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Monday, July 13, 2026

Big lie

From Wikipedia, the free encyclopedia
Adolf Hitler, c.1923, two years before he coined the expression

A big lie (German: große Lüge) is a gross distortion or misrepresentation of the truth, primarily used as a political propaganda technique. The German expression was first used by Adolf Hitler in his book Mein Kampf (1925) to describe how people could be induced to believe so colossal a lie because they would not believe that someone "could have the impudence to distort the truth so infamously". Hitler claimed that the technique had been used by Jews to blame Germany's loss in World War I on German general Erich Ludendorff, who was a prominent nationalist political leader in the Weimar Republic.

According to historian Jeffrey Herf, the Nazis used the idea of the original big lie to turn sentiment against Jews and justify the Holocaust. Herf maintains that Nazi Germany's chief propagandist Joseph Goebbels and the Nazi Party actually used the big lie technique that they described  and that they used it to turn long-standing antisemitism in Europe into mass murder. Herf further argues that the Nazis' big lie was their depiction of Germany as an innocent, besieged nation striking back at "international Jewry", which the Nazis blamed for starting World War I. Nazi propaganda repeatedly claimed that Jews held outsized and secret power in Britain, Russia, and the United States. It further spread claims that the Jews had begun a war of extermination against Germany, and used these to assert that Germany had a right to annihilate the Jews in self-defense.

In the 21st century, the term has been applied to Donald Trump's and his allies' attempts to overturn the result of the 2020 U.S. presidential election, specifically the false claim that the election was stolen through massive voter and electoral fraud. The scale of the claims resulted in Trump supporters attacking the United States Capitol. Later reports indicate that Trump knew he had genuinely lost the election while promoting the narrative. Scholars say that constant repetition across many different forms of media is necessary for the success of the big lie technique, as is a psychological motivation for the public to believe the extreme assertions.

Nazi Germany

Hitler's description

Hitler claimed that Jews had spread the "big lie" that General Erich Ludendorff (pictured) was responsible for the country's loss in World War I.

Hitler's definition is given in Chapter 10 of Adolf Hitler's Mein Kampf (part of a single paragraph in both the German original and James Murphy's translation):

But it remained for the Jews, with their unqualified capacity for falsehood, and their fighting comrades, the Marxists, to impute responsibility for the downfall precisely to the man who alone had shown a superhuman will and energy in his effort to prevent the catastrophe which he had foreseen and to save the nation from that hour of complete overthrow and shame. By placing responsibility for the loss of the world war on the shoulders of Ludendorff they took away the weapon of moral right from the only adversary dangerous enough to be likely to succeed in bringing the betrayers of the Fatherland to Justice.

All this was inspired by the principle  which is quite true within itself  that in the big lie there is always a certain force of credibility; because the broad masses of a nation are always more easily corrupted in the deeper strata of their emotional nature than consciously or voluntarily; and thus in the primitive simplicity of their minds they more readily fall victims to the big lie than the small lie, since they themselves often tell small lies in little matters but would be ashamed to resort to large-scale falsehoods.

It would never come into their heads to fabricate colossal untruths, and they would not believe others could have the impudence to distort the truth so infamously. Even though the facts which prove this to be so may be brought clearly to their minds, they will still doubt and waver and will continue to think there may be some other explanation. For the grossly impudent lie always leaves traces behind it, even after it has been nailed down, a fact which is known to all expert liars in this world and to all who conspire together in the art of lying.

Adolf Hitler, Mein Kampf, vol. I, ch. X

In 1943, The New York Times contributor Edwin James asserted that Hitler's biggest lie was his revisionist claim that Germany was not defeated in war in 1918, but rather was betrayed by internal groups. This stab-in-the-back myth was spread by right-wing groups, including the Nazis.

In enacting the Holocaust

According to historian Jeffrey Herf, the Nazis used the idea of the original big lie to turn sentiment against Jews and justify the Holocaust. Herf maintains that Joseph Goebbels and the Nazi Party actually used the big lie technique that they described  and that they used it to turn long-standing antisemitism in Europe into mass murder. Herf further argues that the Nazis' big lie was their depiction of Germany as an innocent, besieged land striking back at international Jewry, which the Nazis blamed for starting World War I. Nazi propaganda repeatedly claimed that Jews held power behind the scenes in Britain, Russia, and the United States. It further spread claims that the Jews had begun a war of extermination against Germany, and used these to assert that Germany had a right to annihilate the Jews in self-defense.

The Cold War historian Zachary Jonathan Jacobson describes its use:

Adolf Hitler first defined the Big Lie as a deviant tool wielded by Viennese Jews to discredit the Germans' deportment in World War I. Yet, in tragically ironic fashion, it was Hitler and his Nazi regime that actually employed the mendacious strategy. In an effort to rewrite history and blame European Jews for Germany's defeat in World War I, Hitler and his propaganda minister accused them of profiting from the war, consorting with foreign powers and "war shirking" (avoiding conscription). Jews, Hitler contended, were the weak underbelly of the Weimer state that exposed the loyal and true German population to catastrophic collapse. To sell this narrative, Joseph Goebbels insisted "all effective propaganda must be limited to a very few points and must harp on these in slogans until the last member of the public understands."

In short, Nazi fascism hinged on creating one streamlined, overarching lie ... the Nazis built an ideology on a fiction, the notion that Germany's defeat in World War I could be avenged (and reversed) by purging the German population of those purportedly responsible: the Jews.

Goebbels's description

Joseph Goebbels, the head of Nazi Germany's Ministry of Propaganda

Joseph Goebbels also put forth a theory which has come to be commonly associated with the expression "big lie". Goebbels wrote the following paragraph in an article dated 12 January 1941, sixteen years after Hitler first used the phrase. The article, titled "Aus Churchills Lügenfabrik" (English: "From Churchill's Lie Factory") was published in Die Zeit ohne Beispiel:

The essential English leadership secret does not depend on particular intelligence. Rather, it depends on a remarkably stupid thick-headedness. The English follow the principle that when one lies, one should lie big, and stick to it. They keep up their lies, even at the risk of looking ridiculous.

Alleged quotation

The following supposed quotation of Joseph Goebbels has been repeated in numerous books and articles and on thousands of web pages, yet none of them has cited a primary source. According to the research and reasoning of Randall Bytwerk, it is an unlikely thing for Goebbels to have said:

If you tell a lie big enough and keep repeating it, people will eventually come to believe it. The lie can be maintained only for such time as the State can shield the people from the political, economic and/or military consequences of the lie. It thus becomes vitally important for the State to use all of its powers to repress dissent, for the truth is the mortal enemy of the lie, and thus by extension, the truth is the greatest enemy of the State.

U.S. psychological profile of Hitler

The phrase "big lie" was used in a report prepared around 1943 by Walter C. Langer for the United States Office of Strategic Services in describing Hitler's psychological profile. The report was later published in book form as The Mind of Adolf Hitler in 1972. Langer stated of the dictator:

His primary rules were: never allow the public to cool off; never admit a fault or wrong; never concede that there may be some good in your enemy; never leave room for alternatives; never accept blame; concentrate on one enemy at a time and blame him for everything that goes wrong; people will believe a big lie sooner than a little one; and if you repeat it frequently enough people will sooner or later believe it.

A somewhat similar quote appears in the 1943 Analysis of the Personality of Adolph Hitler: With Predictions of His Future Behaviour and Suggestions for Dealing with Him Now and After Germany's Surrender, by Henry A. Murray:

... never to admit a fault or wrong; never to accept blame; concentrate on one enemy at a time; blame that enemy for everything that goes wrong; take advantage of every opportunity to raise a political whirlwind.

Hitler's death

A 1947 U.S. book on the death of Adolf Hitler describes the infectious Soviet disinformation concerning his purported survival as an example of the technique, nodding to German philosopher Hans Vaihinger's 1911 book The Philosophy of 'As if', which ponders the acceptance of lies for utilitarian purposes. The U.S. book asserts that Soviet leadership, "realizing that Communist totalitarian systems and secret police methods require a continuing menace as justification for their existence, decided to keep the ghost of Hitler alive ... as a means of dramatizing the continuing menace of Fascism", bolstering their military.

In his controversial 1968 book, Soviet historian Lev Bezymenski cites the initial announcement of Hitler's death by Nazi Germany as an example of the big lie, as it claimed him to have died while acting as a soldier in the line of duty.

United States

Cold War

A 1964 Senate Internal Security Subcommittee report on The Protocols of the Elders of Zion, a fabricated antisemitic text first published in Russia in 1903, stated that continued circulation of the Protocols could be attributed in part to utilization of "the Hitler technique of the 'big lie'".

By Republicans

The term has been used by prominent American right-wing figures to describe allegations that Trump's victory in the 2016 elections was the result of collusion between his campaign and Russia. Former Attorney General William Barr described those allegations as "a very damaging, big lie" that inhibited the administration's ability to properly deal with Vladimir Putin, a sentiment also echoed by Former House Speaker Newt Gingrich.

By early 2021, Trump and several prominent Republicans started to use the term "the big lie", claiming that it refers to other electoral issues. Trump stated that the term refers to the "Fraudulent Presidential Election of 2020". An opinion piece in the typically center-right Wall Street Journal, as well as Republican politicians Mitch McConnell and Newt Gingrich, referred to "the big lie" as Democratic opposition to what were new and more restrictive voter identification requirements. McConnell's office referred to a Democratic attempt to abolish the filibuster to enact voting rights legislation as "the left's Big Lie [that] there is some evil anti-voting conspiracy sweeping America". Timothy Snyder describes attempts to invert the narrative:

The lie is so big that it reorders the world. And so part of telling the big lie is that you immediately say it's the other side that tells the big lie. Sadly, but it's just a matter of record, all of that is in Mein Kampf.

By January 2022, Republicans were taking actions to impose new voting restrictions and to take complete control of voting and the administrative management of elections, all while a large majority of Republicans continued to believe that the 2020 election had been stolen from them and asserted that democracy was at risk of failing. Extensive press coverage indicated the Republican efforts themselves appeared to present a threat to democracy.

2020 stolen election claims

Donald Trump
According to CNN fact checker Daniel Dale, as of June 9, 2021, former president Donald Trump had issued 132 written statements since leaving office, of which "a third have included lies about the election"  more than any other subject.
To sow election doubt, Trump escalated use of "rigged election" and "election interference" statements in advance of the 2024 election compared to the previous two elections—the statements described as part of a "heads I win; tails you cheated" rhetorical strategy.

During his political career, U.S. president Donald Trump has employed what have been characterized as the firehose of falsehood and big lie propaganda techniques. To support his attempts to overturn the 2020 U.S. presidential election, he and his allies repeatedly and falsely claimed that there had been massive election fraud and that Trump was the true winner of the election. By 2023, major news outlets characterized Trump's claims as not merely falsehoods, but as lies.

U.S. Senators Josh Hawley of Missouri and Ted Cruz of Texas subsequently contested the election results in the Senate. Their effort was characterized as "the big lie" by then President-elect Joe Biden: "I think the American public has a real good, clear look at who they are. They're part of the big lie, the big lie." Republican senators Mitt Romney of Utah and Pat Toomey of Pennsylvania, scholars of fascism Timothy Snyder and Ruth Ben-Ghiat, Russian affairs expert Fiona Hill, and others also used the term "big lie" to refer to Trump's false claims about massive election fraud. By May 2021, many Republicans had come to embrace the false claim and use it as justification to impose new voting restrictions and attempt to take control of the administrative management of elections. Republicans who opposed the claims faced backlash.

In early 2021, The New York Times examined Trump's promotion of "the big lie" for political purposes to subvert the 2020 election, and concluded that the lie encouraged the 2021 attack on the U.S. Capitol. The attack was cited in a resolution to impeach Trump for a second time. During Trump's second impeachment trial, the house managers Jamie Raskin, Joe Neguse, Joaquin Castro, Stacey Plaskett and Madeleine Dean discussed how Trump used "the big lie" to repeatedly make the false claim the election was stolen from him. On October 7, a Senate Judiciary Committee report said that Trump attempted to use the Department of Justice to "lay the foundation of the 'Big Lie'" before the general election and remain in power regardless of election results.

In early 2022, The New York Times presented a detailed analysis of the continuing efforts by Trump and his allies to further promote "the big lie" and related lies in their attempts to overturn and influence future elections, including those in 2022 and 2024. On June 13, 2022, the U.S. House Select Committee on the January 6 Attack presented testimony that Trump knew he lost the 2020 election, but nevertheless promoted the false claim to exploit donors, and, as a result, raked in "half a billion" dollars. In the days following his first indictment on March 30, 2023, he repeatedly posted similar election-related commentary to social media.

Dominion Voting Systems, which provided voting machines to many jurisdictions in the 2020 U.S. elections, filed four major lawsuits related to the big lie that Dominion fixed the election. From Trump's lawyer Rudy Giuliani, Dominion seeks $1.3 billion in damages, alleging that "he and his allies manufactured and disseminated the 'Big Lie', which foreseeably went viral and deceived millions of people into believing that Dominion had stolen their votes and fixed the election." Separately, in Dominion Voting Systems v. Fox News Network it sought $1.6 billion from Fox News. During discovery, Fox News' internal communications were released, indicating that prominent hosts and top executives were aware the network was reporting false statements but continued doing so to retain viewers for financial reasons. On April 18, 2023, Fox News agreed to pay Dominion a $788 million settlement, described by CNN as the "big price" of telling "the Big Lie". Dominion is also suing two other TV networks, Newsmax and One America News Network, for $1.6 billion each, as well as My Pillow and its CEO Mike Lindell for $1.2 billion.

On April 25, 2023, CNN reported that Trump had told a new lie about the 2020 election: "Trump pointedly noted that Biden got more votes than Trump in fewer than a fifth of US counties in 2020. Trump then said, 'Nothing like this has ever happened before. Usually, it's very equal, or  but the winner always had the most counties.'" The statement was described as "complete bunk". Both Bill Clinton (1992 and 1996) and Barack Obama (2008 and 2012) carried "a minority of counties in each of their victories". William H. Frey, a senior fellow at the Brookings Institution, explained:

There is nothing suspicious about winning the presidency with a smaller number of counties. Counties vary widely in size, with large urban and suburban counties  areas where Biden did best  housing far larger populations than most of the outer suburb, small town and rural counties that Trump won.

Trump v. Cable News Network, Inc.
CourtSouthern District of Florida
StartedOctober 3, 2022; 3 years ago
Docket nos.0:22-cv-61842 (S.D. Fla.)
23-14044 (11th Cir.)
Case history
Appealed toUnited States Court of Appeals for the Eleventh Circuit
Outcome
Dismissed on July 28, 2023; 2 years ago by district court. Dismissal affirmed by Eleventh Circuit in November 2025
Court membership
Judge sittingRaag Singhal

On July 28, 2023, a federal district court judge dismissed an October 2022 Trump lawsuit against CNN, stating that CNN's multiple uses of the term "big lie" about Trump's claims of election fraud did not constitute actionable defamation. The judge wrote that CNN's statements were opinion, not factually verifiable statements, and that "no reasonable viewer" would infer that "Trump advocates the persecution and genocide of Jews or any other group of people". The suit was dismissed with prejudice, meaning Trump could not sue again on the same basis.

On December 13, 2024, just over a month after the 2024 United States presidential election, the Public Religion Research Institute published the results of a survey conducted shortly after that election, between November 8 and December 2. The survey focused primarily on the 2024 election but included questions about the claim that the 2020 election was stolen from Trump: 63 percent of Republicans and 31 percent of voters overall still agreed that the 2020 election had been stolen from Trump.

In 2025, research published in Episteme found Trump's use of the "big lie" to support his false stolen election claims as exhibiting multiple rhetoric strategies employed by demagogues. Namely by utilizing a "crisis narrative" and a pervasive sense of grievance, victimization, and persecution. It highlighted that even if supporters did not fully believe the claims, such claims "felt" true, and were left with enough "interpretive openness" that would allow supporters to fill in the gaps. Thus, "they find it an accurate reflection of their own experiences toward a related content, irrespective of evidential support".

People's Republic of China

The Government of China has denied committing human rights abuses against Uyghurs in Xinjiang, and has labelled declarations of Uyghur genocide as a "big lie" perpetrated by hostile forces.

Analysis

Psychologists, psychiatrists and others have explained why the big lie technique works. Dr. Ramani Durvasula, a licensed clinical psychologist and professor of psychology who is an expert on narcissistic personality disorder and narcissistic abuse says that:

Repetition is important, because the Big Lie works through indoctrination. The Big Lie then becomes its own evidence base  if it is repeated enough, people believe it, and the very repetition almost tautologically becomes the support for the Lie. ... Hear something enough it becomes truth. People assume there is an evidence base when the lie is big (it's like a blind spot). ... [People also fail to realize] that there are people in our midst that lack empathy, have no care for the common good, are grandiose, arrogant, and willing to exploit and manipulate people for solely their own egocentric needs. ... [Instead] a sort of halo effect imbues leaders with presumed expertise and power  when that is not at all the case (most if not all megalomaniacal leaders, despots, tyrants, oligarchs share narcissism/psychopathy as a trait).

The importance of repetition in the acceptance of the big lie is stressed by Miriam Bowers-Abbott, an associate professor of logic at Mount Carmel College of Nursing, who states: "What's especially helpful is repetition in a variety of contexts. That is, not just the same words over and over  but integration of an idea in lots of ways. It builds its own little web of support." Such repetition can occur in the physical environment, according to Dr. Matt Blanchard, a clinical psychologist at New York University, who states: "Nothing sells the Big Lie like novelty t-shirts, hats and banners. These items are normally associated with sports teams, not life-and-death political issues. But Trump and his circle have deftly used these items to generate the kind of unbridled loyalty Americans associate with pro football. ... The banners and hats crucially add an air of silliness to everything. If I can buy a novelty hat about it, can it really be so serious? ... It's a genius mindf**k."

Blanchard also notes that people assess information that has a direct impact on their lives differently than more abstract information with less proximity to them. He states that "the act of 'believing' is not just one thing that humans do. Instead, this one word represents a wide range of relationships that humans have with information. We don't truly 'believe' things, so much as provisionally accept information we find useful." Because of this, he states that "most people don't whole-heartedly 'believe' the Big Lie, but they are more than happy to provisionally accept it because... why not? It might be entertaining. It might flatter your identity. It might help you bond with other people in your community. Or it might help you vent some rage. ... '[B]elief' is always predicated on usefulness."

Psychiatrist Bandy X. Lee notes that emotional reasons lie beneath the acceptance of outrageous assertions such as the big lie, stating:

Usually, they are trying to find comfort and to avoid pain. ... This happens in states of lesser health, where one is less inclined to venture into new domains or to seek creative solutions. There is comfort in repetition, and so a people or a nation under duress will gravitate more toward what is repeated to them than what is realistic. Adolf Hitler understood this very well, which is why the American psychologist Walter Langer coined the phrase to describe his method.

Social media also plays a role in such emotional responses, according to Bowers-Abbott, who states:

It was easier to dislodge untruths before social media. In social media, people tend to take public positions. When that position turns out to be wrong, it's embarrassing. And backing down is typically seen as weakness. So they double-down on untrue claims to save face and personal credibility. ... We are way too emotionally attached to being right. It would be better for our culture as a whole to value uncertainty and intellectual humility and curiosity. Those values help us ask questions without the expectation of permanent answers.

Durvasula, Blanchard and Lee agree that it is unlikely that a believer in a big lie can be persuaded through the presentation of factual evidence. Durvasula argues that improvement in critical thinking skills is necessary, stating: "It means ending algorithms that only provide confirmatory news and instead people seeing stories and information that provide other points of view ... creating safe spaces to have these conversations ... encouraging civil discourse with those who hold different opinions, teaching people to find common ground (e.g. love of family) even when belief systems are not aligned." Blanchard says that "[S]preaders of the Big Lie will only be discredited in the eyes of their supporters if they face their greatest fear  accountability. ... They must be seen to lose at the ballot box, they must be arrested when they break the law, they must be sued for every defamation, they must be pursued with every legal tool available in an open society. ... Above all else they must be seen as weak. Only then will their lies lose their usefulness for the millions who once saw something to gain  personally, psychologically, politically, financially  in choosing to believe." Lee notes that when attempting to disabuse someone of a big lie, it is important not to put them on the defensive: "You have to fix the underlying emotional vulnerability that led people to believing it in the first place. For populations, it is usually the pain of not having a place in the world, which socioeconomic inequality exacerbates. Deprivation of health care, education, an ability to make a living, and other avenues for dignity can make a population psychologically vulnerable to those who look to exploit them."

Probability interpretations

From Wikipedia, the free encyclopedia

The word "probability" has been used in a variety of ways since it was first applied to the mathematical study of games of chance. Does probability measure the real, physical, tendency of something to occur, or is it a measure of how strongly one believes it will occur, or does it draw on both these elements? In answering such questions, mathematicians interpret the probability values of probability theory.

There are two broad categories of probability interpretations which can be called "physical" and "evidential" probabilities. Physical probabilities, which are also called objective or frequency probabilities, are associated with random physical systems such as roulette wheels, rolling dice and radioactive atoms. In such systems, a given type of event (such as a die yielding a six) tends to occur at a persistent rate, or "relative frequency", in a long run of trials. Physical probabilities either explain, or are invoked to explain, these stable frequencies. The two main kinds of theory of physical probability are frequentist accounts (such as those of Venn, Reichenbach and von Mises) and propensity accounts (such as those of Popper, Miller, Giere and Fetzer).

Evidential probability, also called Bayesian probability, can be assigned to any statement whatsoever, even when no random process is involved, as a way to represent its rational subjective plausibility, or the degree to which the statement is supported by the available evidence. On most accounts, evidential probabilities are considered to be rational degrees of belief, defined in terms of dispositions to gamble at certain odds. The four main evidential interpretations are the classical (e.g. Laplace's) interpretation, the subjective interpretation (de Finetti and Savage), the epistemic or inductive interpretation (RamseyCox) and the logical interpretation (Keynes and Carnap). There are also evidential interpretations of probability covering groups, which are often labelled as 'intersubjective' (proposed by Gillies and Rowbottom).

Some interpretations of probability are associated with approaches to statistical inference, including theories of estimation and hypothesis testing. The physical interpretation, for example, is taken by followers of "frequentist" statistical methods, such as Ronald Fisher, Jerzy Neyman and Egon Pearson. Statisticians of the opposing, Bayesian school typically accept the frequency interpretation when it makes sense (although not as a definition), but there is less agreement regarding physical probabilities. Bayesians consider the calculation of evidential probabilities to be both valid and necessary in statistics. This article, however, focuses on the interpretations of probability rather than theories of statistical inference.

The terminology of this topic is rather confusing, in part because probabilities are studied within a variety of academic fields. The word "frequentist" is especially tricky. To philosophers it refers to a particular theory of physical probability, one that has more or less been abandoned. To scientists, on the other hand, "frequentist probability" is just another name for physical (or objective) probability. Those who promote Bayesian inference view "frequentist statistics" as an approach to statistical inference that is based on the frequency interpretation of probability, usually relying on the law of large numbers and characterized by what is called 'Null Hypothesis Significance Testing' (NHST). Also the word "objective", as applied to probability, sometimes means exactly what "physical" means here, but is also used of evidential probabilities that are fixed by rational constraints, such as logical and epistemic probabilities.

It is unanimously agreed that statistics depends somehow on probability. But, as to what probability is and how it is connected with statistics, there has seldom been such complete disagreement and breakdown of communication since the Tower of Babel. Doubtless, much of the disagreement is merely terminological and would disappear under sufficiently sharp analysis.

Savage, 1954, p. 2

Philosophy

The philosophy of probability presents problems chiefly in matters of epistemology and the uneasy interface between mathematical concepts and ordinary language as it is used by non-mathematicians. Probability theory is an established field of study in mathematics. It has its origins in correspondence discussing the mathematics of games of chance between Blaise Pascal and Pierre de Fermat in the seventeenth century, and was formalized and rendered axiomatic as a distinct branch of mathematics by Andrey Kolmogorov in the twentieth century. In axiomatic form, mathematical statements about probability theory carry the same sort of epistemological confidence within the philosophy of mathematics as are shared by other mathematical statements.

The mathematical analysis originated in observations of the behaviour of game equipment such as playing cards and dice, which are designed specifically to introduce random and equalized elements; in mathematical terms, they are subjects of indifference. This is not the only way probabilistic statements are used in ordinary human language: when people say that "it will probably rain", they typically do not mean that the outcome of rain versus not-rain is a random factor that the odds currently favor; instead, such statements are perhaps better understood as qualifying their expectation of rain with a degree of confidence. Likewise, when it is written that "the most probable explanation" of the name of Ludlow, Massachusetts "is that it was named after Roger Ludlow", what is meant here is not that Roger Ludlow is favored by a random factor, but rather that this is the most plausible explanation of the evidence, which admits other, less likely explanations.

Thomas Bayes attempted to provide a logic that could handle varying degrees of confidence; as such, Bayesian probability is an attempt to recast the representation of probabilistic statements as an expression of the degree of confidence by which the beliefs they express are held.

Though probability initially had somewhat mundane motivations, its modern influence and use is widespread ranging from evidence-based medicine, through six sigma, all the way to the probabilistically checkable proof and the string theory landscape.

A summary of some interpretations of probability

Classical Frequentist Subjective Propensity
Main hypothesis Principle of indifferenceFrequency of occurrenceDegree of beliefDegree of causal connection
Conceptual basis Hypothetical symmetryPast data and reference classKnowledge and intuitionPresent state of system
Conceptual approach ConjecturalEmpiricalSubjectiveMetaphysical
Single case possible YesNoYesYes
Precise YesNoNoYes
Problems Ambiguity in principle of indifferenceCircular definitionReference class problemDisputed concept

Classical definition

The first attempt at mathematical rigour in the field of probability, championed by Pierre-Simon Laplace, is now known as the classical definition. Developed from studies of games of chance (such as rolling dice) it states that probability is shared equally between all the possible outcomes, provided these outcomes can be deemed equally likely. (3.1)

The theory of chance consists in reducing all the events of the same kind to a certain number of cases equally possible, that is to say, to such as we may be equally undecided about in regard to their existence, and in determining the number of cases favorable to the event whose probability is sought. The ratio of this number to that of all the cases possible is the measure of this probability, which is thus simply a fraction whose numerator is the number of favorable cases and whose denominator is the number of all the cases possible.

Pierre-Simon Laplace, A Philosophical Essay on Probabilities

The classical definition of probability works well for situations with only a finite number of equally-likely outcomes.

This can be represented mathematically as follows: If a random experiment can result in N mutually exclusive and equally likely outcomes and if NA of these outcomes result in the occurrence of the event A, the probability of A is defined by

There are two clear limitations to the classical definition. Firstly, it is applicable only to situations in which there is only a 'finite' number of possible outcomes. But some important random experiments, such as tossing a coin until it shows heads, give rise to an infinite set of outcomes. And secondly, it requires an a priori determination that all possible outcomes are equally likely without falling in a trap of circular reasoning by relying on the notion of probability. (In using the terminology "we may be equally undecided", Laplace assumed, by what has been called the "principle of insufficient reason", that all possible outcomes are equally likely if there is no known reason to assume otherwise, for which there is no obvious justification.)

Frequentism

For frequentists, the probability of the ball landing in any pocket can be determined only by repeated trials in which the observed result converges to the underlying probability in the long run.

Frequentists posit that the probability of an event is its relative frequency over time, (3.4) i.e., its relative frequency of occurrence after repeating a process a large number of times under similar conditions. This is also known as aleatory probability. The events are assumed to be governed by some random physical phenomena, which are either phenomena that are predictable, in principle, with sufficient information (see determinism); or phenomena which are essentially unpredictable. Examples of the first kind include tossing dice or spinning a roulette wheel; an example of the second kind is radioactive decay. In the case of tossing a fair coin, frequentists say that the probability of getting a heads is 1/2, not because there are two equally likely outcomes but because repeated series of large numbers of trials demonstrate that the empirical frequency converges to the limit 1/2 as the number of trials goes to infinity.

If we denote by the number of occurrences of an event in trials, then if we say that .

The frequentist view has its own problems. It is of course impossible to actually perform an infinity of repetitions of a random experiment to determine the probability of an event. But if only a finite number of repetitions of the process are performed, different relative frequencies will appear in different series of trials. If these relative frequencies are to define the probability, the probability will be slightly different every time it is measured. But the real probability should be the same every time. If we acknowledge the fact that we only can measure a probability with some error of measurement attached, we still get into problems as the error of measurement can only be expressed as a probability, the very concept we are trying to define. This renders even the frequency definition circular; see for example “What is the Chance of an Earthquake?

Subjectivism

Subjectivists, also known as Bayesians or followers of epistemic probability, give the notion of probability a subjective status by regarding it as a measure of the 'rational degree of belief' of the individual assessing the uncertainty of a particular situation. Epistemic or subjective probability is sometimes called credence, as opposed to the term chance for a propensity probability.

Some examples of epistemic probability are to assign a probability to the proposition that a proposed law of physics is true or to determine how probable it is that a suspect committed a crime, based on the evidence presented.

The use of Bayesian probability raises the philosophical debate as to whether it can contribute valid justifications of belief. Bayesians point to the work of Ramsey and de Finetti as proving that subjective beliefs must follow the laws of probability if they are to be coherent (rational).

Evidence casts doubt that individual humans routinely apply coherent beliefs, indicating that they often do not adhere to Bayesian probability.

The use of Bayesian probability involves specifying a prior probability. This may be obtained from consideration of whether the required prior probability is greater or lesser than a reference probability associated with an urn model or a thought experiment. The issue is that for a given problem, multiple thought experiments could apply, and choosing one is sometimes a matter of judgement: different people may assign different prior probabilities, known as the reference class problem. The "sunrise problem" provides an example.

Propensity

Propensity theorists think of probability as a physical propensity, or disposition, or tendency of a given type of physical situation to yield an outcome of a certain kind or to yield a long run relative frequency of such an outcome. This kind of objective probability is sometimes called 'chance'.

Propensities, or chances, are not relative frequencies, but purported causes of the observed stable relative frequencies. Propensities are invoked to explain why repeating a certain kind of experiment will generate given outcome types at persistent rates, which are known as propensities or chances. Frequentists are unable to take this approach, since relative frequencies do not exist for single tosses of a coin, but only for large ensembles or collectives (see "single case possible" in the table above). In contrast, a propensitist is able to use the law of large numbers to explain the behaviour of long-run frequencies. This law, which is a consequence of the axioms of probability, says that if (for example) a coin is tossed repeatedly many times, in such a way that its probability of landing heads is the same on each toss, and the outcomes are probabilistically independent, then the relative frequency of heads will be close to the probability of heads on each single toss. This law allows that stable long-run frequencies are a manifestation of invariant single-case probabilities. In addition to explaining the emergence of stable relative frequencies, the idea of propensity is motivated by the desire to make sense of single-case probability attributions in quantum mechanics, such as the probability of decay of a particular atom at a particular time.

The main challenge facing propensity theories is to say exactly what propensity means. (And then, of course, to show that propensity thus defined has the required properties.) At present, unfortunately, none of the well-recognised accounts of propensity comes close to meeting this challenge.

A propensity theory of probability was given by Charles Sanders Peirce. A later propensity theory was proposed by philosopher Karl Popper, who had only slight acquaintance with the writings of C. S. Peirce, however. Popper noted that the outcome of a physical experiment is produced by a certain set of "generating conditions". When we repeat an experiment, as the saying goes, we really perform another experiment with a (more or less) similar set of generating conditions. To say that a set of generating conditions has propensity p of producing the outcome E means that those exact conditions, if repeated indefinitely, would produce an outcome sequence in which E occurred with limiting relative frequency p. For Popper then, a deterministic experiment would have propensity 0 or 1 for each outcome, since those generating conditions would have same outcome on each trial. In other words, non-trivial propensities (those that differ from 0 and 1) only exist for genuinely nondeterministic experiments.

A number of other philosophers, including David Miller and Donald A. Gillies, have proposed propensity theories somewhat similar to Popper's.

Other propensity theorists (e.g. Ronald Giere) do not explicitly define propensities at all, but rather see propensity as defined by the theoretical role it plays in science. They argued, for example, that physical magnitudes such as electrical charge cannot be explicitly defined either, in terms of more basic things, but only in terms of what they do (such as attracting and repelling other electrical charges). In a similar way, propensity is whatever fills the various roles that physical probability plays in science.

What roles does physical probability play in science? What are its properties? One central property of chance is that, when known, it constrains rational belief to take the same numerical value. David Lewis called this the Principal Principle, (3.3 & 3.5) a term that philosophers have mostly adopted. For example, suppose you are certain that a particular biased coin has propensity 0.32 to land heads every time it is tossed. What is then the correct price for a gamble that pays $1 if the coin lands heads, and nothing otherwise? According to the Principal Principle, the fair price is 32 cents.

Logical, epistemic, and inductive probability

It is widely recognized that the term "probability" is sometimes used in contexts where it has nothing to do with physical randomness. Consider, for example, the claim that the extinction of the dinosaurs was probably caused by a large meteorite hitting the earth. Statements such as "Hypothesis H is probably true" have been interpreted to mean that the (presently available) empirical evidence (E, say) supports H to a high degree. This degree of support of H by E has been called the logical, epistemic, or inductive probability of H given E.

The differences between these interpretations are rather small, and may seem inconsequential. One of the main points of disagreement lies in the relation between probability and belief. Logical probabilities are conceived (for example in Keynes' Treatise on Probability) to be objective, logical relations between propositions (or sentences), and hence not to depend in any way upon belief. They are degrees of (partial) entailment, or degrees of logical consequence, not degrees of belief. (They do, nevertheless, dictate proper degrees of belief, as is discussed below.) Frank P. Ramsey, on the other hand, was skeptical about the existence of such objective logical relations and argued that (evidential) probability is "the logic of partial belief". (p 157) In other words, Ramsey held that epistemic probabilities simply are degrees of rational belief, rather than being logical relations that merely constrain degrees of rational belief.

Another point of disagreement concerns the uniqueness of evidential probability, relative to a given state of knowledge. Rudolf Carnap held, for example, that logical principles always determine a unique logical probability for any statement, relative to any body of evidence. Ramsey, by contrast, thought that while degrees of belief are subject to some rational constraints (such as, but not limited to, the axioms of probability) these constraints usually do not determine a unique value. Rational people, in other words, may differ somewhat in their degrees of belief, even if they all have the same information.

Prediction

An alternative account of probability emphasizes the role of prediction – predicting future observations on the basis of past observations, not on unobservable parameters. In its modern form, it is mainly in the Bayesian vein. This was the main function of probability before the 20th century, but fell out of favor compared to the parametric approach, which modeled phenomena as a physical system that was observed with error, such as in celestial mechanics.

The modern predictive approach was pioneered by Bruno de Finetti, with the central idea of exchangeability – that future observations should behave like past observations. This view came to the attention of the Anglophone world with the 1974 translation of de Finetti's book, and has since been propounded by such statisticians as Seymour Geisser.

Axiomatic probability

The mathematics of probability can be developed on an entirely axiomatic basis that is independent of any interpretation: see the articles on probability theory and probability axioms for a detailed treatment.

Complex system

From Wikipedia, the free encyclopedia https://en.wikipedia.org/wiki/Complex_system A complex system is a syst...