When I learned about the Jeffreys' prior in my graduate statistical inference class my professors made it sound sort of like it was interesting mostly for historical reasons rather than because anyone would ever use it. Then when I took Bayesian data analysis, we were never asked to use Jeffreys' priors. Does anyone actually use these in practice. If so (or if not), why or why not? Why do some statisticians not take it seriously?
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1$\begingroup$ I like to use Jeffreys' prior as a default/non-informative prior for the simple binomial model ($p\left(\theta\right)\propto\sqrt{\theta\left(1-\theta\right)}$). It's conjugate with weight equivalent to a single datum and it's a 1$^{\rm{st}}$-order probability-matching prior, so I have a good feeling for what it does to the likelihood function and for how to interpret the resulting credible intervals. $\endgroup$– CyanCommented Aug 10, 2015 at 21:38
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A partial answer to this is found in Gelman et al., Bayesian Data Analysis, 3rd ed.
Jeffreys' principle can be extended to multiparameter models, but the results are more controversial. Simpler approaches based on assuming independent noninformative prior distributions for the components of the vector parameter $\theta$ can give different results than are obtained with Jeffreys' principle. When the number of parameters in a problem is large, we find it useful to abandon pure noninformative prior distributions in favor of hierarchical models, as we discuss in Chapter 5.
When Gelman writes that the results are "controversial," I believe he means that a prior that is noninformative in one dimension tends to become strongly informative in several. If memory serves, this was a claim made in the same section of BDA 2nd ed., but I don't have a copy with me at the moment.
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2$\begingroup$ And with very good reason $\endgroup$ Commented Aug 8, 2015 at 0:39