# 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 Nth harmonic number, and ${\displaystyle K_{h}}$ is the number of occurrences of the non-zero binary vector ${\displaystyle h}$ among the columns in ${\displaystyle Z}$. 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.