynamical formulation of quantum mechanics, Kahler structure, and Fisher information [article]

Marcel Reginatto
1999 arXiv   pre-print
the Fisher information metric.  ...  The Schrodinger equation can be derived using the minimum Fisher information principle.  ...  CROSS-ENTROPY AND FISHER INFORMATION Let P (y i ) be a probability density which is a function of n continuous coordinates y i , and let P (y i + ∆y i ) be the density that results from a small change  ... 
arXiv:quant-ph/9909065v1 fatcat:2rhe6cgtyjerpdktoho7xb5uiu

The information metric on the moduli space of instantons with global symmetries

Emanuel Malek, Jeff Murugan, Jonathan P. Shock
2016 Physics Letters B  
In this note we revisit Hitchin's prescription Hitchin of the Fisher metric as a natural measure on the moduli space of instantons that encodes the space-time symmetries of a classical field theory.  ...  Motivated by the idea of the moduli space of supersymmetric instantons as an emergent space in the sense of the gauge/gravity duality, we extend the prescription to encode also global symmetries of the  ...  In addition we are grateful for insight from Tslil Clingman, Johanna Erdmenger, Hiroaki Matsueda, Andrew Singleton, Koenraad Schalm and David Tong.  ... 
doi:10.1016/j.physletb.2015.12.044 fatcat:ogwcm7rfffezfdq6rs7tp7usnu

Information geometry in quantum field theory: lessons from simple examples [article]

Johanna Erdmenger, Kevin T. Grosvenor, Ro Jefferson
2020 arXiv   pre-print
By studying their Fisher metrics, we derive some general lessons that may have important implications for the application of information geometry in holography.  ...  We then investigate what information the Fisher metric retains about the physics of the underlying theory by studying the geometry for both the classical 2d Ising model and the corresponding 1d free fermion  ...  By the Fisher metric here, we mean the generalization of the definition of the Fisher metric in which the real-valued probability distribution p is simply replaced with a density matrix ρ and the integral  ... 
arXiv:2001.02683v2 fatcat:hxs4kom33zduxepet3cvuotrie

Bankruptcy analysis with self-organizing maps in learning metrics

S. Kaski, J. Sinkkonen, J. Peltonen
2001 IEEE Transactions on Neural Networks  
Kiviluoto for his valuable backround information on bankruptcy analysis and his help regarding the interpretation the data, and the Academy of Finland for financial support.  ...  ACKNOWLEDGMENT The authors would like to thank Finnvera Ltd. and particularly P. Bergius for the data set, K.  ...  The metric is based on the Fisher information matrix, which results from a local approximation of the Kullback-Leibler divergence between the posterior densities at close-by points in the primary data  ... 
doi:10.1109/72.935102 pmid:18249924 fatcat:zuyrxbunyzbotamcv6v42xceeq

Fisher information metric and Poisson kernels

Mitsuhiro Itoh, Yuichi Shishido
2008 Differential geometry and its applications  
We show that this map is embedding and the pull-back metric of the Fisher information metric by this embedding coincides with the original metric of X up to constant provided X is a rank one symmetric  ...  A complete Riemannian manifold X with negative curvature satisfying −b 2 K X −a 2 < 0 for some constants a, b, is naturally mapped in the space of probability measures on the ideal boundary ∂X by assigning  ...  The authors would also like to thank to a referee for useful comments.  ... 
doi:10.1016/j.difgeo.2007.11.027 fatcat:djuoyazscrf45fefh2z2vcduza

A novel information geometry method for estimating parameters of the Weibull wind speed distribution

M Kurban, E Dokur, S Ceyhan
2018 Proceedings of the Estonian Academy of Sciences  
Non-Euclidean geometry and the Riemannian metric called the Fisher metric or the information metric are used in this approach.  ...  Differential equations derived from the Fisher information matrix are solved for the Weibull statistical manifold by the shooting method.  ...  ACKNOWLEDGEMENT The publication costs of this article were partially covered by the Estonian Academy of Sciences.  ... 
doi:10.3176/proc.2018.1.01 fatcat:6ebbs76sibdvrfneyisvkwydhm

A new distribution metric for image segmentation

Romeil Sandhu, Tryphon Georgiou, Allen Tannenbaum, Joseph M. Reinhardt, Josien P. W. Pluim
2008 Medical Imaging 2008: Image Processing  
Moreover, we briefly provide a theoretical comparison between the popular Fisher Information metric, from which the Bhattacharyya distance originates, with the newly proposed similarity metric.  ...  Forming a natural geodesic, our metric quantifies "distance" for two density functionals as the standard deviation of the difference between logarithms of those distributions.  ...  Information on the National Centers for Biomedical Computing can be obtained from  ... 
doi:10.1117/12.769010 dblp:conf/miip/SandhuGT08 fatcat:heh366uvtvdvnl5rqqcvidu4de

Information Geometry and Classical Cramér-Rao Type Inequalities [article]

Kumar Vijay Mishra, M. Ashok Kumar
2021 arXiv   pre-print
We examine the role of information geometry in the context of classical Cramér-Rao (CR) type inequalities.  ...  The classical deterministic CR inequality is derived from Kullback-Leibler (KL)-divergence.  ...  M. acknowledges support from the National Academies of Sciences, Engineering, and Medicine via Army Research Laboratory Harry Diamond Distinguished Postdoctoral Fellowship.  ... 
arXiv:2104.01061v3 fatcat:augyhh34xjfjpawc6ia62am53m
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