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

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 ${K}$ is the number of non-zero columns in $Z$ , $m_{k}$ is the number of ones in column $k$ of $Z$ , $H_{N}$ is the $N$ -th harmonic number, and $K_{1}^{(i)}$ is the number of new dishes sampled by the $i$ -th customer. The parameter $\alpha$ 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 $\mathrm {Poisson} (\alpha )$ dishes. The $i$ -th customer then takes dishes that have been previously sampled with probability $m_{k}/i$ , where $m_{k}$ is the number of people who have already sampled dish $k$ . He also takes $\mathrm {Poisson} (\alpha /i)$ new dishes. Therefore, $z_{nk}$ 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. $\operatorname {lof} (Z)$ 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.