# How to determine whether or not a prior is improper?

Since priors seem to defined in terms of what they are proportional to, rather than what they are equal to, I'm a little confused on how to tell whether or not a prior is improper. I think I have a general understanding of the concept, but I'm not sure how to apply it to a specific example, and every explanation I can find online seems pretty "hand-wavy" to me.

I've calculated Jeffrey's prior for the variance $\theta$ of a normal distribution, where the mean is known to be 0, and found that:

$\pi_{J}(\theta) \space \propto \space \frac{1}{\theta\sqrt{2}}$

Now, will calculating $\int_{0}^{\infty} \frac{1}{\theta\sqrt{2}} \space d\theta$ and checking whether or not it's finite tell me whether or not $\pi_{J}(\theta)$ is improper? Or is there something more I must check?

• note: I removed an extra minus sign in your integral and wrote the integral from $0$ to $+\infty$, rather than from $-\infty$ to $+\infty$. Dec 5 '16 at 14:03