Information Geometric Duality of <i>ϕ</i>-Deformed Exponential Families
In the world of generalized entropies—which, for example, play a role in physical systems with sub- and super-exponential phase space growth per degree of freedom—there are two ways for implementing constraints in the maximum entropy principle: linear and escort constraints. Both...
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MDPI AG
2019-01-01
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| Series: | Entropy |
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| Online Access: | https://www.mdpi.com/1099-4300/21/2/112 |
| _version_ | 1798039254058663936 |
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| author | Jan Korbel Rudolf Hanel Stefan Thurner |
| author_facet | Jan Korbel Rudolf Hanel Stefan Thurner |
| author_sort | Jan Korbel |
| collection | DOAJ |
| description | In the world of generalized entropies—which, for example, play a role in physical systems with sub- and super-exponential phase space growth per degree of freedom—there are two ways for implementing constraints in the maximum entropy principle: linear and escort constraints. Both appear naturally in different contexts. Linear constraints appear, e.g., in physical systems, when additional information about the system is available through higher moments. Escort distributions appear naturally in the context of multifractals and information geometry. It was shown recently that there exists a fundamental duality that relates both approaches on the basis of the corresponding deformed logarithms (deformed-log duality). Here, we show that there exists another duality that arises in the context of information geometry, relating the Fisher information of <inline-formula> <math display="inline"> <semantics> <mi>ϕ</mi> </semantics> </math> </inline-formula>-deformed exponential families that correspond to linear constraints (as studied by J.Naudts) to those that are based on escort constraints (as studied by S.-I. Amari). We explicitly demonstrate this information geometric duality for the case of <inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <mi>c</mi> <mo>,</mo> <mi>d</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>-entropy, which covers all situations that are compatible with the first three Shannon⁻Khinchin axioms and that include Shannon, Tsallis, Anteneodo⁻Plastino entropy, and many more as special cases. Finally, we discuss the relation between the deformed-log duality and the information geometric duality and mention that the escort distributions arising in these two dualities are generally different and only coincide for the case of the Tsallis deformation. |
| first_indexed | 2024-04-11T21:51:20Z |
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| id | doaj.art-7247ab584055465db07b0a8e1bcf5035 |
| institution | Directory Open Access Journal |
| issn | 1099-4300 |
| language | English |
| last_indexed | 2024-04-11T21:51:20Z |
| publishDate | 2019-01-01 |
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| series | Entropy |
| spelling | doaj.art-7247ab584055465db07b0a8e1bcf50352022-12-22T04:01:14ZengMDPI AGEntropy1099-43002019-01-0121211210.3390/e21020112e21020112Information Geometric Duality of <i>ϕ</i>-Deformed Exponential FamiliesJan Korbel0Rudolf Hanel1Stefan Thurner2Section for Science of Complex Systems, CeMSIIS, Medical University of Vienna, Spitalgasse 23, A-1090 Vienna, AustriaSection for Science of Complex Systems, CeMSIIS, Medical University of Vienna, Spitalgasse 23, A-1090 Vienna, AustriaSection for Science of Complex Systems, CeMSIIS, Medical University of Vienna, Spitalgasse 23, A-1090 Vienna, AustriaIn the world of generalized entropies—which, for example, play a role in physical systems with sub- and super-exponential phase space growth per degree of freedom—there are two ways for implementing constraints in the maximum entropy principle: linear and escort constraints. Both appear naturally in different contexts. Linear constraints appear, e.g., in physical systems, when additional information about the system is available through higher moments. Escort distributions appear naturally in the context of multifractals and information geometry. It was shown recently that there exists a fundamental duality that relates both approaches on the basis of the corresponding deformed logarithms (deformed-log duality). Here, we show that there exists another duality that arises in the context of information geometry, relating the Fisher information of <inline-formula> <math display="inline"> <semantics> <mi>ϕ</mi> </semantics> </math> </inline-formula>-deformed exponential families that correspond to linear constraints (as studied by J.Naudts) to those that are based on escort constraints (as studied by S.-I. Amari). We explicitly demonstrate this information geometric duality for the case of <inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <mi>c</mi> <mo>,</mo> <mi>d</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>-entropy, which covers all situations that are compatible with the first three Shannon⁻Khinchin axioms and that include Shannon, Tsallis, Anteneodo⁻Plastino entropy, and many more as special cases. Finally, we discuss the relation between the deformed-log duality and the information geometric duality and mention that the escort distributions arising in these two dualities are generally different and only coincide for the case of the Tsallis deformation.https://www.mdpi.com/1099-4300/21/2/112generalized entropy<i>ϕ</i>-deformed familyFisher informationinformation geometryCramér–Rao bound(<i>c</i>,<i>d</i>)-entropy |
| spellingShingle | Jan Korbel Rudolf Hanel Stefan Thurner Information Geometric Duality of <i>ϕ</i>-Deformed Exponential Families Entropy generalized entropy <i>ϕ</i>-deformed family Fisher information information geometry Cramér–Rao bound (<i>c</i>,<i>d</i>)-entropy |
| title | Information Geometric Duality of <i>ϕ</i>-Deformed Exponential Families |
| title_full | Information Geometric Duality of <i>ϕ</i>-Deformed Exponential Families |
| title_fullStr | Information Geometric Duality of <i>ϕ</i>-Deformed Exponential Families |
| title_full_unstemmed | Information Geometric Duality of <i>ϕ</i>-Deformed Exponential Families |
| title_short | Information Geometric Duality of <i>ϕ</i>-Deformed Exponential Families |
| title_sort | information geometric duality of i ϕ i deformed exponential families |
| topic | generalized entropy <i>ϕ</i>-deformed family Fisher information information geometry Cramér–Rao bound (<i>c</i>,<i>d</i>)-entropy |
| url | https://www.mdpi.com/1099-4300/21/2/112 |
| work_keys_str_mv | AT jankorbel informationgeometricdualityofiphideformedexponentialfamilies AT rudolfhanel informationgeometricdualityofiphideformedexponentialfamilies AT stefanthurner informationgeometricdualityofiphideformedexponentialfamilies |