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Herd immunity (also called herd effect, community immunity, population immunity, or social immunity) is a form of indirect protection from infectious disease that occurs when a large percentage of a population has become immune to an infection, whether through previous infections or vaccination, thereby providing a measure of protection for individuals who are not immune. In a population in which a large proportion of individuals possess immunity, such people being unlikely to contribute to disease transmission, chains of infection are more likely to be disrupted, which either stops or slows the spread of disease. The greater the proportion of immune individuals in a community, the smaller the probability that non-immune individuals will come into contact with an infectious individual, helping to shield non-immune individuals from infection.

Individuals can become immune by recovering from an earlier infection or through vaccination. Some individuals cannot become immune due to medical reasons, such as an immunodeficiency or immunosuppression, and in this group herd immunity is a crucial method of protection. Once a certain threshold has been reached, herd immunity gradually eliminates a disease from a population. This elimination, if achieved worldwide, may result in the permanent reduction in the number of infections to zero, called eradication. Herd immunity created via vaccination contributed to the eventual eradication of smallpox in 1977 and has contributed to the reduction of the frequencies of other diseases. Herd immunity does not apply to all diseases, just those that are contagious, meaning that they can be transmitted from one individual to another. Tetanus, for example, is infectious but not contagious, so herd immunity does not apply.

The term "herd immunity" was first used in 1923. It was recognized as a naturally occurring phenomenon in the 1930s when it was observed that after a significant number of children had become immune to measles, the number of new infections temporarily decreased, including among susceptible children. Mass vaccination to induce herd immunity has since become common and proved successful in preventing the spread of many infectious diseases. Opposition to vaccination has posed a challenge to herd immunity, allowing preventable diseases to persist in or return to communities that have inadequate vaccination rates.

Anscombe's quartet comprises four data sets that have nearly identical simple descriptive statistics, yet have very different distributions and appear very different when graphed. Each dataset consists of eleven (x,y) points. They were constructed in 1973 by the statistician Francis Anscombe to demonstrate both the importance of graphing data when analyzing it, and the effect of outliers and other influential observations on statistical properties. He described the article as being intended to counter the impression among statisticians that "numerical calculations are exact, but graphs are rough."

In mathematics, the theory of optimal stopping or early stopping is concerned with the problem of choosing a time to take a particular action, in order to maximise an expected reward or minimise an expected cost. Optimal stopping problems can be found in areas of statistics, economics, and mathematical finance (related to the pricing of American options). A key example of an optimal stopping problem is the secretary problem. Optimal stopping problems can often be written in the form of a Bellman equation, and are therefore often solved using dynamic programming.

The Howland will forgery trial was a U.S. court case in 1868 to decide Henrietta Howland Robinson's contest of the will of Sylvia Ann Howland. It is famous for the forensic use of mathematics by Benjamin Peirce as an expert witness.

Many probability distributions that are important in theory or applications have been given specific names.

Alexey Yakovlevich Chervonenkis (Russian: Алексей Яковлевич Червоненкис; 7 September 1938 – 22 September 2014) was a Soviet and Russian mathematician, and, with Vladimir Vapnik, was one of the main developers of the Vapnik–Chervonenkis theory, also known as the "fundamental theory of learning" an important part of computational learning theory. Chervonenkis held joint appointments with the Russian Academy of Sciences and Royal Holloway, University of London.

Alexey Chervonenkis got lost in Losiny Ostrov National Park on 22 September 2014, and later during a search operation was found dead near Mytishchi, a suburb of Moscow. He had died of hypothermia.

The following is a list of algorithms along with one-line descriptions for each.

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.

In mathematics, Buffon's needle problem is a question first posed in the 18th century by Georges-Louis Leclerc, Comte de Buffon:

Suppose we have a floor made of parallel strips of wood, each the same width, and we drop a needle onto the floor. What is the probability that the needle will lie across a line between two strips?

Buffon's needle was the earliest problem in geometric probability to be solved; it can be solved using integral geometry. The solution for the sought probability p, in the case where the needle length l is not greater than the width t of the strips, is

${\displaystyle p={\frac {2}{\pi }}{\frac {l}{t}}.}$

This can be used to design a Monte Carlo method for approximating the number π, although that was not the original motivation for de Buffon's question.

Extreme learning machines are feedforward neural networks for classification, regression, clustering, sparse approximation, compression and feature learning with a single layer or multiple layers of hidden nodes, where the parameters of hidden nodes (not just the weights connecting inputs to hidden nodes) need not be tuned. These hidden nodes can be randomly assigned and never updated (i.e. they are random projection but with nonlinear transforms), or can be inherited from their ancestors without being changed. In most cases, the output weights of hidden nodes are usually learned in a single step, which essentially amounts to learning a linear model. The name "extreme learning machine" (ELM) was given to such models by its main inventor Guang-Bin Huang.

According to their creators, these models are able to produce good generalization performance and learn thousands of times faster than networks trained using backpropagation. In literature, it also shows that these models can outperform support vector machines (SVM) and SVM provides suboptimal solutions in both classification and regression applications.