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 method used to display numerical data that spans a broad range of values, especially when there are significant differences among the magnitudes of the numbers involved. Unlike a linear scale where each u ...

that measures

information Information is an Abstraction, abstract concept that refers to something which has the power Communication, to inform. At the most fundamental level, it pertains to the Interpretation (philosophy), interpretation (perhaps Interpretation (log ...

entropy Entropy is a scientific concept, most commonly associated with states of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynamics, where it was first recognized, to the micros ...

, based on base 10

logarithm In mathematics, the logarithm of a number is the exponent by which another fixed value, the base, must be raised to produce that number. For example, the logarithm of to base is , because is to the rd power: . More generally, if , the ...

s and powers of 10. One hartley is the information content of an event if the

probability Probability is a branch of mathematics and statistics concerning events and numerical descriptions of how likely they are to occur. The probability of an event is a number between 0 and 1; the larger the probability, the more likely an e ...

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) or numeral is a single symbol used alone (such as "1"), or in combinations (such as "15"), to represent numbers in positional notation, such as the common base 10. The name "digit" originate ...

(or dit), assuming ''

a priori ('from the earlier') and ('from the later') are Latin phrases used in philosophy to distinguish types of knowledge, Justification (epistemology), justification, or argument by their reliance on experience. knowledge is independent from any ...

'' 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. His legacy includes t ...

. 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 communication. 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 represented as ...

, which is the information content of an event if the

of that event occurring is .

Natural logarithm The natural logarithm of a number is its logarithm to the base of a logarithm, base of the e (mathematical constant), mathematical constant , which is an Irrational number, irrational and Transcendental number, transcendental number approxima ...

s and powers of e define the nat. One ban corresponds to ln(10) nat = log₂(10) Sh, or approximately 2.303 nat, 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, internationally known by the abbreviation SI (from French ), is the modern form of the metric system and the world's most widely used system of measurement. It is the only system of measurement with official st ...

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 Syste ...

''. Though there is no associated

SI unit The International System of Units, internationally known by the abbreviation SI (from French ), is the modern form of the metric system and the world's most widely used system of units of measurement, system of measurement. It is the only system ...

information entropy In information theory, the entropy of a random variable quantifies the average level of uncertainty or information associated with the variable's potential states or possible outcomes. This measures the expected amount of information needed ...

is part of the

International System of Quantities The International System of Quantities (ISQ) is a standard system of Quantity, quantities used in physics and in modern science in general. It includes basic quantities such as length and mass and the relationships between those quantities. This ...

, defined by International Standard

IEC 80000-13 ISO/IEC 80000, ''Quantities and units'', is an international standard describing the International System of Quantities (ISQ). It was developed and promulgated jointly by the International Organization for Standardization (ISO) and the Intern ...

of the

International Electrotechnical Commission The International Electrotechnical Commission (IEC; ) is an international standards organization that prepares and publishes international standards for all electrical, electronics, electronic and related technologies. IEC standards cover a va ...

History

The term ''hartley'' is named after

, 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. He was highly influential in the development of theoretical computer ...

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 Bletchley Park estate, estate in Bletchley, Milton Keynes (Buckinghamshire), that became the principal centre of Allies of World War II, Allied World War II cryptography, code-breaking during the S ...

using the

Banburismus Banburismus was a Cryptanalysis, cryptanalytic process developed by Alan Turing at Bletchley Park in United Kingdom, Britain during the Second World War. It was used by Bletchley Park's Hut 8 to help break German ''Kriegsmarine'' (naval) message ...

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 Banbury is an historic market town and civil parish on the River Cherwell in Oxfordshire, South East England. The parish had a population of 54,335 at the 2021 Census. Banbury is a significant commercial and retail centre for the surrounding ...

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 ( or ) is a method of statistical inference in which Bayes' theorem is used to calculate a probability of a hypothesis, given prior evidence, and update it as more information becomes available. Fundamentally, Bayesian infer ...

. 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 ...

, notably as a measure of information in

Bayes factor The Bayes factor is a ratio of two competing statistical models represented by their evidence, and is used to quantify the support for one model over the other. The models in question can have a common set of parameters, such as a null hypothesis ...

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 event A taking place in the presence of B, and the odds of A in the absence of B ...

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:

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 independent, non-governmental, international standard development organization composed of representatives from the national standards organizations of member countries. M ...

(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'' was the in-house scientific journal for scientists of Bell Labs, published yearly by the IEEE society. The journal was originally established as ''The Bell System Technical Journal'' (BSTJ) in New York by the Am ...

, 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, book ...

, 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 the Biometrika Trust. The editor-in-chief is Paul Fearnhead (Lancaster University). The principal focus of this journal is theoretical statistics. It was ...

, 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'' is a peer-reviewed, academic journal of philosophy, owned by the British Society for the Philosophy of Science and published by University of Chicago Press. The journal publishes work that uses p ...

, 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 Units of level

History

Usage as a unit of odds

See also

Notes

References