Hartley (unit): Difference between revisions
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{{Fundamental info units}} |
{{Fundamental info units}} |
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The '''hartley''' (symbol '''Hart'''), also called a '''ban''', or a '''dit''' (short for '''d'''ecimal dig'''it'''), is a [[logarithmic unit]] which measures [[information]] or [[information entropy|entropy]], based on base 10 [[logarithm]]s and powers of 10, rather than the powers of 2 and [[binary logarithm|base 2 logarithms]] which define the [[bit]], or [[shannon (unit)|shannon]]. One ban or hartley is the information content of an event if the [[probability]] of that event occurring is 1/10.<ref |
The '''hartley''' (symbol '''Hart'''), also called a '''ban''', or a '''dit''' (short for '''d'''ecimal dig'''it'''),<ref name="Lukoff_1979"/> is a [[logarithmic unit]] which measures [[information]] or [[information entropy|entropy]], based on base 10 [[logarithm]]s and powers of 10, rather than the powers of 2 and [[binary logarithm|base 2 logarithms]] which define the [[bit]], or [[shannon (unit)|shannon]]. One ban or hartley is the information content of an event if the [[probability]] of that event occurring is 1/10.<ref name="IEC"/> It is therefore equal to the information contained in one [[decimal digit]] (or dit), assuming ''[[A priori probability|a priori]]'' equiprobability of each possible value. |
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As a [[bit]] corresponds to a [[binary digit]], a ban corresponds to a [[decimal digit]]. A '''deciban''' is one tenth of a ban; the name is formed from ''ban'' by the [[SI prefix]] ''[[deci-]]''. |
As a [[bit]] corresponds to a [[binary digit]], a ban corresponds to a [[decimal digit]]. A '''deciban''' is one tenth of a ban; the name is formed from ''ban'' by the [[SI prefix]] ''[[deci-]]''. |
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== History == |
== History == |
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The term ''hartley'' is named after [[Ralph Hartley]], who suggested in 1928 to measure information using a logarithmic base equal to the number of distinguishable states in its representation, which would be the base 10 for a decimal digit.<ref |
The term ''hartley'' is named after [[Ralph Hartley]], who suggested in 1928 to measure information using a logarithmic base equal to the number of distinguishable states in its representation, which would be the base 10 for a decimal digit.<ref name="Hartley_1928"/><ref name="Reza_1994"/> |
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The ''ban'' and the ''deciban'' were invented by [[Alan Turing]] with [[I. J. Good]] in 1940, to measure the amount of information that could be deduced by the codebreakers at [[Bletchley Park]] using the [[Banburismus]] procedure, towards determining each day's unknown setting of the German naval [[Enigma machine|Enigma]] cipher machine. The name was inspired by the enormous sheets of card, printed in the town of [[Banbury]] about 30 miles away, that were used in the process.<ref name=" |
The ''ban'' and the ''deciban'' were invented by [[Alan Turing]] with [[I. J. Good]] in 1940, to measure the amount of information that could be deduced by the codebreakers at [[Bletchley Park]] using the [[Banburismus]] procedure, towards determining each day's unknown setting of the German naval [[Enigma machine|Enigma]] cipher machine. The name was inspired by the enormous sheets of card, printed in the town of [[Banbury]] about 30 miles away, that were used in the process.<ref name="Good_1979"/> |
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[[I. J. Good|Jack Good]] argued that the sequential summation of ''decibans'' to build up a measure of the weight of evidence in favour of a hypothesis, is essentially [[Bayesian inference]].<ref name |
[[I. J. Good|Jack Good]] argued that the sequential summation of ''decibans'' to build up a measure of the weight of evidence in favour of a hypothesis, is essentially [[Bayesian inference]].<ref name="Good_1979"/> [[Donald A. Gillies]], however, argued the ''ban'' is, in effect, the same as [[Karl Popper|Karl Popper's]] measure of the severity of a test.<ref name="Gillies_1990"/> |
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==Usage as a unit of odds== |
==Usage as a unit of odds== |
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The deciban is a particularly useful unit for [[log-odds]], notably as a measure of information in [[Bayes factor]]s, [[odds ratio]]s (ratio of odds, so log is difference of log-odds), or [[weight of evidence|weights of evidence]]. 10 decibans corresponds to odds of 10:1; 20 decibans to 100:1 odds, etc. According to [[I. J. Good]], a change in a weight of evidence of 1 deciban (i.e., a change in the odds from evens to about 5:4) is about as finely as humans can reasonably be expected to quantify their degree of belief in a hypothesis.<ref |
The deciban is a particularly useful unit for [[log-odds]], notably as a measure of information in [[Bayes factor]]s, [[odds ratio]]s (ratio of odds, so log is difference of log-odds), or [[weight of evidence|weights of evidence]]. 10 decibans corresponds to odds of 10:1; 20 decibans to 100:1 odds, etc. According to [[I. J. Good]], a change in a weight of evidence of 1 deciban (i.e., a change in the odds from evens to about 5:4) is about as finely as humans can reasonably be expected to quantify their degree of belief in a hypothesis.<ref name="Good_1985"/> |
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Odds corresponding to integer decibans can often be well-approximated by simple integer ratios; these are collated below. Value to two decimal places, simple approximation (to within about 5%), with more accurate approximation (to within 1%) if simple one is inaccurate: |
Odds corresponding to integer decibans can often be well-approximated by simple integer ratios; these are collated below. Value to two decimal places, simple approximation (to within about 5%), with more accurate approximation (to within 1%) if simple one is inaccurate: |
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==References== |
==References== |
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{{reflist |
{{reflist|refs= |
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<ref name="Lukoff_1979">{{cite book |author-last=Lukoff |author-first=Herman |author-link=Herman Lukoff |title=From Dits to Bits: A personal history of the electronic computer |date=1979 |publisher=Robotics Press |location=Portland, Oregon, USA |isbn=0-89661-002-0 |lccn=79-90567}}</ref> |
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<ref name="IEC">{{cite web |title=IEC 80000-13:2008 |publisher=[[International Organization for Standardization]] (ISO) |url=http://www.iso.org/iso/catalogue_detail?csnumber=31898 |access-date=2013-07-21}}</ref> |
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<ref name="Hartley_1928">{{cite journal |author-last=Hartley |author-first=Ralph Vinton Lyon |author-link=Ralph Vinton Lyon Hartley |title=Transmission of Information |date=July 1928 |volume=VII |issue=3 |journal=[[Bell System Technical Journal]] |pages=535-563 |url=http://dotrose.com/etext/90_Miscellaneous/transmission_of_information_1928b.pdf |access-date=2008-03-27}}</ref> |
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<ref name="Reza_1994">{{cite book |author-last=Reza |author-first=Fazlollah M. |author-link=Fazlollah M. Reza |title=An Introduction to Information Theory |location=New York |publisher=[[Dover Publications]] |date=1994 |isbn=0-486-68210-2}}</ref> |
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<ref name="Good_1979">{{cite journal |author-last=Good |author-first=Irving John |author-link=Irving John Good |title=Studies in the History of Probability and Statistics. XXXVII A. M. Turing's statistical work in World War II |journal=[[Biometrika]] |date=1979 |volume=66 |issue=2 |pages=393-396 |doi=10.1093/biomet/66.2.393 |mr=0548210}}</ref> |
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<ref name="Gillies_1990">{{cite journal |author-last=Gillies |author-first=Donald A. |author-link=Donald A. Gillies |date=1990 |title=The Turing-Good Weight of Evidence Function and Popper's Measure of the Severity of a Test |periodical=[[British Journal for the Philosophy of Science]] |volume=41 |issue=1 |pages=143-146 |mr=055678 |jstor=688010 |doi=10.1093/bjps/41.1.143}}</ref> |
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<ref name="Good_1985">{{cite journal |title=Weight of Evidence: A Brief Survey |author-last=Good |author-first=Irving John |author-link=Irving John Good |date=1985 |journal=Bayesian Statistics |volume=2 |pages=253 |url=http://www.waterboards.ca.gov/water_issues/programs/tmdl/docs/303d_policydocs/207.pdf |access-date=2012-12-13}}</ref> |
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}} |
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[[Category:Units of information]] |
[[Category:Units of information]] |
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Revision as of 09:17, 3 August 2018
|
Units of information |
| Information-theoretic |
|---|
| Data storage |
| Quantum information |
The hartley (symbol Hart), also called a ban, or a dit (short for decimal digit),[1] is a logarithmic unit which measures information or entropy, based on base 10 logarithms and powers of 10, rather than the powers of 2 and base 2 logarithms which define the bit, or shannon. One ban or hartley is the information content of an event if the probability of that event occurring is 1/10.[2] It is therefore equal to the information contained in one decimal digit (or dit), assuming a priori equiprobability of each possible value.
As a bit corresponds to a binary digit, a ban corresponds to a decimal digit. A deciban is one tenth of a ban; the name is formed from ban by the SI prefix deci-.
One hartley corresponds to log2(10) bit = ln(10) nat, or approximately 3.322 Sh,[a] or 2.303 nat. A deciban is about 0.332 Sh.
Though not an SI unit, the hartley is part of the International System of Quantities, defined by International Standard IEC 80000-13 of the International Electrotechnical Commission. It is named after Ralph Hartley.
History
The term hartley is named after Ralph Hartley, who suggested in 1928 to measure information using a logarithmic base equal to the number of distinguishable states in its representation, which would be the base 10 for a decimal digit.[3][4]
The ban and the deciban were invented by Alan Turing with I. J. Good in 1940, to measure the amount of information that could be deduced by the codebreakers at Bletchley Park using the Banburismus procedure, towards determining each day's unknown setting of the German naval Enigma cipher machine. The name was inspired by the enormous sheets of card, printed in the town of Banbury about 30 miles away, that were used in the process.[5]
Jack Good argued that the sequential summation of decibans to build up a measure of the weight of evidence in favour of a hypothesis, is essentially Bayesian inference.[5] Donald A. Gillies, however, argued the ban is, in effect, the same as Karl Popper's measure of the severity of a test.[6]
Usage as a unit of odds
The deciban is a particularly useful unit for log-odds, notably as a measure of information in Bayes factors, odds ratios (ratio of odds, so log is difference of log-odds), or weights of evidence. 10 decibans corresponds to odds of 10:1; 20 decibans to 100:1 odds, etc. According to I. J. Good, a change in a weight of evidence of 1 deciban (i.e., a change in the odds from evens to about 5:4) is about as finely as humans can reasonably be expected to quantify their degree of belief in a hypothesis.[7]
Odds corresponding to integer decibans can often be well-approximated by simple integer ratios; these are collated below. Value to two decimal places, simple approximation (to within about 5%), with more accurate approximation (to within 1%) if simple one is inaccurate:
| decibans | exact value |
approx. value |
approx. ratio |
accurate ratio |
probability |
|---|---|---|---|---|---|
| 0 | 100/10 | 1 | 1:1 | 50% | |
| 1 | 101/10 | 1.26 | 5:4 | 56% | |
| 2 | 102/10 | 1.58 | 3:2 | 8:5 | 61% |
| 3 | 103/10 | 2.00 | 2:1 | 67% | |
| 4 | 104/10 | 2.51 | 5:2 | 71.5% | |
| 5 | 105/10 | 3.16 | 3:1 | 19:6, 16:5 | 76% |
| 6 | 106/10 | 3.98 | 4:1 | 80% | |
| 7 | 107/10 | 5.01 | 5:1 | 83% | |
| 8 | 108/10 | 6.31 | 6:1 | 19:3, 25:4 | 86% |
| 9 | 109/10 | 7.94 | 8:1 | 89% | |
| 10 | 1010/10 | 10 | 10:1 | 91% |
See also
Notes
- ^ This value, approximately 10/3, but slightly less, can be understood simply because : 3 decimal digits are slightly less information than 10 binary digits, so 1 decimal digit is slightly less than 10/3 binary digits.
References
- ^ Lukoff, Herman (1979). From Dits to Bits: A personal history of the electronic computer. Portland, Oregon, USA: Robotics Press. ISBN 0-89661-002-0. LCCN 79-90567.
- ^ "IEC 80000-13:2008". International Organization for Standardization (ISO). Retrieved 2013-07-21.
- ^ Hartley, Ralph Vinton Lyon (July 1928). "Transmission of Information" (PDF). Bell System Technical Journal. VII (3): 535–563. Retrieved 2008-03-27.
- ^ Reza, Fazlollah M. (1994). An Introduction to Information Theory. New York: Dover Publications. ISBN 0-486-68210-2.
- ^ a b Good, Irving John (1979). "Studies in the History of Probability and Statistics. XXXVII A. M. Turing's statistical work in World War II". Biometrika. 66 (2): 393–396. doi:10.1093/biomet/66.2.393. MR 0548210.
- ^ Gillies, Donald A. (1990). "The Turing-Good Weight of Evidence Function and Popper's Measure of the Severity of a Test". British Journal for the Philosophy of Science. 41 (1): 143–146. doi:10.1093/bjps/41.1.143. JSTOR 688010. MR 0055678.
- ^ Good, Irving John (1985). "Weight of Evidence: A Brief Survey" (PDF). Bayesian Statistics. 2: 253. Retrieved 2012-12-13.