# Transforming prior distribution in inference for binomial N parameter

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

• I might start by looking for an expression for the distribution of $\mu, \theta$ i.e. the prior $p(\mu, \theta)$ as a change of variable, remembering $\theta \in [0,1]$ – Henry Dec 14 '20 at 10:05

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 : $$\begin{array}{lcl} p(\mu) &\propto & 1 / \mu\\ N & \sim & \mathcal{P}(\mu) \\ \theta & \sim & \mathcal{U}([0, 1]) \end{array}$$