The following is a problem from Bayesian Data Analysis 2nd ed, p. 97. Andrew Gelman has not included its solution in the guide on his website and it has been driving me crazy all day. Literally all day.
For some data $y$, modeled as a binomial distribution with population $N$ and probability $\theta$ parameters, both of which are unknown. The problem sets up the question with this information: (1) Setting a prior on $N$ is difficult, as it only takes on positive natural numbers, so it is treated as $\Pr(N|\mu) = Poisson(\mu)$, where $\mu$ is unknown. (2) To define the prior on $(N, \theta)$, we have $\lambda=\mu\theta$. (The logic here is that it may be easier to formulate a prior considering the unconditional expectation of the observations, rather than the mean of the unobserved $N$.) (3) A potential noninformative prior is $p(\lambda, \theta) \varpropto 1/\lambda$.
The part of the problem that I am hung up on is how to transform the variables and determine $p(N, \theta)$.
The approach that I have attempted is to write $p(N,\theta|\lambda)p(\lambda, \theta)$, and eliminate the unwanted $\lambda$ via integration, that is $p(N,\theta)=\int_0^\infty C\mu^N/(exp(\mu)\lambda N!)d\lambda$, and substituting out $\mu$ with the relation $\mu=\lambda/\theta$. This approach reduces to $p(N,\theta)=C/(N+1)$, where $C$ is the constant of proportionality introduced from (3).
This result concerns me, because it implies that the joint probability of some values of $\theta$ and $N$ only depends on $N$, and not on $\theta$. Furthermore, some vague bells are ringing from my quite decrepit multivariable calculus, attempting to remind me about Jacobians and coordinate transformations, but I am uncertain that this integration approach is even appropriate.
I appreciate your help and insight.