Hangover

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Hangover

Hangover
Other names	veisalgia from Norwegian: kveis, "discomfort following overindulgence", and Greek: ἄλγος álgos, "pain"
The Day After by Edvard Munch, 1894–95

A hangover is the experience of various unpleasant physiological and psychological effects usually following the consumption of alcohol, such as wine, beer, and liquor. Hangovers can last for several hours or for more than 24 hours. Typical symptoms of a hangover may include headache, drowsiness, weakness, concentration problems, dry mouth, dizziness, fatigue, muscle ache, gastrointestinal distress (e.g., nausea, vomiting, diarrhea), absence of hunger, light sensitivity, depression, sweating, hyper-excitability, high blood pressure, irritability, and anxiety.

While the causes of a hangover are still poorly understood, several factors are known to be involved including acetaldehyde accumulation, changes in the immune system and glucose metabolism, dehydration, metabolic acidosis, disturbed prostaglandin synthesis, increased cardiac output, vasodilation, sleep deprivation, and malnutrition. Beverage-specific effects of additives or by-products such as congeners in alcoholic beverages also play an important role. The symptoms usually occur after the intoxicating effect of the alcohol begins to wear off, generally the morning after a night of heavy drinking.

Though many possible remedies and folk cures have been suggested, there is no compelling evidence to suggest that any are effective for preventing or treating hangovers. Avoiding alcohol or drinking in moderation are the most effective ways to avoid a hangover. The socioeconomic consequences of hangovers include workplace absenteeism, impaired job performance, reduced productivity and poor academic achievement. A hangover may also impair performance during potentially dangerous daily activities such as driving a car or operating heavy machinery.

Signs and symptoms

One of the signs of a severe hangover is a headache.

A painting from 1681 showing a person affected by vomiting, a typical symptom of alcohol hangover

An alcohol hangover is associated with a variety of symptoms that may include drowsiness, headache, concentration problems, dry mouth, dizziness, gastrointestinal complaints, fatigue, sweating, nausea, hyper-excitability, anxiety, and a feeling of general discomfort that may last more than 24 hours. Alcohol hangover symptoms develop when blood alcohol concentration falls considerably and peak when it returns to almost zero. Hangover symptoms validated in controlled studies include general malaise, thirst, headache, feeling dizzy or faint, tiredness, loss of appetite, nausea, stomach ache, and feeling as though one's heart is racing. Some symptoms such as changes in sleep pattern and gastrointestinal distress are attributed to direct effects of the alcohol intoxication, or withdrawal symptoms. Drowsiness and impaired cognitive function are the two dominant features of alcohol hangover.

Causes

Hangovers are usually caused by alcohol consumption, but other components of alcoholic beverages might contribute to hangover symptoms or make a hangover worse. For example, congeners are compounds, other than ethyl alcohol, that are produced during fermentation. These substances contribute to the taste and smell of alcoholic beverages. Darker spirits, such as bourbon, which tend to have higher levels of congeners than clear spirits, could worsen hangover symptoms for some people. Sulfites are compounds that are added to wine as preservatives. People who have a sensitivity to sulfites may experience a headache after drinking wine.

Several pathophysiological changes may give rise to the alcohol hangover including increased levels of acetaldehyde, hormonal alterations of the cytokine pathways and decrease of the availability of glucose. Additional associated phenomena are dehydration, metabolic acidosis, disturbed prostaglandin synthesis, increased cardiac output, vasodilation, sleep deprivation and insufficient eating.

Pathophysiology

Alcohol flush reaction as a result of the accumulation of acetaldehyde, the first metabolite of alcohol

Main article: Pharmacology of ethanol

After being ingested, the ethanol in alcoholic beverages is first converted to acetaldehyde by the enzyme alcohol dehydrogenase and then to acetic acid by oxidation and egestion process. These reactions also convert nicotinamide adenine dinucleotide (NAD⁺) to its reduced form NADH in a redox reaction. By causing an imbalance of the NAD⁺/NADH redox system, alcoholic beverages make normal bodily functions more difficult. Consequences of the alcohol induced redox changes in the human body include increased triglyceride production, increased amino acid catabolism, inhibition of the citric acid cycle, lactic acidosis, ketoacidosis, hyperuricemia, disturbance in cortisol and androgen metabolism and increased fibrogenesis. The metabolism of glucose and insulin are also influenced. However, recent studies showed no significant correlation between hangover severity and the concentrations of various hormones, electrolytes, free fatty acids, triglycerides, lactate, ketone bodies, cortisol, and glucose in blood and urine samples.

Alcohol also induces the CYP2E1 enzyme, which metabolizes ethanol and other substances into more reactive toxins. In particular, in binge drinking the enzyme is activated and plays a role in creating a harmful condition known as oxidative stress which can lead to cell death.

Acetaldehyde

Acetaldehyde, the first by-product of ethanol, is between 10 and 30 times more toxic than alcohol itself and can remain at an elevated plateau for many hours after initial ethanol consumption. In addition, certain genetic factors can amplify the negative effects of acetaldehyde. For example, some people (predominantly East Asians) have a mutation in their alcohol dehydrogenase gene that makes this enzyme unusually fast at converting ethanol to acetaldehyde. In addition, about half of all East Asians convert acetaldehyde to acetic acid more slowly (via acetaldehyde dehydrogenase), causing a higher buildup of acetaldehyde than normally seen in other groups. The high concentration of acetaldehyde causes the alcohol flush reaction, colloquially known as the "Asian Flush". Since the alcohol flush reaction is highly uncomfortable and the possibility of hangovers is immediate and severe, people with this gene variant are less likely to become alcoholics.

Acetaldehyde may also influence glutathione peroxidase, a key antioxidant enzyme, and increases the susceptibility to oxidative stress. Likewise, acetic acid (or the acetate ion) can cause additional problems. One study found that injecting sodium acetate into rats caused them to have nociceptive behavior (headaches). In addition, there is a biochemical explanation for this finding. High acetate levels cause adenosine to accumulate in many parts of the brain. But when the rats were given caffeine, which blocks the action of adenosine, they no longer experienced headaches.

Congeners

In addition to ethanol and water, most alcoholic drinks also contain congeners, either as flavoring or as a by-product of fermentation and the wine aging process. While ethanol is by itself sufficient to produce most hangover effects, congeners may potentially aggravate hangover and other residual effects to some extent. Congeners include substances such as amines, amides, acetones, acetaldehydes, polyphenols, methanol, histamines, fusel oil, esters, furfural, and tannins, many but not all of which are toxic. One study in mice indicates that fusel oil may have a mitigating effect on hangover symptoms, while some whiskey congeners such as butanol protect the stomach against gastric mucosal damage in the rat. Different types of alcoholic beverages contain different amounts of congeners. In general, dark liquors have a higher concentration while clear liquors have a lower concentration. Whereas vodka has virtually no more congeners than pure ethanol, bourbon has a total congener content 37 times higher than that found in vodka.

Several studies have examined whether certain types of alcohol cause worse hangovers. All four studies concluded that darker liquors, which have higher congeners, produced worse hangovers. One even showed that hangovers were worse and more frequent with darker liquors. In a 2006 study, an average of 14 standard drinks (330 ml each) of beer was needed to produce a hangover, but only 7 to 8 drinks was required for wine or liquor (note that one standard drink has the same amount of alcohol regardless of type). Another study ranked several drinks by their ability to cause a hangover as follows (from low to high): distilled ethanol diluted with fruit juice, beer, vodka, gin, white wine, whisky, rum, red wine and brandy.

One potent congener is methanol. It is naturally formed in small quantities during fermentation and it can be accidentally concentrated by improper distillation techniques. Metabolism of methanol produces some extremely toxic compounds, such as formaldehyde and formic acid, which may play a role in the severity of hangover. Ethanol slows the conversion of methanol into its toxic metabolites so that most of the methanol can be excreted harmlessly in the breath and urine without forming its toxic metabolites. This may explain the temporary postponement of symptoms reported in the common remedy of drinking more alcohol to relieve hangover symptoms. Since methanol metabolism is effectively inhibited by consumption of alcohol, methanol accumulates during drinking and only begins to be metabolized once ethanol has been cleared. This delayed action makes it an attractive candidate explanation for delayed post-intoxication symptoms and correlations between methanol concentrations and the presence of hangover symptoms that have been found in studies.

Vitamin and electrolyte loss

The metabolic processes required for alcohol elimination deplete essential vitamins and electrolytes. Furthermore, alcohol is a diuretic, causing excretion of electrolytes through urination. After a night of drinking, the resulting lack of key B and C vitamins, as well as potassium, magnesium, and zinc may cause fatigue, aching and other hangover-like symptoms.

Dehydration

Ethanol has a dehydrating effect by causing increased urine production (diuresis), which could cause thirst, dry mouth, dizziness and may lead to an electrolyte imbalance. Studies suggest that electrolyte changes play only a minor role in the genesis of the alcohol hangover and are caused by dehydration effects. Drinking water may help relieve symptoms as a result of dehydration but it is unlikely that rehydration significantly reduces the presence and severity of alcohol hangover. Alcohol's effect on the stomach lining can account for nausea because alcohol stimulates the production of hydrochloric acid in the stomach.

Low blood sugar

Studies show that alcohol hangover is associated with a decrease in blood glucose concentration (less than 70 mg/dl), but the relationship between blood glucose concentration and hangover severity is unclear. Also known as insulin shock, hypoglycemia can lead to coma or even death.

Immune system

In current research, the significant relationship between immune factors and hangover severity is the most convincing among all factors so far studied. An imbalance of the immune system, in particular of cytokine metabolism has been identified as playing a role in the pathophysiology of the hangover state. Especially the hangover symptoms nausea, headache, and fatigue have been suggested to be mediated by changes in the immune system. The concentration of several cytokines have been found to be significantly increased in the blood after alcohol consumption. These include interleukin 12 (IL-12), interferon gamma (IFNγ) and interleukin 10 (IL-10). Some pharmacological studies such as on tolfenamic acid and Opuntia ficus-indica (OFI) have also indicated an involvement of the immune system. These studies suggest that the presence and severity of hangover symptoms can probably be reduced by administration of a cyclooxygenase inhibitor such as aspirin or ibuprofen.

Person-related factors

Several factors which do not in themselves cause alcohol hangover are known to influence its severity. These factors include personality, genetics, health status, age, sex, associated activities during drinking such as smoking, the use of other drugs, physical activity such as dancing, as well as sleep quality and duration.

• Genetics: alleles associated with aldehyde dehydrogenase (ALDH) and flushing phenotypes (alcohol flush reaction) in Asians are known genetic factors that influence alcohol tolerance and the development of hangover effects. Existing data shows that drinkers with genotypes known to lead to acetaldehyde accumulation are more susceptible to hangover effects. The fact that about 25% of heavy drinkers claim that they have never had a hangover is also an indication that genetic variation plays a role in individual differences of hangover severity.
• Age: some people experience hangovers as getting worse as one ages. This is thought to be caused by declining supplies of alcohol dehydrogenase, the enzyme involved in metabolizing alcohol. Although it is actually unknown whether hangover symptoms and severity change with age, research shows that drinking patterns change across ages, and heavy drinking episodes that may result in hangover are much less often experienced as age increases.
• Sex: at the same number of drinks, women are more prone to hangover than men, and this is likely explained by sex differences in the pharmacokinetics of alcohol. Women attain a higher blood alcohol concentration (BAC) than men at the same number of drinks. At equivalent BACs, men and women appear to be indistinguishable with respect to most hangover effects.
• Cigarette smoking: acetaldehyde which is absorbed from cigarette smoking during alcohol consumption is regarded as a contributor to alcohol hangover symptoms.

Management

Main article: Hangover remedies

See also: Alcohol intoxication § Management

"There is no magic potion for beating hangovers—and only time can help. A person must wait for the body to finish clearing the toxic byproducts of alcohol metabolism, to rehydrate, to heal irritated tissue, and to restore immune and brain activity to normal. There is no way to speed up the brain’s recovery from alcohol use—drinking coffee, taking a shower, or having an alcoholic beverage the next morning will not cure a hangover." says NIAAA.

Within the limited amount of serious study on the subject, there is debate about whether a hangover may be prevented or at least mitigated. There is also a vast body of folk medicine and simple quackery. A four-page literature review in the British Medical Journal concludes: "No compelling evidence exists to suggest that any conventional or complementary intervention is effective for preventing or treating alcohol hangover. The most effective way to avoid the symptoms of alcohol induced hangover is to avoid drinking." Most remedies do not significantly reduce overall hangover severity. Some compounds reduce specific symptoms such as vomiting and headache, but are not effective in reducing other common hangover symptoms such as drowsiness and fatigue.

Potentially beneficial

• Rehydration: Drinking water before going to bed or during hangover may relieve dehydration-associated symptoms such as thirst, dizziness, dry mouth, and headache.
• Tolfenamic acid, an inhibitor of prostaglandin synthesis, in a 1983 study reduced headache, nausea, vomiting, irritation but had no effect on tiredness in 30 people.
• Pyritinol: A 1973 study found that large doses (several hundred times the recommended daily intake) of Pyritinol, a synthetic Vitamin B₆ analog, can help to reduce hangover symptoms. Possible side effects of pyritinol include hepatitis (liver damage) due to cholestasis and acute pancreatitis.
• Yeast-based extracts: The difference in the change for discomfort, restlessness, and impatience were statistically significant but no significant differences on blood chemistry parameters, blood alcohol or acetaldehyde concentrations have been found, and it did not significantly improve general well-being.

Unsupported remedies

See also: Hangover remedies

Kudzu roots (Pueraria lobata), a common ingredient in herbal hangover remedies, may have harmful effects when combined with alcohol.

Recommendations for foods, drinks and activities to relieve hangover symptoms abound. The ancient Romans, on the authority of Pliny the Elder, favored raw owl's eggs or fried canary, while the "prairie oyster" restorative, introduced at the 1878 Paris World Exposition, calls for raw egg yolk mixed with Worcestershire sauce, Tabasco sauce, salt and pepper. By 1938, the Ritz-Carlton Hotel provided a hangover remedy in the form of a mixture of Coca-Cola and milk (Coca-Cola itself having been invented, by some accounts, as a hangover remedy). Alcoholic writer Ernest Hemingway relied on tomato juice and beer. Other purported hangover cures include cocktails such as Bloody Mary or Black Velvet (consisting of equal parts champagne and stout). A 1957 survey by an American folklorist found widespread belief in the efficacy of heavy fried foods, tomato juice and sexual activity.

Other untested or discredited treatments include:

• Hair of the dog: The belief is that consumption of further alcohol after the onset of a hangover will relieve symptoms, based upon the theory that the hangover represents a form of alcohol withdrawal and that by satiating the body's need for alcohol the symptoms will be relieved. Social drinkers and alcoholics claim that drinking more alcohol gives relief from hangover symptoms, but research shows that the use of alcohol as a hangover cure seems to predict current or future problem drinking and alcohol use disorder, through negative reinforcement and the development of physical dependence. While the practice is popular in tradition and promoted by many sellers of alcoholic beverages, medical opinion holds that the practice merely postpones the symptoms, and courts addiction. Favored choices include a Corpse Reviver, Fernet Branca and Bloody Mary.
• Kudzu (葛): the root ("lobata") been touted in the West as a remedy, though it's the flower ("flos") that is actually used in traditional remedies. A 2007 review finds evidence for the flower being potentially useful, but the root being likely harmful, as the latter is an inhibitor of ALDH2.
• Artichoke: Research shows that artichoke extract does not prevent the signs and symptoms of alcohol-induced hangover.
• Sauna or steam-bath: Medical opinion holds this may be dangerous, as the combination of alcohol and hyperthermia increases the likelihood of dangerous abnormal heart rhythms.
• Oxygen: There have been anecdotal reports from those with easy access to a breathing oxygen supply—medical staff, and military pilots—that oxygen can also reduce the symptoms of hangovers sometimes caused by alcohol consumption. The theory is that the increased oxygen flow resulting from oxygen therapy improves the metabolic rate, and thus increases the speed at which toxins are broken down. However, one source states that (in an aviation context) oxygen has no effect on physical impairment caused by hangover.
• Fructose and glucose: Glucose and fructose significantly inhibit the metabolic changes produced by alcohol intoxication; nevertheless they have no significant effect on hangover severity.
• Vitamin B₆: No effects on alcohol metabolism, peak blood alcohol and glucose concentrations have been found and psychomotor function is not significantly improved when using Vitamin B₆ supplements.
• Caffeinated drinks: Caffeine narrows blood vessels and raises blood pressure, worsening the hangover conditions. It also slows down rehydration.
• Consuming solely one type of alcoholic beverage ("Grape or grain but never the twain"), or consuming different types in a specific order ("Beer before wine and you'll feel fine; wine before beer and you'll feel queer"): These do not have any significant effect on the intensity of a subsequent hangover when controlling for BAC.

Common misconceptions

Drinking water for managing hangover

The consumption of alcohol is known to increase urine production, leading to dehydration—a common contributor to hangover symptoms. However, dehydration represents only one aspect of hangover symptoms. Excessive alcohol intake can induce multiple physiological effects, including systemic inflammation, gastrointestinal irritation, disruptions in sleep cycles, and hypoglycemia. Additionally, the metabolism of alcohol generates acetaldehyde, a toxic byproduct that may damage cells and tissues.

Products such as sports drinks, vitamin supplements, or rehydration solutions are frequently promoted as remedies for alcohol-related effects. While these may aid in rehydration, they generally fail to alleviate other hangover symptoms, such as inflammation or metabolic disturbances.

To reduce hangover severity, experts recommend consuming water or non-alcoholic beverages intermittently while drinking alcohol. This practice not only helps maintain hydration but also promotes moderation by slowing the pace of alcohol consumption, thereby extending the time available for the body to metabolize alcohol.

"Hair of the dog that bit" method

The practice colloquially referred to as "hair of the dog"—consuming alcohol the morning after excessive drinking—is occasionally cited as a hangover remedy. While temporarily elevating blood alcohol levels may mask symptoms such as anxiety or restlessness, this approach does not resolve the underlying causes of a hangover. Experts note that such relief is short-lived, as symptoms reemerge once alcohol is metabolized.

According to the National Institute on Alcohol Abuse and Alcoholism (NIAAA), a hangover may represent a mild form of alcohol withdrawal, with symptoms peaking in severity when blood alcohol concentration (BAC) returns to baseline. Continued alcohol consumption to mitigate discomfort delays this process, prolonging physiological recovery. Medical professionals caution that this method risks exacerbating dehydration, metabolic strain, and other hangover-related effects, ultimately extending the body’s exposure to alcohol’s toxic byproducts.

Painkillers for managing hangover

The use of over-the-counter (OTC) analgesics to alleviate hangover symptoms, such as headaches, may carry significant health risks when combined with alcohol consumption. Non-steroidal anti-inflammatory drugs (NSAIDs), including ibuprofen, naproxen, and aspirin, elevate the risk of gastrointestinal bleeding by approximately 37% when consumed alongside even one alcoholic beverage daily, as reported by the National Institute on Alcohol Abuse and Alcoholism (NIAAA).

Combining alcohol with medications containing acetaminophen (paracetamol) poses additional hazards, particularly to liver function. The NIAAA notes that concurrent use can induce hepatotoxicity, as both substances are metabolized by the liver. The U.S. Food and Drug Administration (FDA) explicitly advises against alcohol consumption while taking acetaminophen-containing products due to this heightened risk of liver damage.

Medical professionals emphasize caution when mixing alcohol with any medication, including those without explicit warnings against alcohol interaction. Adverse effects may still occur, underscoring the importance of consulting healthcare guidelines or providers before combining substances.

Epidemiology

Hangovers occur commonly.

• A 1990 study of students at a rural New England university found that 25% had experienced a hangover in the previous week and 8% reported missing classes.
• Fifteen percent of men and women who have consumed alcohol experience hangovers at least monthly and 10% of British men reported hangover-related problems at work at least monthly.
• An estimated 9.23% (11.6 million workers) of the U.S. labor force work with a hangover.
• About 23% of drinkers do not report any hangover after drinking to intoxication.

Society and culture

"Exercise against a hangover"

A somewhat dated French idiomatic expression for hangover is "mal aux cheveux", literally "sore hair" (or "[even] my hair hurts"). Some terms for 'hangover' are derived from names for liquor, for example, in Chile a hangover is known as a caña from a Spanish slang term for a glass of beer. Similar is the Irish 'brown bottle flu' derived from the type of bottle common to beer. In German, the hangover is known as a "Kater", literally a tomcat. The most likely origin of the term is a pun on catarrh. Nowadays, "Kater" often is understood literally, resulting in cat motifs as depictions of hangovers.

Alcohol hangover has considerable economic consequences. In Finland, a country with a population of 5 million persons, over 1 million workdays are lost each year because of hangovers. The average annual opportunity cost due to hangovers are estimated as 2000 (USD) per working adult. The socioeconomic implications of an alcohol hangover include workplace absenteeism, impaired job performance, reduced productivity and poor academic achievement. Potentially dangerous daily activities such as driving a car or operating heavy machinery are also negatively influenced.

In mid-2017, it was reported that one company in the UK allows sick days when hung over.

As of 2019, South Korea's 'hangover-cure product' market, which comes in various formulations such as beverages, pills, and jelly, is a 250 billion won (US$213 million) industry.

Research

Psychological research of alcohol hangover is growing rapidly. The Alcohol Hangover Research Group had its inaugural meeting in June 2010 as part of the Research Society on Alcoholism (RSA) 33rd Annual Scientific Meeting in San Antonio, Texas, USA.

In 2012, Éduc'alcool, a Quebec-based non-profit organization that aims to educate the public on the responsible use of alcohol, published a report noting hangovers have long-lasting effects that inhibit the drinker's capabilities a full 24 hours after heavy drinking.

Information theory

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Information_theory

Information theory is the mathematical study of the quantification, storage, and communication of information. The field was established and formalized by Claude Shannon in the 1940s, though early contributions were made in the 1920s through the works of Harry Nyquist and Ralph Hartley. It is at the intersection of electronic engineering, mathematics, statistics, computer science, neurobiology, physics, and electrical engineering.

A key measure in information theory is entropy. Entropy quantifies the amount of uncertainty involved in the value of a random variable or the outcome of a random process. For example, identifying the outcome of a fair coin flip (which has two equally likely outcomes) provides less information (lower entropy, less uncertainty) than identifying the outcome from a roll of a die (which has six equally likely outcomes). Some other important measures in information theory are mutual information, channel capacity, error exponents, and relative entropy. Important sub-fields of information theory include source coding, algorithmic complexity theory, algorithmic information theory and information-theoretic security.

Applications of fundamental topics of information theory include source coding/data compression (e.g. for ZIP files), and channel coding/error detection and correction (e.g. for DSL). Its impact has been crucial to the success of the Voyager missions to deep space, the invention of the compact disc, the feasibility of mobile phones and the development of the Internet and artificial intelligence. The theory has also found applications in other areas, including statistical inference, cryptography, neurobiology, perception, signal processing, linguistics, the evolution and function of molecular codes (bioinformatics), thermal physics, molecular dynamics, black holes, quantum computing, information retrieval, intelligence gathering, plagiarism detection, pattern recognition, anomaly detection, the analysis of music, art creation, imaging system design, study of outer space, the dimensionality of space, and epistemology.

Overview

Information theory studies the transmission, processing, extraction, and utilization of information. Abstractly, information can be thought of as the resolution of uncertainty. In the case of communication of information over a noisy channel, this abstract concept was formalized in 1948 by Claude Shannon in a paper entitled A Mathematical Theory of Communication, in which information is thought of as a set of possible messages, and the goal is to send these messages over a noisy channel, and to have the receiver reconstruct the message with low probability of error, in spite of the channel noise. Shannon's main result, the noisy-channel coding theorem, showed that, in the limit of many channel uses, the rate of information that is asymptotically achievable is equal to the channel capacity, a quantity dependent merely on the statistics of the channel over which the messages are sent.

Coding theory is concerned with finding explicit methods, called codes, for increasing the efficiency and reducing the error rate of data communication over noisy channels to near the channel capacity. These codes can be roughly subdivided into data compression (source coding) and error-correction (channel coding) techniques. In the latter case, it took many years to find the methods Shannon's work proved were possible.

A third class of information theory codes are cryptographic algorithms (both codes and ciphers). Concepts, methods and results from coding theory and information theory are widely used in cryptography and cryptanalysis, such as the unit ban.

Historical background

Main article: History of information theory

The landmark event establishing the discipline of information theory and bringing it to immediate worldwide attention was the publication of Claude E. Shannon's classic paper "A Mathematical Theory of Communication" in the Bell System Technical Journal in July and October 1948. Historian James Gleick rated the paper as the most important development of 1948, noting that the paper was "even more profound and more fundamental" than the transistor. He came to be known as the "father of information theory". Shannon outlined some of his initial ideas of information theory as early as 1939 in a letter to Vannevar Bush.

Prior to this paper, limited information-theoretic ideas had been developed at Bell Labs, all implicitly assuming events of equal probability. Harry Nyquist's 1924 paper, Certain Factors Affecting Telegraph Speed, contains a theoretical section quantifying "intelligence" and the "line speed" at which it can be transmitted by a communication system, giving the relation W = K log m (recalling the Boltzmann constant), where W is the speed of transmission of intelligence, m is the number of different voltage levels to choose from at each time step, and K is a constant. Ralph Hartley's 1928 paper, Transmission of Information, uses the word information as a measurable quantity, reflecting the receiver's ability to distinguish one sequence of symbols from any other, thus quantifying information as H = log Sⁿ = n log S, where S was the number of possible symbols, and n the number of symbols in a transmission. The unit of information was therefore the decimal digit, which since has sometimes been called the hartley in his honor as a unit or scale or measure of information. Alan Turing in 1940 used similar ideas as part of the statistical analysis of the breaking of the German second world war Enigma ciphers.

Much of the mathematics behind information theory with events of different probabilities were developed for the field of thermodynamics by Ludwig Boltzmann and J. Willard Gibbs. Connections between information-theoretic entropy and thermodynamic entropy, including the important contributions by Rolf Landauer in the 1960s, are explored in Entropy in thermodynamics and information theory.

In Shannon's revolutionary and groundbreaking paper, the work for which had been substantially completed at Bell Labs by the end of 1944, Shannon for the first time introduced the qualitative and quantitative model of communication as a statistical process underlying information theory, opening with the assertion:

"The fundamental problem of communication is that of reproducing at one point, either exactly or approximately, a message selected at another point."

With it came the ideas of:

• The information entropy and redundancy of a source, and its relevance through the source coding theorem;
• The mutual information, and the channel capacity of a noisy channel, including the promise of perfect loss-free communication given by the noisy-channel coding theorem;
• The practical result of the Shannon–Hartley law for the channel capacity of a Gaussian channel; as well as
• The bit—a new way of seeing the most fundamental unit of information.

Quantities of information

Main article: Quantities of information

Information theory is based on probability theory and statistics, where quantified information is usually described in terms of bits. Information theory often concerns itself with measures of information of the distributions associated with random variables. One of the most important measures is called entropy, which forms the building block of many other measures. Entropy allows quantification of measure of information in a single random variable.

Another useful concept is mutual information defined on two random variables, which describes the measure of information in common between those variables, which can be used to describe their correlation. The former quantity is a property of the probability distribution of a random variable and gives a limit on the rate at which data generated by independent samples with the given distribution can be reliably compressed. The latter is a property of the joint distribution of two random variables, and is the maximum rate of reliable communication across a noisy channel in the limit of long block lengths, when the channel statistics are determined by the joint distribution.

The choice of logarithmic base in the following formulae determines the unit of information entropy that is used. A common unit of information is the bit or shannon, based on the binary logarithm. Other units include the nat, which is based on the natural logarithm, and the decimal digit, which is based on the common logarithm.

In what follows, an expression of the form p log p is considered by convention to be equal to zero whenever p = 0. This is justified because ${\displaystyle \underset{p\to 0^{+}}{lim}p\log p=0}$ for any logarithmic base.

Entropy of an information source

Based on the probability mass function of a source, the Shannon entropy H, in units of bits per symbol, is defined as the expected value of the information content of the symbols.

The amount of information conveyed by an individual source symbol ${\displaystyle x_i}$ with probability ${\displaystyle p_i}$ is known as its self-information or surprisal, ${\displaystyle I(p_i)}$. This quantity is defined as:

${\displaystyle I(p_i)=-{\log }_2(p_i)}$

A less probable symbol has a larger surprisal, meaning its occurrence provides more information. The entropy ${\displaystyle H}$ is the weighted average of the surprisal of all possible symbols from the source's probability distribution:

${\displaystyle H(X)={\mathbb{E}}_X[I(x)]=\sum _i{p}_{i}I(p_i)=-\sum _i{p}_i{\log }_2(p_i)}$

Intuitively, the entropy ${\displaystyle H(X)}$ of a discrete random variable X is a measure of the amount of uncertainty associated with the value of ${\displaystyle X}$ when only its distribution is known. A high entropy indicates the outcomes are more evenly distributed, making the result harder to predict.

For example, if one transmits 1000 bits (0s and 1s), and the value of each of these bits is known to the receiver (has a specific value with certainty) ahead of transmission, no information is transmitted. If, however, each bit is independently and equally likely to be 0 or 1, 1000 shannons of information (more often called bits) have been transmitted.

The entropy of a Bernoulli trial as a function of success probability, often called the binary entropy function, H_b(p). The entropy is maximized at 1 bit per trial when the two possible outcomes are equally probable, as in an unbiased coin toss.

Properties

A key property of entropy is that it is maximized when all the messages in the message space are equiprobable. For a source with n possible symbols, where $p_i=\frac{1}{n}$ for all ${\displaystyle i}$, the entropy is given by:

${\displaystyle H(X)={\log }_2(n)}$

This maximum value represents the most unpredictable state.

For a source that emits a sequence of ${\displaystyle N}$ symbols that are independent and identically distributed (i.i.d.), the total entropy of the message is ${\displaystyle N\cdot H}$ bits. If the source data symbols are identically distributed but not independent, the entropy of a message of length ${\displaystyle N}$ will be less than ${\displaystyle N\cdot H}$.

Units

The choice of the logarithmic base in the entropy formula determines the unit of entropy used:

• A base-2 logarithm (as shown in the main formula) measures entropy in bits per symbol. This unit is also sometimes called the shannon in honor of Claude Shannon.
• A Natural logarithm (base e) measures entropy in nats per symbol. This is often used in theoretical analysis as it avoids the need for scaling constants (like ln 2) in derivations.
• Other bases are also possible. A base-10 logarithm measures entropy in decimal digits, or hartleys, per symbol. A base-256 logarithm measures entropy in bytes per symbol, since 2⁸ = 256.

Binary Entropy Function

The special case of information entropy for a random variable with two outcomes (a Bernoulli trial) is the binary entropy function. This is typically calculated using a base-2 logarithm, and its unit is the shannon. If one outcome has probability p, the other has probability 1 − p. The entropy is given by:

${\displaystyle H_{\mathrm{b}}(p)=-p{\log }_{2}p-(1-p){\log }_2(1-p)}$

This function is depicted in the plot shown above, reaching its maximum of 1 bit when p = 0.5, corresponding to the highest uncertainty.

Joint entropy

The joint entropy of two discrete random variables X and Y is merely the entropy of their pairing: (X, Y). This implies that if X and Y are independent, then their joint entropy is the sum of their individual entropies.

For example, if (X, Y) represents the position of a chess piece—X the row and Y the column, then the joint entropy of the row of the piece and the column of the piece will be the entropy of the position of the piece.

${\displaystyle H(X,Y)={\mathbb{E}}_{X,Y}[-\log p(x,y)]=-\sum _{x,y}p(x,y)\log p(x,y)}$

Despite similar notation, joint entropy should not be confused with cross-entropy.

Conditional entropy (equivocation)

The conditional entropy or conditional uncertainty of X given random variable Y (also called the equivocation of X about Y) is the average conditional entropy over Y:

${\displaystyle H(X|Y)={\mathbb{E}}_Y[H(X|y)]=-\sum _{y\in Y}p(y)\sum _{x\in X}p(x|y)\log p(x|y)=-\sum _{x,y}p(x,y)\log p(x|y).}$

Because entropy can be conditioned on a random variable or on that random variable being a certain value, care should be taken not to confuse these two definitions of conditional entropy, the former of which is in more common use. A basic property of this form of conditional entropy is that:

${\displaystyle H(X|Y)=H(X,Y)-H(Y).}$

Mutual information (transinformation)

Mutual information measures the amount of information that can be obtained about one random variable by observing another. It is important in communication where it can be used to maximize the amount of information shared between sent and received signals. The mutual information of X relative to Y is given by:

${\displaystyle I(X;Y)={\mathbb{E}}_{X,Y}[SI(x,y)]=\sum _{x,y}p(x,y)\log \frac{p(x,y)}{p(x)p(y)}}$

where SI (Specific mutual Information) is the pointwise mutual information.

A basic property of the mutual information is that:

${\displaystyle I(X;Y)=H(X)-H(X|Y).}$

That is, knowing $Y$, we can save an average of I(X; Y) bits in encoding ${\displaystyle X}$ compared to not knowing ${\displaystyle Y}$.

Mutual information is symmetric:

${\displaystyle I(X;Y)=I(Y;X)=H(X)+H(Y)-H(X,Y).}$

Mutual information can be expressed as the average Kullback–Leibler divergence (information gain) between the posterior probability distribution of ${\displaystyle X}$ given the value of $Y$ and the prior distribution on ${\displaystyle X}$:

${\displaystyle I(X;Y)={\mathbb{E}}_{p(y)}[D_{\mathrm{K}\mathrm{L}}(p(X|Y=y)\Vert p(X))].}$

In other words, this is a measure of how much, on the average, the probability distribution on ${\displaystyle X}$ will change if we are given the value of $Y$. This is often recalculated as the divergence from the product of the marginal distributions to the actual joint distribution:

${\displaystyle I(X;Y)=D_{\mathrm{K}\mathrm{L}}(p(X,Y)\Vert p(X)p(Y)).}$

Mutual information is closely related to the log-likelihood ratio test in the context of contingency tables and the multinomial distribution and to Pearson's χ² test: mutual information can be considered a statistic for assessing independence between a pair of variables, and has a well-specified asymptotic distribution.

Kullback–Leibler divergence (information gain)

The Kullback–Leibler divergence (or information divergence, information gain, or relative entropy) is a way of comparing two distributions: a "true" probability distribution ⁠${\displaystyle p(X)}$⁠, and an arbitrary probability distribution ⁠${\displaystyle q(X)}$⁠. If we compress data in a manner that assumes ⁠${\displaystyle q(X)}$⁠ is the distribution underlying some data, when, in reality, ⁠${\displaystyle p(X)}$⁠ is the correct distribution, the Kullback–Leibler divergence is the number of average additional bits per datum necessary for compression. It is thus defined

${\displaystyle D_{\mathrm{K}\mathrm{L}}(p(X)\Vert q(X))=\sum _{x\in X}-p(x)\log q(x)-\sum _{x\in X}-p(x)\log p(x)=\sum _{x\in X}p(x)\log \frac{p(x)}{q(x)}.}$

Although it is sometimes used as a 'distance metric', KL divergence is not a true metric since it is not symmetric and does not satisfy the triangle inequality (making it a semi-quasimetric).

Another interpretation of the KL divergence is the "unnecessary surprise" introduced by a prior from the truth: suppose a number ${\displaystyle X}$ is about to be drawn randomly from a discrete set with probability distribution ⁠${\displaystyle p(x)}$⁠. If Alice knows the true distribution ⁠${\displaystyle p(x)}$⁠, while Bob believes (has a prior) that the distribution is ⁠${\displaystyle q(x)}$⁠, then Bob will be more surprised than Alice, on average, upon seeing the value of ${\displaystyle X}$. The KL divergence is the (objective) expected value of Bob's (subjective) surprisal minus Alice's surprisal, measured in bits if the log is in base 2. In this way, the extent to which Bob's prior is "wrong" can be quantified in terms of how "unnecessarily surprised" it is expected to make him.

Directed Information

Directed information, ${\displaystyle I(X^n\to Y^n)}$, is an information theory measure that quantifies the information flow from the random process ${\displaystyle X^n=\{X_1,X_2,\dots ,X_n\}}$ to the random process ${\displaystyle Y^n=\{Y_1,Y_2,\dots ,Y_n\}}$. The term directed information was coined by James Massey and is defined as:

${\displaystyle I(X^n\to Y^n)\triangleq \sum _{i=1}^{n}I(X^i;Y_i|Y^{i-1})}$,

where ${\displaystyle I(X^i;Y_i|Y^{i-1})}$ is the conditional mutual information ${\displaystyle I(X_1,X_2,...,X_i;Y_i|Y_1,Y_2,...,Y_{i-1})}$.

In contrast to mutual information, directed information is not symmetric. The ${\displaystyle I(X^n\to Y^n)}$ measures the information bits that are transmitted causally^{[clarification needed]} from ${\displaystyle X^n}$ to ${\displaystyle Y^n}$. The Directed information has many applications in problems where causality plays an important role such as capacity of channel with feedback, capacity of discrete memoryless networks with feedback, gambling with causal side information, compression with causal side information, real-time control communication settings, and in statistical physics.

Other quantities

Other important information theoretic quantities include the Rényi entropy and the Tsallis entropy (generalizations of the concept of entropy), differential entropy (a generalization of quantities of information to continuous distributions), and the conditional mutual information. Also, pragmatic information has been proposed as a measure of how much information has been used in making a decision.

Coding theory

Main article: Coding theory

A picture showing scratches on the readable surface of a CD-R. Music and data CDs are coded using error correcting codes and thus can still be read even if they have minor scratches using error detection and correction.

Coding theory is one of the most important and direct applications of information theory. It can be subdivided into source coding theory and channel coding theory. Using a statistical description for data, information theory quantifies the number of bits needed to describe the data, which is the information entropy of the source.

• Data compression (source coding): There are two formulations for the compression problem:
  • Lossless data compression: the data must be reconstructed exactly;
  • Lossy data compression: allocates bits needed to reconstruct the data, within a specified fidelity level measured by a distortion function. This subset of information theory is called rate–distortion theory.
• Error-correcting codes (channel coding): While data compression removes as much redundancy as possible, an error-correcting code adds just the right kind of redundancy (i.e., error correction) needed to transmit the data efficiently and faithfully across a noisy channel.

This division of coding theory into compression and transmission is justified by the information transmission theorems, or source–channel separation theorems that justify the use of bits as the universal currency for information in many contexts. However, these theorems only hold in the situation where one transmitting user wishes to communicate to one receiving user. In scenarios with more than one transmitter (the multiple-access channel), more than one receiver (the broadcast channel) or intermediary "helpers" (the relay channel), or more general networks, compression followed by transmission may no longer be optimal.

Source theory

Any process that generates successive messages can be considered a source of information. A memoryless source is one in which each message is an independent identically distributed random variable, whereas the properties of ergodicity and stationarity impose less restrictive constraints. All such sources are stochastic. These terms are well studied in their own right outside information theory.

Rate

Information rate is the average entropy per symbol. For memoryless sources, this is merely the entropy of each symbol, while, in the case of a stationary stochastic process, it is:

${\displaystyle r=\underset{n\to \mathrm{\infty }}{lim}H(X_n|X_{n-1},X_{n-2},X_{n-3},\dots );}$

that is, the conditional entropy of a symbol given all the previous symbols generated. For the more general case of a process that is not necessarily stationary, the average rate is:

${\displaystyle r=\underset{n\to \mathrm{\infty }}{lim}\frac{1}{n}H(X_1,X_2,\dots X_n);}$

that is, the limit of the joint entropy per symbol. For stationary sources, these two expressions give the same result.

The information rate is defined as:

${\displaystyle r=\underset{n\to \mathrm{\infty }}{lim}\frac{1}{n}I(X_1,X_2,\dots X_n;Y_1,Y_2,\dots Y_n);}$

It is common in information theory to speak of the "rate" or "entropy" of a language. This is appropriate, for example, when the source of information is English prose. The rate of a source of information is related to its redundancy and how well it can be compressed, the subject of source coding.

Channel capacity

Main article: Channel capacity

Communications over a channel is the primary motivation of information theory. However, channels often fail to produce exact reconstruction of a signal; noise, periods of silence, and other forms of signal corruption often degrade quality.

Consider the communications process over a discrete channel. A simple model of the process is shown below:

${\displaystyle \underset{\text{Message}}{\overset{W}{\to }}\begin{array}{|c|}\hline \text{Encoder}\\ f_n\\ \hline\end{array}\underset{\genfrac{}{}{0ex}{}{\mathrm{E}\mathrm{n}\mathrm{c}\mathrm{o}\mathrm{d}\mathrm{e}\mathrm{d}}{\mathrm{s}\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{e}\mathrm{n}\mathrm{c}\mathrm{e}}}{\overset{X^n}{\to }}\begin{array}{|c|}\hline \text{Channel}\\ p(y|x)\\ \hline\end{array}\underset{\genfrac{}{}{0ex}{}{\mathrm{R}\mathrm{e}\mathrm{c}\mathrm{e}\mathrm{i}\mathrm{v}\mathrm{e}\mathrm{d}}{\mathrm{s}\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{e}\mathrm{n}\mathrm{c}\mathrm{e}}}{\overset{Y^n}{\to }}\begin{array}{|c|}\hline \text{Decoder}\\ g_n\\ \hline\end{array}\underset{\genfrac{}{}{0ex}{}{\mathrm{E}\mathrm{s}\mathrm{t}\mathrm{i}\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{d}}{\mathrm{m}\mathrm{e}\mathrm{s}\mathrm{s}\mathrm{a}\mathrm{g}\mathrm{e}}}{\overset{\hat{W}}{\to }}}$

Here ${\displaystyle X}$ represents the space of messages transmitted, and $Y$ the space of messages received during a unit time over our channel. Let p(y|x) be the conditional probability distribution function of $Y$ given ${\displaystyle X}$. We will consider p(y|x) to be an inherent fixed property of our communications channel (representing the nature of the noise of our channel). Then the joint distribution of ${\displaystyle X}$ and $Y$ is completely determined by our channel and by our choice of f(x), the marginal distribution of messages we choose to send over the channel. Under these constraints, we would like to maximize the rate of information, or the signal, we can communicate over the channel. The appropriate measure for this is the mutual information, and this maximum mutual information is called the channel capacity and is given by:

${\displaystyle C=\underset{f}{max}I(X;Y).}$

This capacity has the following property related to communicating at information rate R (where R is usually bits per symbol). For any information rate R < C and coding error ε > 0, for large enough N, there exists a code of length N and rate ≥ R and a decoding algorithm, such that the maximal probability of block error is ≤ ε; that is, it is always possible to transmit with arbitrarily small block error. In addition, for any rate R > C, it is impossible to transmit with arbitrarily small block error.

Channel coding is concerned with finding such nearly optimal codes that can be used to transmit data over a noisy channel with a small coding error at a rate near the channel capacity.

Capacity of particular channel models

• A continuous-time analog communications channel subject to Gaussian noise—see Shannon–Hartley theorem.
• A binary symmetric channel (BSC) with crossover probability p is a binary input, binary output channel that flips the input bit with probability p. The BSC has a capacity of 1 − H_b(p) bits per channel use, where H_b is the binary entropy function to the base-2 logarithm:

• A binary erasure channel (BEC) with erasure probability p is a binary input, ternary output channel. The possible channel outputs are 0, 1, and a third symbol 'e' called an erasure. The erasure represents complete loss of information about an input bit. The capacity of the BEC is 1 − p bits per channel use.

Channels with memory and directed information

In practice many channels have memory. Namely, at time ${\displaystyle i}$ the channel is given by the conditional probability${\displaystyle P(y_i|x_i,x_{i-1},x_{i-2},...,x_1,y_{i-1},y_{i-2},...,y_1)}$. It is often more comfortable to use the notation ${\displaystyle x^i=(x_i,x_{i-1},x_{i-2},...,x_1)}$ and the channel become ${\displaystyle P(y_i|x^i,y^{i-1})}$. In such a case the capacity is given by the mutual information rate when there is no feedback available and the Directed information rate in the case that either there is feedback or not (if there is no feedback the directed information equals the mutual information).

Fungible information

Fungible information is the information for which the means of encoding is not important. Classical information theorists and computer scientists are mainly concerned with information of this sort. It is sometimes referred as speakable information.

Applications to other fields

Intelligence uses and secrecy applications

Information theoretic concepts apply to cryptography and cryptanalysis. Turing's information unit, the ban, was used in the Ultra project, breaking the German Enigma machine code and hastening the end of World War II in Europe. Shannon himself defined an important concept now called the unicity distance. Based on the redundancy of the plaintext, it attempts to give a minimum amount of ciphertext necessary to ensure unique decipherability.

Information theory leads us to believe it is much more difficult to keep secrets than it might first appear. A brute force attack can break systems based on asymmetric key algorithms or on most commonly used methods of symmetric key algorithms (sometimes called secret key algorithms), such as block ciphers. The security of all such methods comes from the assumption that no known attack can break them in a practical amount of time.

Information theoretic security refers to methods such as the one-time pad that are not vulnerable to such brute force attacks. In such cases, the positive conditional mutual information between the plaintext and ciphertext (conditioned on the key) can ensure proper transmission, while the unconditional mutual information between the plaintext and ciphertext remains zero, resulting in absolutely secure communications. In other words, an eavesdropper would not be able to improve his or her guess of the plaintext by gaining knowledge of the ciphertext but not of the key. However, as in any other cryptographic system, care must be used to correctly apply even information-theoretically secure methods; the Venona project was able to crack the one-time pads of the Soviet Union due to their improper reuse of key material.

Pseudorandom number generation

Pseudorandom number generators are widely available in computer language libraries and application programs. They are, almost universally, unsuited to cryptographic use as they do not evade the deterministic nature of modern computer equipment and software. A class of improved random number generators is termed cryptographically secure pseudorandom number generators, but even they require random seeds external to the software to work as intended. These can be obtained via extractors, if done carefully. The measure of sufficient randomness in extractors is min-entropy, a value related to Shannon entropy through Rényi entropy; Rényi entropy is also used in evaluating randomness in cryptographic systems. Although related, the distinctions among these measures mean that a random variable with high Shannon entropy is not necessarily satisfactory for use in an extractor and so for cryptography uses.

Seismic exploration

One early commercial application of information theory was in the field of seismic oil exploration. Work in this field made it possible to strip off and separate the unwanted noise from the desired seismic signal. Information theory and digital signal processing offer a major improvement of resolution and image clarity over previous analog methods.

Semiotics

Semioticians Doede Nauta [nl] and Winfried Nöth both considered Charles Sanders Peirce as having created a theory of information in his works on semiotics. Nauta defined semiotic information theory as the study of "the internal processes of coding, filtering, and information processing."

Concepts from information theory such as redundancy and code control have been used by semioticians such as Umberto Eco and Ferruccio Rossi-Landi [it] to explain ideology as a form of message transmission whereby a dominant social class emits its message by using signs that exhibit a high degree of redundancy such that only one message is decoded among a selection of competing ones.

Integrated process organization of neural information

Quantitative information theoretic methods have been applied in cognitive science to analyze the integrated process organization of neural information in the context of the binding problem in cognitive neuroscience. In this context, either an information-theoretical measure, such as functional clusters (Gerald Edelman and Giulio Tononi's functional clustering model and dynamic core hypothesis (DCH)) or effective information (Tononi's integrated information theory (IIT) of consciousness), is defined (on the basis of a reentrant process organization, i.e. the synchronization of neurophysiological activity between groups of neuronal populations), or the measure of the minimization of free energy on the basis of statistical methods (Karl J. Friston's free energy principle (FEP), an information-theoretical measure which states that every adaptive change in a self-organized system leads to a minimization of free energy, and the Bayesian brain hypothesis).

Miscellaneous applications

Information theory also has applications in the search for extraterrestrial intelligence, black holes, bioinformatics, and gambling.