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 $Z$:

: $p(Z) = \frac\exp\\prod_^K \frac$

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 $K_h$ is the number of occurrences of the non-zero binary vector $h$ among the columns in $Z$. 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 $\mathrm(\alpha)$ 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 $\mathrm(\alpha / i)$ new dishes. Therefore, $z_$ 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. $\operatorname(Z)$ 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

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

Bayesian
Bayesian statistics