Theory of information and its value:
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| Other Authors: | , |
| Format: | Book |
| Language: | English |
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Cham, Switzerland
Springer
[2020]
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| Subjects: | |
| Online Access: | Inhaltsverzeichnis Klappentext |
| Item Description: | Geistiger Schöpfer nur auf dem Buchdeckel |
| Physical Description: | xxii, 419 Seiten. - Illustrationen |
| ISBN: | 9783030228323 |
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Record in the Search Index
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| adam_text | Contents 1 2 3 Definition of information and entropy in the absence of noise............ 1.1 Definition of entropy in the case of équiprobable outcomes.......... 1.2 Entropy and its properties in the case of non-equiprobable outcomes............................................................................................ 1.3 Conditional entropy. Hierarchical additivity............................... 1.4 Asymptotic equivalence of non-equiprobable and équiprobable outcomes.................................. ........ ................................................. 1.5 Asymptotic equiprobability and entropie stability.......................... 1.6 Definition of entropy of a continuous random variable.................. 1.7 Properties of entropy in the generalized version. Conditional entropy.............................................................................................. Encoding of discrete information in the absence of noise and penalties.............................................................................................. 2.1 Main principles of encoding discrete information.......................... 2.2 Main theorems for encoding without noise. Independent identically distributed messages....................................................... 2.3 Optimal encoding by Huffman. Examples...................................... 2.4 Errors of encoding without noise in the case of a finite code sequence length................................................................................ Encoding in the presence of penalties. First variational problem ....
3.1 Direct method of computing information capacity of a message for one example................................................................................ 3.2 Discrete channel without noise and its capacity.............................. 3.3 Solution of the first variational problem. Thermodynamic parameters and potentials................................................................. 3.4 Examples of application of general methods for computation of channel capacity ............................................................................... 3.5 Methods of potentials in the case of a large number of parameters 3.6 Capacity of a noiseless channel with penalties in a generalized version .............................................................................................. 1 3 5 8 12 16 22 28 35 36 40 44 48 53 54 56 58 65 70 74 xix
XX 4 5 6 7 Contents First asymptotic theorem and related results....................................... 4.1 Potential Г or the cumulant generating function............................ 4.2 Some asymptotic results of statistical thermodynamics. Stability of the canonical distribution............................................................. 4.3 Asymptotic equivalence of two types of constraints...................... 4.4 Some theorems about the characteristic potential .......................... Computation of entropy for special cases. Entropy of stochastic processes...................................................................................................... 5.1 Entropy of a segment of a stationary discrete process and entropy rate .................................................................................................... 5.2 Entropy of a Markov chain............................................................... 5.3 Entropy rate of part of the components of a discrete Markov process and of a conditional Markov process.................................. 5.4 Entropy of Gaussian random variables............................................ 5.5 Entropy of a stationary sequence. Gaussian sequence.................... 5.6 Entropy of stochastic processes in continuous time. General concepts and relations................... 5.7 Entropy of a Gaussian process in continuous time........................... 5.8 Entropy of a stochastic point process .............................................. 5.9 Entropy of a discrete Markov process in continuoustime................
5.10 Entropy of diffusion Markov processes........................................... 5.11 Entropy of a composite Markov process, a conditionalprocess, and some components of a Markov process.................................... Information in the presence of noise. Shannon’s amount of information............................................................................................ 6.1 Information losses under degenerate transformations and simple noise.................................................................................................. 6.2 Mutual information for discrete random variables.......................... 6.3 Conditional mutual information. Hierarchical additivity of information......................................................................................... 6.4 Mutual information in the general case........................................... 6.5 Mutual information for Gaussian variables..................................... 6.6 Information rate of stationary and stationary-connected processes. Gaussian processes......................................................... 6.7 Mutual information of components of a Markov process.............. Message transmission in the presence of noise. Second asymptotic theorem and its various formulations..................................................... 7.1 Principles of information transmission and information reception in the presence of noise..................................................................... 7.2 Random code and the mean probability of error............................ 7.3
Asymptotic zero probability of decoding error. Shannon’s theorem (second asymptotic theorem)..................................... 7.4 Asymptotic formula for the probability of error.............................. 77 78 82 89 94 103 104 107 113 123 128 134 137 144 153 157 161 173 173 178 181 187 189 196 202 217 218 221 225 228
Contents xxi 7.5 7.6 Enhanced estimators for optimal decoding..................................... 232 Some general relations between entropies and mutual informations for encoding and decoding......................................... 243 8 9 Channel capacity. Important particular cases of channels.................. 8.1 Definition of channel capacity......................................................... 8.2 Solution of the second variational problem. Relations for channel capacity and potential....................................................................... 8.3 The type of optimal distribution and the partition function............ 8.4 Symmetric channels ......................................................................... 8.5 Binary channels................................................................................ 8.6 Gaussian channels............................. 8.7 Stationary Gaussian channels........................................................... 8.8 Additive channels............................................................................ Definition of the value of information..................................................... 9.1 Reduction of average cost under uncertainty reduction.................. 9.2 Value of Hartley’s information amount. An example...................... 9.3 Definition of the value of Shannon’s information amount and a-information .................................................................................. 9.4 Solution of the third variational problem. The corresponding
potentials............................................................................................ 9.5 Solution of a variational problem under severaladditional assumptions........................................................................................ 9.6 Value of Boltzmann’s information amount..................................... 9.7 Another approach to defining the value of Shannon’s information 249 249 252 259 262 264 267 277 284 289 290 294 300 304 313 318 321 10 Value of Shannon’s information for the most important Bayesian systems........................................................................................................ 10.1 Two-state system.............................................................................. 10.2 Systems with translation invariant cost function ......................... 10.3 Gaussian Bayesian systems............................................................. 10.4 Stationary Gaussian systems ........................................................... 11 Asymptotic results about the value of information. Third asymptotic theorem.................................................................................. 11.1 On the distinction between the value functions of different types of information. Preliminary forms......................... ........................ 11.2 Theorem about asymptotic equivalence of the value functions of different types of information........................................................... 11.3 Rate of convergence between the values of Shannon’s and Hartley’s information
....................................................................... 11.4 Alternative forms of the main result. Generalizations and special cases.................................................................................................. 11.5 Generalized Shannon’s theorem...................................................... 327 327 331 338 346 353 354 357 369 379 385
Contents xxii 12 A Information theory and the second law of thermodynamics.............. 12.1 Information about a physical system being in thermodynamic equilibrium. The generalized second law of thermodynamics .... 12.2 Influx of Shannon’s information and transformation of heat into work ................................................................................................... 12.3 Energy costs of creating and recording information. An example . 12.4 Energy costs of creating and recording information. General formulation......................................................................................... 12.5 Energy costs in physical channels................................................... 391 392 395 399 403 405 Some matrix (operator) identities........................................................... 409 A. 1 Rules for operator transfer from left to right.................................... 409 A.2 Determinant of a block matrix......................................................... 410 References............................................................................................................ 413 Index.............................................................. 417
Ruslan L Stratonovkh Edited by: Roman V. Belavkin · Panos M. Pardalos · Jose C. Principe Theory of Information and its Value This English version of Ruslan L. Stratonovichs Theory ofInformation (1975) builds on theory and provides methods, techniques, and concepts toward utilizing critical ap plications. Unifying theories of information, optimization, and statistical physics, the value of information theory has gained recognition in data science, machine learning, and artificial intelligence. With the emergence of a data-driven economy, progress in machine learning, artificial intelligence algorithms, and increased computational re sources, the need for comprehending information is essential. This book is even more relevant today than when it was first published in 1975. It extends the classic work of R.L. Stratonovich, one of the original developers of the symmetrized version of stochastic calculus and filtering theory, to name just two topics. Each chapter begins with basic, fundamental ideas, supported by clear examples; the material then advances to great detail and depth. The reader is not required to be familiar with the more difficult and specific material. Rather, the treasure trove of examples of stochastic processes and problems makes this book accessible to a wide readership of researchers, postgraduates, and undergraduate students in mathematics, engineering, physics and computer science who are specializing in information theory, data analysis, or machine learning.
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| language | English |
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| spelling | Stratonovič, Ruslan L. 1930- Verfasser (DE-588)1063979455 aut Theory of information and its value Ruslan L. Stratonovich ; Roman V. Belavkin, Panos M. Pardalos, Jose C. Principe (editors) Cham, Switzerland Springer [2020] © 2020 xxii, 419 Seiten. - Illustrationen txt rdacontent n rdamedia nc rdacarrier Geistiger Schöpfer nur auf dem Buchdeckel Informationstheorie (DE-588)4026927-9 gnd rswk-swf Informationstheorie (DE-588)4026927-9 s DE-604 Belavkin, Roman V. (DE-588)1206550384 edt Pardalos, Panos M. 1954- (DE-588)115385827 edt Príncipe, José C. 1950- Sonstige (DE-588)138325022 oth Erscheint auch als Online-Ausgabe 978-3-030-22833-0 Digitalisierung UB Regensburg - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=031847308&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Regensburg - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=031847308&sequence=000003&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext |
| spellingShingle | Stratonovič, Ruslan L. 1930- Theory of information and its value Informationstheorie (DE-588)4026927-9 gnd |
| subject_GND | (DE-588)4026927-9 |
| title | Theory of information and its value |
| title_auth | Theory of information and its value |
| title_exact_search | Theory of information and its value |
| title_full | Theory of information and its value Ruslan L. Stratonovich ; Roman V. Belavkin, Panos M. Pardalos, Jose C. Principe (editors) |
| title_fullStr | Theory of information and its value Ruslan L. Stratonovich ; Roman V. Belavkin, Panos M. Pardalos, Jose C. Principe (editors) |
| title_full_unstemmed | Theory of information and its value Ruslan L. Stratonovich ; Roman V. Belavkin, Panos M. Pardalos, Jose C. Principe (editors) |
| title_short | Theory of information and its value |
| title_sort | theory of information and its value |
| topic | Informationstheorie (DE-588)4026927-9 gnd |
| topic_facet | Informationstheorie |
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