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?