What is a good example of a non-informative prior for the uniform distribution?

Question

I recently noticed that for non-informative priors, people usually use something like a uniform prior, which works for many different distributions. However, assuming that your likelihood is nothing more than a uniform $\frac{1}{\theta}$ for a parameter theta, what is a non-informative prior that works here? The first thing coming to mind is the Jeffrey's prior, but that yields $-\frac{1}{\theta^2}$, which when trying to calculate the posterior mean / variance leads to a divergent series when trying to sum across the product of the prior and likelihood. Does anyone have any ideas of what is a good non-informative prior?

Answer 1

I will assume your model is a uniform distribution on the interval $(0, \theta)$. So let $X_1, \dotsc, X_n$ iid with that distribution, with $\theta>0$. Then the likelihood function can be written $$ L(\theta) = \theta^{-n} \cdot \mathbb{1}(\theta \ge T) $$ where $T=\max_{i=1}^n X_i$. The first idea is the Jeffrey' prior, and your statement of that cannot be right (it is negative!). What I get is $n/\theta$ for $\theta \ge T$. That may look strange, first, it depends on the data through $T$. But in this case that isn't a problem, since the likelihood is zero for $\theta < T$, so the prior on that interval is unimportant, it will always be multiplied with zero when calculating the posterior. Second, it is improper, but, as long as $n\ge 2$ it leads to a proper posterior (which do have the pareto form). A detailed development is here.

The posterior is $$ f(\theta | T) = \frac{n-1}{\theta}\cdot\left(\frac{T}{\theta}\right)^{n-1} $$ for $\theta \ge T$.

For more information see this paper. A more thorough paper is this.