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🔗 Statistics

In statistics, a Pólya urn model (also known as a Pólya urn scheme or simply as Pólya's urn), named after George Pólya, is a type of statistical model used as an idealized mental exercise framework, unifying many treatments.

In an urn model, objects of real interest (such as atoms, people, cars, etc.) are represented as colored balls in an urn or other container. In the basic Pólya urn model, the urn contains x white and y black balls; one ball is drawn randomly from the urn and its color observed; it is then returned in the urn, and an additional ball of the same color is added to the urn, and the selection process is repeated. Questions of interest are the evolution of the urn population and the sequence of colors of the balls drawn out.

This endows the urn with a self-reinforcing property sometimes expressed as the rich get richer.

Note that in some sense, the Pólya urn model is the "opposite" of the model of sampling without replacement, where every time a particular value is observed, it is less likely to be observed again, whereas in a Pólya urn model, an observed value is more likely to be observed again. In both of these models, the act of measurement has an effect on the outcome of future measurements. (For comparison, when sampling with replacement, observation of a particular value has no effect on how likely it is to observe that value again.) In a Pólya urn model, successive acts of measurement over time have less and less effect on future measurements, whereas in sampling without replacement, the opposite is true: After a certain number of measurements of a particular value, that value will never be seen again.

One of the reasons for interest in this particular rather elaborate urn model (i.e. with duplication and then replacement of each ball drawn) is that it provides an example in which the count (initially x black and y white) of balls in the urn is not concealed, which is able to approximate the correct updating of subjective probabilities appropriate to a different case in which the original urn content is concealed while ordinary sampling with replacement is conducted (without the Pólya ball-duplication). Because of the simple "sampling with replacement" scheme in this second case, the urn content is now static, but this greater simplicity is compensated for by the assumption that the urn content is now unknown to an observer. A Bayesian analysis of the observer's uncertainty about the urn's initial content can be made, using a particular choice of (conjugate) prior distribution. Specifically, suppose that an observer knows that the urn contains only identical balls, each coloured either black or white, but he does not know the absolute number of balls present, nor the proportion that are of each colour. Suppose that he holds prior beliefs about these unknowns: for him the probability distribution of the urn content is well approximated by some prior distribution for the total number of balls in the urn, and a beta prior distribution with parameters (x,y) for the initial proportion of these which are black, this proportion being (for him) considered approximately independent of the total number. Then the process of outcomes of a succession of draws from the urn (with replacement but without the duplication) has approximately the same probability law as does the above Pólya scheme in which the actual urn content was not hidden from him. The approximation error here relates to the fact that an urn containing a known finite number m of balls of course cannot have an exactly beta-distributed unknown proportion of black balls, since the domain of possible values for that proportion are confined to being multiples of ${\displaystyle 1/m}$, rather than having the full freedom to assume any value in the continuous unit interval, as would an exactly beta distributed proportion. This slightly informal account is provided for reason of motivation, and can be made more mathematically precise.

This basic Pólya urn model has been enriched and generalized in many ways.

🔗 Mathematics

Bellard's formula is used to calculate the nth digit of π in base 16.

Bellard's formula was discovered by Fabrice Bellard in 1997. It is about 43% faster than the Bailey–Borwein–Plouffe formula (discovered in 1995). It has been used in PiHex, the now-completed distributed computing project.

One important application is verifying computations of all digits of pi performed by other means. Rather than having to compute all of the digits twice by two separate algorithms to ensure that a computation is correct, the final digits of a very long all-digits computation can be verified by the much faster Bellard's formula.

Formula:

${\displaystyle {\begin{aligned}\pi ={\frac {1}{2^{6}}}\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{2^{10n}}}\,\left(-{\frac {2^{5}}{4n+1}}\right.&{}-{\frac {1}{4n+3}}+{\frac {2^{8}}{10n+1}}-{\frac {2^{6}}{10n+3}}\left.{}-{\frac {2^{2}}{10n+5}}-{\frac {2^{2}}{10n+7}}+{\frac {1}{10n+9}}\right)\end{aligned}}}$

🔗 Biography

William Angus Wallace (born 31 October 1948) is a Scottish orthopaedic surgeon. He is Professor of Orthopaedic and Accident Surgery at the Faculty of Medicine & Health Sciences of the University of Nottingham. He came to widespread public notice for a life-saving surgery he performed using improvised equipment on a British Airways flight in 1995, and for treating Wayne Rooney before the 2006 FIFA World Cup.

🔗 Military history 🔗 Military history/Military science, technology, and theory 🔗 Military history/Weaponry 🔗 Military history/Russian, Soviet and CIS military history 🔗 Military history/Post-Cold War

The Poseidon (Russian: Посейдон, "Poseidon", NATO reporting name Kanyon), previously known by Russian codename Status-6 (Russian: Статус-6), is an autonomous, nuclear-powered, and nuclear-armed unmanned underwater vehicle under development by Rubin Design Bureau, capable of delivering both conventional and nuclear payloads.

The Poseidon is one of the six new Russian strategic weapons announced by Russian President Vladimir Putin on 1 March 2018.

🔗 Business

Topgrading is a corporate hiring and interviewing methodology that is intended to identify preferred candidates for a particular position. In the methodology, prospective employees undergo a 12-step process that includes extensive interviews, the creation of detailed job scorecards, research into job history, coaching, and more. After being interviewed and reference-checked, job candidates are grouped into one of three categories: A Players, B Players, or C Players. A Players have the most potential for high performance in their role while B and C Players may require more work to be successful. The methodology has been used by major corporations and organizations like General Electric, Lincoln Financial, Honeywell, Barclays, and the American Heart Association.

🔗 Philosophy 🔗 Philosophy/Logic 🔗 Philosophy/Ancient philosophy 🔗 Philosophy/Epistemology

"I know that I know nothing" is a saying derived from Plato's account of the Greek philosopher Socrates. Socrates himself was never recorded as having said this phrase, and scholars generally agree that Socrates only ever asserted that he believed that he knew nothing, having never claimed that he knew that he knew nothing. It is also sometimes called the Socratic paradox, although this name is often instead used to refer to other seemingly paradoxical claims made by Socrates in Plato's dialogues (most notably, Socratic intellectualism and the Socratic fallacy).

This saying is also connected or conflated with the answer to a question Socrates (according to Xenophon) or Chaerephon (according to Plato) is said to have posed to the Pythia, the Oracle of Delphi, in which the oracle stated something to the effect of "Socrates is the wisest person in Athens." Socrates, believing the oracle but also completely convinced that he knew nothing, was said to have concluded that nobody knew anything, and that he was only wiser than others because he was the only person who recognized his own ignorance.

🔗 Transport 🔗 Trains

The Cosmopolitan Railway was a proposed global railroad network advocated by William Gilpin, formerly the first territorial governor of Colorado (1861–62), in his 1890 treatise Cosmopolitan Railway: Compacting and Fusing Together All the World's Continents. Gilpin named his capital city of Denver as the "railroad centre of the West".

🔗 Espionage 🔗 Internet 🔗 Russia 🔗 Russia/mass media in Russia 🔗 Russia/Russian, Soviet, and CIS military history 🔗 Russia/politics and law of Russia

Russian web brigades, also called Russian trolls, Russian bots, or more recently Kremlin Bots (after the Kremlin in Moscow) / Kremlins (a pejorative allusion to Gremlin) are state-sponsored anonymous Internet political commentators and trolls linked to the Government of Russia. Participants report that they are organized into teams and groups of commentators that participate in Russian and international political blogs and Internet forums using sockpuppets, social bots and large-scale orchestrated trolling and disinformation campaigns to promote pro-Putin and pro-Russian propaganda. Articles on the Russian Wikipedia concerning the MH17 crash and the 2014 Ukraine conflict were targeted by Russian internet propaganda outlets.

🔗 Cognitive science

Soar is a cognitive architecture, originally created by John Laird, Allen Newell, and Paul Rosenbloom at Carnegie Mellon University. (Rosenbloom continued to serve as co-principal investigator after moving to Stanford University, then to the University of Southern California's Information Sciences Institute.) It is now maintained and developed by John Laird's research group at the University of Michigan.

The goal of the Soar project is to develop the fixed computational building blocks necessary for general intelligent agents – agents that can perform a wide range of tasks and encode, use, and learn all types of knowledge to realize the full range of cognitive capabilities found in humans, such as decision making, problem solving, planning, and natural language understanding. It is both a theory of what cognition is and a computational implementation of that theory. Since its beginnings in 1983 as John Laird’s thesis, it has been widely used by AI researchers to create intelligent agents and cognitive models of different aspects of human behavior. The most current and comprehensive description of Soar is the 2012 book, The Soar Cognitive Architecture.

🔗 International relations 🔗 France 🔗 Law 🔗 Politics 🔗 Basque 🔗 Islands 🔗 Spain

Pheasant Island (French: Île des Faisans/Île de la Conférence, Spanish: Isla de los Faisanes, Basque: Konpantzia) is an uninhabited river island in the Bidasoa river, located between France and Spain, whose administration alternates between both nations.