Identifying a Poisson-binomial hierarchy

This is motivated by a problem where the count model is more complicated than the Poisson. But the general question is better understood with this simple model. Suppose I have a textbook hierarchical model:

X|Y \sim \text{Binomial}(Y, p),
Y \sim \text{Poisson}(\lambda).

Then it’s well known that X \sim \text{Poisson}(\lambda p). If you try and fit this model in Stan you’ll get a lot of divergences and the wrong parameter estimates, because only the product \lambda p is identified. The same problem crops up if you replace the Poisson with a negative binomial (under a mean/ inverse overdispersion parametrisation, neg_binomial_2).

My question is: is there a way to reparametrise the model so as to identify \lambda and p?

Apologies in advance if this is a silly question.

I’m tempted to say if X is the only data you have then there’s no way you can separately estimate \lambda and p unless you place informative priors on them. You can have a single parameter that represents there product though if you just want to make inferences about the product or do predictive inference.

We run in to this on pharmacokinetic models where the volume of plasma divided by the bioavailability of a drug is identifiable, but the separate parameters are not. Usually in that situation, we have some prior information on those parameters though.