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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 family of urn models that can be used to interpret many commonly used statistical models. The model represents objects of interest (such as atoms, people, cars, etc.) as colored balls in an urn.
The Dirichlet-multinomial distribution can also be motivated via an urn model for positive integer values of the vector , known as the Polya urn model.Specifically, imagine an urn containing balls of colors numbering for the ith color, where random draws are made.
In Ars Conjectandi (1713), Jacob Bernoulli considered the problem of determining, given a number of pebbles drawn from an urn, the proportions of different colored pebbles within the urn. This problem was known as the inverse probability problem, and was a topic of research in the eighteenth century, attracting the attention of Abraham de ...
Yet another way to visualize the Dirichlet process and Chinese restaurant process is as a modified Pólya urn scheme sometimes called the Blackwell–MacQueen sampling scheme. Imagine that we start with an urn filled with black balls. Then we proceed as follows: Each time we need an observation, we draw a ball from the urn.
In the case when the 3 parameters , and are positive integers, the Beta negative binomial can also be motivated by an urn model - or more specifically a basic Pólya urn model. Consider an urn initially containing α {\displaystyle \alpha } red balls (the stopping color) and β {\displaystyle \beta } blue balls.
The beta-binomial distribution can also be motivated via an urn model for positive integer values of α and β, known as the Pólya urn model. Specifically, imagine an urn containing α red balls and β black balls, where random draws are made. If a red ball is observed, then two red balls are returned to the urn.
The urn was found in 2014 wrapped in textiles, which themselves were considered an extremely rare survival. The hoard is estimated to have been created around 900AD, and was discovered by ...
Initially, the urn contains α 1 balls of color 1, α 2 balls of color 2, and so on. Now perform N draws from the urn, where after each draw, the ball is placed back into the urn with an additional ball of the same color. In the limit as N approaches infinity, the proportions of different colored balls in the urn will be distributed as Dir(α 1 ...