The hartley (symbol Hart), also called a ban, or a dit (short for decimal digit),
is a
logarithmic unit
A logarithmic scale (or log scale) is a way of displaying numerical data over a very wide range of values in a compact way—typically the largest numbers in the data are hundreds or even thousands of times larger than the smallest numbers. Such a ...
that measures
information
Information is an abstract concept that refers to that which has the power to inform. At the most fundamental level information pertains to the interpretation of that which may be sensed. Any natural process that is not completely random ...
or
entropy
Entropy is a scientific concept, as well as a measurable physical property, that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynam ...
, based on base 10
logarithm
In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number to the base is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 o ...
s and powers of 10. One hartley is the information content of an event if the
probability
Probability is the branch of mathematics concerning numerical descriptions of how likely an Event (probability theory), event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and ...
of that event occurring is .
It is therefore equal to the information contained in one
decimal digit
A numerical digit (often shortened to just digit) is a single symbol used alone (such as "2") or in combinations (such as "25"), to represent numbers in a positional numeral system. The name "digit" comes from the fact that the ten digits (Latin ...
(or dit), assuming ''
a priori
("from the earlier") and ("from the later") are Latin phrases used in philosophy to distinguish types of knowledge, justification, or argument by their reliance on empirical evidence or experience. knowledge is independent from current ex ...
'' equiprobability of each possible value. It is named after
Ralph Hartley
Ralph Vinton Lyon Hartley (November 30, 1888 – May 1, 1970) was an American electronics researcher. He invented the Hartley oscillator and the Hartley transform, and contributed to the foundations of information theory.
Biography
Hartley was ...
.
If
base 2 logarithms and powers of 2 are used instead, then the unit of information is the
shannon or
bit
The bit is the most basic unit of information in computing and digital communications. The name is a portmanteau of binary digit. The bit represents a logical state with one of two possible values. These values are most commonly represente ...
, which is the information content of an event if the
probability
Probability is the branch of mathematics concerning numerical descriptions of how likely an Event (probability theory), event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and ...
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-
''Deci'' (symbol d) is a decimal unit prefix in the metric system denoting a factor of one tenth. Proposed in 1793, and adopted in 1795, the prefix comes from the Latin , meaning "tenth". Since 1960, the prefix is part of the International System ...
''.
Though there is no associated
SI unit
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 ...
,
information entropy
In information theory, the entropy of a random variable is the average level of "information", "surprise", or "uncertainty" inherent to the variable's possible outcomes. Given a discrete random variable X, which takes values in the alphabet \ ...
is part of the
International System of Quantities
The International System of Quantities (ISQ) consists of the quantities used in physics and in modern science in general, starting with basic quantities such as length and mass, and the relationships between those quantities. This system underlie ...
, 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
Ralph Vinton Lyon Hartley (November 30, 1888 – May 1, 1970) was an American electronics researcher. He invented the Hartley oscillator and the Hartley transform, and contributed to the foundations of information theory.
Biography
Hartley was ...
, 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
Alan Mathison Turing (; 23 June 1912 – 7 June 1954) was an English mathematician, computer scientist, logician, cryptanalyst, philosopher, and theoretical biologist. Turing was highly influential in the development of theoretical com ...
with
Irving John "Jack" Good in 1940, to measure the amount of information that could be deduced by the codebreakers at
Bletchley Park
Bletchley Park is an English country house and estate in Bletchley, Milton Keynes ( Buckinghamshire) that became the principal centre of Allied code-breaking during the Second World War. The mansion was constructed during the years following ...
using the
Banburismus
Banburismus was a cryptanalytic process developed by Alan Turing at Bletchley Park in Britain during the Second World War. It was used by Bletchley Park's Hut 8 to help break German ''Kriegsmarine'' (naval) messages enciphered on Enigma machin ...
procedure, towards determining each day's unknown setting of the German naval
Enigma
Enigma may refer to:
*Riddle, someone or something that is mysterious or puzzling
Biology
*ENIGMA, a class of gene in the LIM domain
Computing and technology
* Enigma (company), a New York-based data-technology startup
* Enigma machine, a family ...
cipher machine. The name was inspired by the enormous sheets of card, printed in the town of
Banbury
Banbury is a historic market town on the River Cherwell in Oxfordshire, South East England. It had a population of 54,335 at the 2021 Census.
Banbury is a significant commercial and retail centre for the surrounding area of north Oxfordshire ...
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, 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
In statistics, the logit ( ) function is the quantile function associated with the standard logistic distribution. It has many uses in data analysis and machine learning, especially in data transformations.
Mathematically, the logit is the i ...
, notably as a measure of information in
Bayes factor
The Bayes factor is a ratio of two competing statistical models represented by their marginal likelihood, and is used to quantify the support for one model over the other. The models in questions can have a common set of parameters, such as a nu ...
s,
odds ratio
An odds ratio (OR) is a statistic that quantifies the strength of the association between two events, A and B. The odds ratio is defined as the ratio of the odds of A in the presence of B and the odds of A in the absence of B, or equivalently (due ...
s (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
The bit is the most basic unit of information in computing and digital communications. The name is a portmanteau of binary digit. The bit represents a logical state with one of two possible values. These values are most commonly represente ...
*
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
The International Organization for Standardization (ISO ) is an international standard development organization composed of representatives from the national standards organizations of member countries. Membership requirements are given in Ar ...
(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
Dover Publications, also known as Dover Books, is an American book publisher founded in 1941 by Hayward and Blanche Cirker. It primarily reissues books that are out of print from their original publishers. These are often, but not always, books ...
, 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
''British Journal for the Philosophy of Science'' (''BJPS'') is a peer-reviewed, academic journal of philosophy, owned by the British Society for the Philosophy of Science (BSPS) and published by University of Chicago Press. The journal publishes ...
, 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