I'm trying to understand the definition of a Beta process, as given in the paper:
www.ece.duke.edu/~lcarin/Paisley\_BP-FA\_ICML.pdf


The problem is that from the definition it follows that every Dirichlet process is also a beta process, which seems, ahm, wrong. Can you help me figure out what I don't understand?


This is the definition from the paper:
"Let $H\_0$ be a continuous probability measure and α a positive scalar. Then for all disjoint, inﬁnitesimal partitions, $B\_1 ,\ldots, B\_K$, the beta
process is generated as follows,
 $H(B\_k) \sim Beta(\alpha H\_0 (B\_k ), \alpha(1 − H\_0(B\_k )))$


with K → ∞ and $H\_0(B\_k)$ → 0 for $k = 1,\ldots,K$. This process is denoted $H \sim BP(\alpha H\_0 )$."


This is the definition for a Dirichlet Process (DP):


If $X \sim DP(\alpha H\_0)$ where $\alpha$ is a scalar and $H\_0$ is a probability distribution, then for every finite partition $A\_1,\ldots,A\_K$ it follows that
$(X(A\_1),\ldots,X(A\_K)) \sim DIR(\alpha H\_0(A\_1), \ldots,\alpha H\_0(A\_K))$.


So let's assume that I have $X\sim DP(\alpha H\_0)$. Given any partition $B\_1 ,\ldots, B\_K$, and any $k = 1 \ldots K$, I can define a partition $A\_1 = B\_k, A\_2 = \Omega -B\_k$ and from the DP definition it follows that 


$(X(A\_1),X(A\_2)) \sim DIR(\alpha H\_0(A\_1), \alpha H\_0(A\_2))$
which is equivalent to saying that
$X(B\_k) \sim Beta(\alpha H\_0(B\_k), \alpha(1-H\_0(B\_k)))$


hence $X\sim BP(\alpha H\_0)$. Where is my mistake?



Difference between Beta Process and Dirichlet process
One cannot conclude $X\sim BP(\alpha H\_0)$ just by knowing the marginal distribution of each $X(B\_k)$, separately.
Your calculation is not wrong as the univariate marginal distribution and conditional distribution of a Dirichlet distribution are [Beta distributed](http://en.wikipedia.org/wiki/Dirichlet_process#Stick-breaking_construction). 


In particular, in a Dirichlet process, samples correspond to the density function $$f(\theta)=\sum \beta\_i \delta\_{\theta\_i}$$
(Here the $\beta\_i$ are constructed as $\beta\_i=\beta\_i'\prod\_{j < i}(1-\beta\_j')$, and $\beta\_j'\sim \text{Beta}(1,\alpha)=Y$)


While in a Beta process, given an **infinitesimal** partition $(B\_1,...,B\_K)$ with $K\to \infty$ and $H(B\_k)\to 0$ the samples correspond to the density function
$$H(B)=\sum \pi\_i\delta\_{B\_i}$$ where $\pi\_i\sim \text{Beta}(\alpha H\_0(B\_i),\alpha (1-H\_0(B\_i)))$


I hope you can see the difference. One more thing, the reason why the Dirichlet process is defined in terms of finitely dimensional distributions is because [Kolmogorov extension theorem](http://en.wikipedia.org/wiki/Kolmogorov_extension_theorem) guarantees that it defines a stochastic process. Unfortunately the Beta process, does not verify the conditions of this theorem, and as a continuous time Levy process must be defined directly in the infinitesimal limit.


