The key feature of a truly objective prior is that it is invariant under change of variables. I understand this concept, however, I'm having a hard time finding a simple 1D or 2D example of when you might need to use something like Jeffreys prior instead of just a uniform distribution, other than the case where your support is infinite. Anybody have an good examples? Also, if I'm ever interested in estimating a latent variable instead of a parameter, does this ever become an issue?
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A simple example is when data are binomial. If you place a uniform prior on $p$ from 0 to 1, then the posterior distribution of $p$ is $Beta(x+1,n-x+1)$, so we can see that a flat prior on p is actually not objective in the sense that it is equivalent to assuming an additional success and an additional failure before collecting data, which pushes the posterior toward $p=.5$. In this case, the Jeffreys prior is $Beta(.5,.5)$, which is not uniform, contributes half of a success and half of a failure to the posterior distribution. So, yes, the Jeffreys prior is invariant to transformation, but that does not necessarily mean that it is objective in terms of the information it contributes to the posterior distribution.
Also, in this situation, a uniform prior on $logit(p)$ is equivalent to an improper $Beta(0,0)$ prior on $p$ which is not uniform from 0 to 1.
