Consider a pair of RVs $X$ and $Y$, with the following conditional distributions:
$$X | Y=y \sim Binom(L, y)$$
$$Y | X=x \sim Beta(\alpha + x, \nu)$$
where $L$, $\alpha$, and $\nu$; are all positive ($L$ is an integer of course). Is there a name for the joint distribution of $(X,Y)$? Or perhaps for the marginal distribution of $Y$? I think that if $x$ is eliminated from the shape "parameter" of the beta distribution, then $X$ is beta-binomial distributed. But in the above bivariate model, $X=x$ affects the shape parameter for the conditional distribution of $Y$, so I do not think $X$ is beta-binomial distributed. Apologies if the above makes no sense; I am not very knowledgeable about probability and statistics.