I am looking for a variant of a Chinese restaurant process (CRP) close to the following. The $n+1$'th customer joins an existing table of size $b$ with probability $bp$ for some parameter $p$ and creates the new table with probability $1 - p\sum_{b\in B} b = 1 - n p$. The total number of customers $N$ is known and $p<1/N$.
Is there such a model, perhaps with a difference dependence of joining probability on $|b|$ (we are not sure what the data supports) but without normalization by $n+1$ as in the common CRP? We are interested in predicting the number of clusters in this model from a partial observation of the process of length $M<N$.
Since $N$ is fixed it seems to be similar to a random graph on N nodes with random edges, where connected components are clusters or "tables". But it differs in that that when creating a random graph incrementally, adding a vertex $n+1$ and drawing $n$ independent edges with probability $p$, it may happen that two or more existing clusters get merged together. In the process that we need, selecting a table is exclusive.
