Indian buffet process

In the mathematical theory of probability, the Indian buffet process (IBP) is a stochastic process defining a probability distribution over sparse binary matrices with a finite number of rows and an infinite number of columns. This distribution is suitable to use as a prior for models with potentially infinite number of features. The form of the prior ensures that only a finite number of features will be present in any finite set of observations but more features may appear as more data points are observed.

## Indian buffet process prior

Let Z be an N×K binary matrix indicating the presence or absence of a latent feature. The IBP places the following prior on Z:

p(Z)=αK+∏i=1NK1(i)!exp⁡{−αHN}∏k=1K+(N−mk)!(mk−1)!N!

where K is the number of non-zero columns in Z, mk is the number of ones in column k of Z, HN is the N-th harmonic number, and K1(i) is the number of new dishes sampled by the i-th customer. The parameter α controls the expected number of features present in each observation.

In the Indian buffet process, the rows of Z correspond to customers and the columns correspond to dishes in an infinitely long buffet. The first customer takes the first Poisson(α) dishes. The i-th customer then takes dishes that have been previously sampled with probability mk/i, where mk is the number of people who have already sampled dish k. He also takes Poisson(α/i) new dishes. Therefore, znk is one if customer n tried the k-th dish and zero otherwise.

This process is infinitely exchangeable for an equivalence class of binary matrices defined by a left-ordered many-to-one function. lof⁡(Z) is obtained by ordering the columns of the binary matrix Z from left to right by the magnitude of the binary number expressed by that column, taking the first row as the most significant bit.

## See also

* Chinese restaurant process

## References

* T.L. Griffiths and Z. Ghahramani The Indian Buffet Process: An Introduction and Review, Journal of Machine Learning Research, pp. 1185–1224, 2011.

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