Why balloons are better than balls (in urn schemes)

November 18, 2011

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The below is taken from a work in progress:

The Polya urn is a heuristic associated with Dirichlet process mixtures. We present the scheme in a modified format, using balloons instead of balls, where the probability of drawing a balloon from the urn is proportional to its volume. Balloons are preferred because their volume may be adjusted by fractional amounts, whereas a ball count may be adjusted only in whole amounts.

The Polya urn initially contains n uniquely colored baloons, each filled with the same volume of air. At each draw, a single baloon is selected at random from the collection of balloons within the urn, and its color is recorded. If the recorded color had been observed in previous draws, the baloon is inflated by an amount equal to its original volume. Finally, the balloon is returned to the urn. Hence, a balloon drawn from the Polya urn is more likely be observed in subsequent draws. ‘The rich get richer’ is a fitting mnemonic for the Polya urn scheme.

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