TITLE: Vague Gamma prior?
QUESTION [17 upvotes]: I'm looking at a MCMC algorithm where the author takes a Gamma(shape = 0.001, rate = 0.001) prior distribution, which they refer to as a vague prior. For all my searching, I am struggling to see how this is vague. The density seems to spread probability almost uniformly when you get away from 0, but the overwhelming amount of probability sits around 0. 
Does anyone understand why this is considered vague, rather than almost constant at 0?

REPLY [4 votes]: Gamma priors can also be used for the conjugate prior to the precision (inverse variance) as would be the case in a standard Bayesian regression model. In this case this prior would be considered vague where you have a reasonably number of parameters/observations that the variance parameters governs (more than 8 is usually enough for this prior to be vague). This would usually be fine for the error of your model, as the variance parameter would govern the error of $N$ observations, where $N$ would usually be large enough for the prior to effectively be vague. In this case the parameters of the conditional Gamma distribution are not really effected by the addition of 0.001 as they will be reasonably large already
However you are right in the sense that this prior is not completely vague and this is something that is more noticeable in other situations. Take a mixed effects model where you have $J$ groups and the levels are governed by a variance parameter. In this case $J$ is not necessary large and therefore the parameters of the conditional Gamma distribution are likewise small. These are therefore effected by the large mass at $0$ and the addition of 0.001 to each of the parameters of your conditional distribution.
However this isn't really a problem if you make the Gamma distribution hyperparameters small enough. It is however a problem that your posterior is now effected by the mass at $0$. In this case there becomes a tendancy for the posterior to some what match the prior. There are several solutions to this which are discussed by Gelman (2006). He also does a much better job than me of describing the problem and solution. If you want to know more about this problem then I suggest reading this paper. In practise however many statisticians will continue to use these priors because unless there is a low number of parameters/observations that the variance parameters governs, then the choice of this prior will not affect the results noticeably