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 ${\displaystyle Z}$  be an ${\displaystyle N\times K}$  binary matrix indicating the presence or absence of a latent feature. The IBP places the following prior on ${\displaystyle Z}$ :

${\displaystyle p(Z)={\frac {\alpha ^{K^{+}}}{\prod _{i=1}^{N}K_{1}^{(i)}!}}\exp\{-\alpha H_{N}\}\prod _{k=1}^{K^{+}}{\frac {(N-m_{k})!(m_{k}-1)!}{N!}}}$ 

where ${\displaystyle {K}}$  is the number of non-zero columns in ${\displaystyle Z}$ , ${\displaystyle m_{k}}$  is the number of ones in column ${\displaystyle k}$  of ${\displaystyle Z}$ , ${\displaystyle H_{N}}$  is the ${\displaystyle N}$ -th harmonic number, and ${\displaystyle K_{1}^{(i)}}$  is the number of new dishes sampled by the ${\displaystyle i}$ -th customer. The parameter ${\displaystyle \alpha }$  controls the expected number of features present in each observation.

In the Indian buffet process, the rows of ${\displaystyle Z}$  correspond to customers and the columns correspond to dishes in an infinitely long buffet. The first customer takes the first ${\displaystyle \mathrm {Poisson} (\alpha )}$  dishes. The ${\displaystyle i}$ -th customer then takes dishes that have been previously sampled with probability ${\displaystyle m_{k}/i}$ , where ${\displaystyle m_{k}}$  is the number of people who have already sampled dish ${\displaystyle k}$ . He also takes ${\displaystyle \mathrm {Poisson} (\alpha /i)}$  new dishes. Therefore, ${\displaystyle z_{nk}}$  is one if customer ${\displaystyle n}$  tried the ${\displaystyle k}$ -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. ${\displaystyle \operatorname {lof} (Z)}$  is obtained by ordering the columns of the binary matrix ${\displaystyle Z}$  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.