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.)
In the basic Pólya urn model, the experimenter puts x white and y black balls into an urn.
At each step, one ball is drawn uniformly at random 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.
Thus the urn has a self-reinforcing property ("the rich get richer").
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.
Questions of interest are the evolution of the urn population and the sequence of colors of the balls drawn out.
This can be proved by drawing the Pascal's triangle of all possible configurations.
In particular, starting with one white and one black ball (i.e.,
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 they do not know the absolute number of balls present, nor the proportion that are of each colour.
Suppose that they hold prior beliefs about these unknowns: for them 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 them) 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 them.
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
, 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 generalized in many ways.
Polya's Urn is a quintessential example of an exchangeable process.
We proceed to draw balls at random from the urn.
, because more black balls have been added to the urn.
[2] Recall that a (finite or infinite) sequence of random variables is called exchangeable if its joint distribution is invariant under permutations of indices.
On the first draw the number of balls in the urn is
Now we must show that if the order of black and white balls is permuted, there is no change to the probability.
, since this is the number of balls in the urn at that round.
With the same argument, we can calculate the probability for white balls.
white balls drawn in some order) the final probability will be equal to the following expression, where we take advantage of commutativity of multiplication in the numerator:
This probability is not related to the order of seeing black and white balls and only depends on the total number of white balls and the total number of black balls.
[2] According to the De Finetti's theorem, there must be a unique prior distribution such that the joint distribution of observing the sequence is a Bayesian mixture of the Bernoulli probabilities.