Pivotal quantities question in Bayesian Analysis, why the prior distribution is $p(\theta)\propto 1/\theta$?

Question

I'm reading the classical Gelman's Bayesian Data Analysis and on page 54 he states

My questions are:

1. Why $p(\theta)\propto 1/\theta$? It shouldn't be $p(\theta)=\frac{y}{\theta}$?
2. For me $p(\log(\theta))=\frac{1}{\log(\theta)}$, I didn't understand why he wrote $p(\log\theta)\propto 1$

Answer 1

Gelman's notation, using the same letter $p$ for various different functions, is obnoxious.

The expression $\propto\dfrac1\theta$ is implied by $=\dfrac1\theta.$ To say that something${}\propto\dfrac 1\theta$ means that something${}=\dfrac{\text{constant}}\theta,$ and "constant" means not depending on $\theta.$ So if${}=\dfrac y\theta$ is correct, then${}\propto\dfrac1\theta$ is correct.

Next recall that $\left(1\cdot d\log\theta=\dfrac{d\theta}\theta\right)$ so that is where the "$1$" comes from.