PolyA Urn Model (polyA jar Model)

PolyA urn modelfrom Wikipedia, the free encyclopedia

In statistics,PolyA Urn Model(Also known asPolyA urn SchemeOr simplyPó Lya's urn), Named after George P ólya, is a type of statistical model used as an idealized mental exercise to understand the nature of certain statistical distributions.

In an urn model, objects of real interest (such as atoms, people, cars, etc .) are represented as colored bils in an urn or other container. in the Basic Urn Model, the urn containsXWhite andYBlack bils; one ball is drawn randomly from the urn and its color observed; it is then placed back in the urn, and the selection process is repeated. questions can then be asked about the probability of drawing one color or another, or some other properties. (remove the black or white balls, put them back, and then randomly retrieve them)

The polyA Urn Model differs only in that, when a ball of a particle color is drawn, that ball is put back along with a new ball of the same color. thus, unlike in the basic model, the contents of the urn change over time, with a self-reinforcing property sometimes expressedThe rich get richer. (Random removal of the black or white balls, and then find another ball of the same color as the ball, and put it back together with the original ball. Then random extraction of the new ball)

Note that in some sense, the polyA Urn Model is the "Opposite" of the model of sampling without replacement? -- Seems to be). When sampling without replacement, every time a particle value is observed, it is less likely to be observed again, whereas in a polyA urn model, an observed value isMoreLikely 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 particle value has no effect on how likely it is to observe that value again .) note also that in a polyA 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 particle value, that value will never be seen again.

Distributions related to the polyA urn

• Beta-binomial distribution: the distribution of the number of successful draws (trials), eg. Number of white bils extraction of white ball, givenNDraws from a polyA urn.
• Multivariate polyA distribution (also known as the Dirichlet compound multinomial distribution): the distribution over the number of bils of each color, givenNDraws from a polyA urn where there areKDifferent colors instead of only two.
• Martingales and the beta distribution: LetWAndBBe the number of white and black bils initially in the urn, andNWThe number of white bils currently in the urn afterNDraws. Then the sequence of values for is a martingale and converges to the beta distribution.
• Dirichlet process, Chinese restaurant process: Imagine a modified polyA urn scheme as follows. we start with an urn with α black bils. when drawing a ball from the urn, if we draw a black ball, put the ball back along with a new ball of a new non-black color randomly generated from a uniform distribution, and consider the newly generated color to be the "value" of the draw. otherwise, put the ball back along with another ball of the same color, as for the standard polyA urn scheme. the colors of an infinite sequence of draws from this modified polyA urn scheme follow a Chinese restaurant process. if, instead of generating a new color, we draw a random value from a given base distribution and use that value to label the ball, the labels of an infinite sequence of draws follow a Dirichlet process.

 

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