Reference priors

We conclude this discussion on priors by mentioning `reference analysis,' which is an area of active research among statisticians. The intention is, similarly to that for other priors motivated by basic principles, that of ``characterizing a `non-informative' or `objective' prior distribution, representing `prior ignorance,' `vague prior knowledge,' and `letting the data speak for themselves' " (Bernardo and Smith 1994). However, ``the problem is more complex than the apparent intuitive immediacy of these words and phrases would suggest'' (Bernardo and Smith 1994, p. 298):

``Put bluntly: data cannot ever speak entirely for themselves: every prior specification has some informative posterior or predictive implications; and `vague' is itself much too vague an idea to be useful. There is no `objective' prior that represents ignorance.

On the other hand, we recognize that there is often a pragmatically important need for a form of prior to posterior analysis capturing, in some well-defined sense, the notion of the prior having a minimal effect, relative to the data, on the final inference. Such a reference analysis might be required as an approximation to actual beliefs; more typically, it might be required as a limiting `what if?' baseline in considering a range of prior to posterior analyses, or as a default option when there are insufficient resources for detailed elicitation of actual prior knowledge.

...From the approach we adopt, it will be clear that the reference prior component of the analysis is simply a mathematical tool. It has considerable pragmatic importance in implementing a reference analysis, whose role and character will be precisely defined, but it is not a privileged, `unique non-informative' or `objective' prior."

The curious reader may take a look at the (Bernardo and Smith 1994) and references cited therein, as well as at Bernardo (1997).