Transforming prior distribution in inference for binomial N parameter

Question

I'm struggling with question 6 in the Exercises to Chapter 3 (page 80) of Bayesian Data Analysis by Andrew Gelman.

http://www.stat.columbia.edu/~gelman/book/BDA3.pdf

We have data Y modeled as independent binomial data, with both $N$ and $θ$ unknown, as per Raftery's 1988 paper "Inference for the binomial N parameter: A hierarchical Bayes approach".

$Y∼Bin(N,θ)$ and

$N∼Poisson(μ)$, where $λ=μθ$

The (noninformative) prior distribution of $λ,θ$ is $p(λ,θ) \propto λ^{-1}$

The question 6(a) asks you to transform to determine $p(N,θ)$.

It's similar to the following question, but I haven't been able to use that to get to the answer.

Bayesian Aproach: Infering the N and $\theta$ values from a binomial distribution

Answer 1

Here is what I got (I'm not very sure about it). I think in that exercise, $N$ is supposed to follow a Poisson distribution with random expectation $\mu$. The (improper) joint distribution of $\mu, \theta$ is defined on the transformation $(\lambda = \mu \theta, \theta)$ by $$p(\mu, \lambda) \propto 1/\lambda .$$ In order to get the joint distribution of $(\mu, \theta)$ you would need to use the fact that $$p(\mu, \theta) = p(\lambda, \theta) \mid\det\frac{\partial(\lambda, \theta)}{\partial(\mu, \theta)}\mid$$

Here, $\mid\det\frac{\partial(\lambda, \theta)}{\partial(\mu, \theta)}\mid = \theta$ such that the improper distribution of $(\mu, \theta)$ is $p(\mu, \theta) \propto 1 / \mu$ so the prior is : $$\begin{array}{lcl} p(\mu) &\propto & 1 / \mu\\ N & \sim & \mathcal{P}(\mu) \\ \theta & \sim & \mathcal{U}([0, 1]) \end{array}$$