Improper distribution is the distribution which doesn't integrate to 1. For example, Improper Uniform Prior Distribution for all positive integers. This distribution integrate to infinity. Is it okay to use this kind of distribution as a prior in Bayesian estimation, when it is really not a probability density?
I think most Bayesians, like Luca has said, don't have problems using improper priors as long as the posterior is well-defined. Others recommend regarding improper priors only as limits of proper priors, and yet others are (were?) against using improper priors at all.
Not being a Bayesian, I can't say what the up-to-date consense would be. In a quick Google search I found a tutorial according to which "Use of improper priors makes model selection and hypothesis testing difficult."
If somebody can elaborate on the topic, that will be great.
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