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

{\displaystyle Z}

be an

N

×

K

{\displaystyle N\times K}

binary matrix indicating the presence or absence of a latent feature. The IBP places the following prior on

Z

{\displaystyle Z}

:

p

(

Z

)

=

α

K

+

∏

i

=

1

N

K

1

(

i

)

!

exp

⁡

{

−

α

H

N

}

∏

k

=

1

K

+

(

N

−

m

k

)

!

(

m

k

−

1

)

!

N

!

{\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

K

+

{\displaystyle {K^{+}}}

is the number of non-zero columns in

Z

{\displaystyle Z}

,

m

k

{\displaystyle m_{k}}

is the number of ones in column

k

{\displaystyle k}

of

Z

{\displaystyle Z}

,

H

N

{\displaystyle H_{N}}

is the

N

{\displaystyle N}

-th harmonic number, and

K

1

(

i

)

{\displaystyle K_{1}^{(i)}}

is the number of new dishes sampled by the

i

{\displaystyle i}

-th customer. The parameter

α

{\displaystyle \alpha }

controls the expected number of features present in each observation.