# Bayesian Nonparametric Latent feature model

For quite a long time I've been trying to understand the paper "Bayesian Nonparametric Latent feature model" (by Zoubin Ghahramani et al.) [http://mlg.eng.cam.ac.uk/zoubin/papers/GhaGriSol06.pdf].

In a nut shell the following model is described:

1. We have $$N$$ objects.
2. A matrix $$X$$ of a size $$N \times D$$, where the row $$x_i$$ of this matrix consists of measurements of $$D$$ observable properties of the $$i$$-th object
3. Each object is represented by a vector of latent features $$f_i$$ and the properties $$x_i$$ are generated from a distribution determined by those latent feature model.

$$F=[f_1^T f_2^T ... f_N^T]^T$$ indicates the latent feature values of all $$N$$ objects.

$$p(F)$$ a prior over features

$$P(X|F)$$ a distribution over observed property matrices conditioned on those features

The matrix $$F$$ can be break into two components:

1. A binary matrix $$Z$$ indicating which features are possessed by each object, with $$z_{ik}=1$$ if object $$i$$ has the feature $$k$$ and $$0$$ otherwise.

2. A second matrix $$V$$ indicating the value of each feature for each object.

$$F$$ is an element-wise product of $$Z$$ and $$V$$.

In the paper it is written that if the latent variable $$f_i \in \{1, ..., K\}$$ and $$K$$ then this is a description of finite mixture model.

A really don't see how. Could someone please clarify this to me