The hartley (symbol Hart), also called a ban, or a dit (short for decimal digit),
is a
logarithmic unit that measures
information or
entropy, based on base 10
logarithms and powers of 10. One hartley is the information content of an event if the
probability of that event occurring is .
It is therefore equal to the information contained in one
decimal digit (or dit), assuming ''
a priori'' equiprobability of each possible value. It is named after
Ralph Hartley.
If
base 2 logarithms and powers of 2 are used instead, then the unit of information is the
shannon or
bit, which is the information content of an event if the
probability of that event occurring is .
Natural logarithm
The natural logarithm of a number is its logarithm to the base of the mathematical constant , which is an irrational and transcendental number approximately equal to . The natural logarithm of is generally written as , , or sometimes, if ...
s and powers of
e define the
nat
Nat or NAT may refer to:
Computing
* Network address translation (NAT), in computer networking
Organizations
* National Actors Theatre, New York City, U.S.
* National AIDS trust, a British charity
* National Archives of Thailand
* National As ...
.
One ban corresponds to ln(10)
nat
Nat or NAT may refer to:
Computing
* Network address translation (NAT), in computer networking
Organizations
* National Actors Theatre, New York City, U.S.
* National AIDS trust, a British charity
* National Archives of Thailand
* National As ...
= log
2(10)
Sh, or approximately 2.303
nat
Nat or NAT may refer to:
Computing
* Network address translation (NAT), in computer networking
Organizations
* National Actors Theatre, New York City, U.S.
* National AIDS trust, a British charity
* National Archives of Thailand
* National As ...
, or 3.322 bit (3.322 Sh). A deciban is one tenth of a ban (or about 0.332 Sh); the name is formed from ''ban'' by the
SI prefix
The International System of Units, known by the international abbreviation SI in all languages and sometimes pleonastically as the SI system, is the modern form of the metric system and the world's most widely used system of measurement. E ...
''
deci-''.
Though there is no associated
SI unit,
information entropy is part of the
International System of Quantities, defined by International Standard
IEC 80000-13
ISO 80000 or IEC 80000 is an international standard introducing the International System of Quantities (ISQ).
It was developed and promulgated jointly by the International Organization for Standardization (ISO) and the International Electrote ...
of the
International Electrotechnical Commission
The International Electrotechnical Commission (IEC; in French: ''Commission électrotechnique internationale'') is an international standards organization that prepares and publishes international standards for all electrical, electronic and r ...
.
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.
The ''ban'' and the ''deciban'' were invented by
Alan Turing with
Irving John "Jack" 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.
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
Bayesian inference is a method of statistical inference in which Bayes' theorem is used to update the probability for a hypothesis as more evidence or information becomes available. Bayesian inference is an important technique in statistics, a ...
.
Donald A. Gillies
Donald A. Gillies (; born 1944) is a British philosopher and historian of science and mathematics. He is an Emeritus Professor in the Department of Science and Technology Studies at University College London.
Career
After undergraduate studie ...
, however, argued the ''ban'' is, in effect, the same as
Karl Popper's measure of the severity of a test.
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 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.
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:
See also
*
bit
*
decibel
The decibel (symbol: dB) is a relative unit of measurement equal to one tenth of a bel (B). It expresses the ratio of two values of a power or root-power quantity on a logarithmic scale. Two signals whose levels differ by one decibel have a po ...
Notes
References
{{reflist, refs=
[{{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]
[{{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]
[{{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
The ''Bell Labs Technical Journal'' is the in-house scientific journal for scientists of Nokia Bell Labs, published yearly by the IEEE society. The managing editor is Charles Bahr.
The journal was originally established as the ''Bell System Techn ...
, pages=535–563 , url=http://dotrose.com/etext/90_Miscellaneous/transmission_of_information_1928b.pdf , access-date=2008-03-27
[{{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]
[{{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
''Biometrika'' is a peer-reviewed scientific journal published by Oxford University Press for thBiometrika Trust The editor-in-chief is Paul Fearnhead (Lancaster University). The principal focus of this journal is theoretical statistics. It was es ...
, date=1979 , volume=66 , issue=2 , pages=393–396 , doi=10.1093/biomet/66.2.393 , mr=0548210
[{{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]
[{{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]
[{{cite book , title=Digitale Rechenautomaten – Eine Einführung , language=de , trans-title=Digital Computers – An Introduction , chapter=1.8.1 Begriffe aus der Informationstheorie , trans-chapter=1.8.1 Terms used in information theory , author-first=Rainer , author-last=Klar , publisher= Walter de Gruyter & Co. / {{ill, G. J. Göschen'sche Verlagsbuchhandlung, de, G. J. Göschen’sche Verlagsbuchhandlung , publication-place=Berlin, Germany , series=Sammlung Göschen , volume=1241/1241a , date=1970-02-01 , isbn=3-11-083160-0 , id={{ISBN, 978-3-11-083160-3. Archiv-Nr. 7990709. , page=35 , edition=1 , url=https://books.google.com/books?id=QnqVDwAAQBAJ&pg=PA35 , access-date=2020-04-13 , url-status=live , archive-url=https://web.archive.org/web/20200418205642/https://books.google.de/books?redir_esc=y&hl=de&id=QnqVDwAAQBAJ&q=dit#v=snippet&q=dit&f=false , archive-date=2020-04-18 (205 pages) (NB. A 2019 reprint of the first edition is available under {{ISBN, 3-11002793-3, 978-3-11002793-8. A reworked and expanded 4th edition exists as well.)]
[{{anchor, Klar-1989{{cite book , title=Digitale Rechenautomaten – Eine Einführung in die Struktur von Computerhardware , language=de , trans-title=Digital Computers – An Introduction into the structure of computer hardware , chapter=1.9.1 Begriffe aus der Informationstheorie , trans-chapter=1.9.1 Terms used in information theory , author-first=Rainer , author-last=Klar , publisher= Walter de Gruyter & Co. , publication-place=Berlin, Germany , series=Sammlung Göschen , volume=2050 , date=1989 , orig-year=1988-10-01 , isbn=3-11011700-2 , id={{ISBN, 978-3-11011700-4 , page=57 , edition=4th reworked (320 pages)]
Units of information
Logarithmic scales of measurement