Are there situations where improper priors can be avoided via a prior on a subset of the real line and a transformation?

Question

There are many situations where improper priors are "permissable" (Berger, 2009). In many cases, these improper priors are improper because they are "flat" on the real line. A well known example is inference on a normal mean, where Jeffery's prior turns out to be $\pi(\mu)= 1$.

$[a,b] \subset \mathbb{R}$ and $\mathbb{R}$ are in bijection with each other. Because it's easy to write down such bijections explicitly, I'm wondering it is ever possible to map the problem of inference on the real line with an improper prior to a problem of inference on $[a,b]$ with a proper one.

My intuition is that the answer is no, but improper priors are still so strange for me I'm searching for ways to grasp them.

Answer 1

When one considers a (differentiable) bijection $\psi:(a,b)\to\mathbb R$, and an $\sigma$-finite measure $\pi$ over $\mathbb R$, the measure $\nu$ defined by $$\nu(A)=\pi(\Psi(A)) $$over measurable sets $A\subset(a,b)$ has a density with respect to the Lebesgue measure $\lambda$ equal to $$\frac{\text{d}\nu}{\text{d}\lambda}(x)=\frac{\text{d}\pi}{\text{d}\lambda}(\Psi(x))\times\left|\frac{\text{d}\Psi}{\text{d}x}(x)\right|$$ which means that the density of the transformed measure is no longer constant. In any case, $$\nu\{(a,b)\}=\pi(\Psi\{(a,b)\})=\pi(\mathbb R)=\infty $$