Jeffreys noninformative prior,

The construction of a noninformative prior is a nontrivial task, requiring analysis of likelihood functions for prospective data. The construction is simplest when the likelihood l 6) for a single parameter is data-translated in some coordinate d>(0) then the noninformative prior density takes the form p (t)) = const, over the permitted range of special form of a more general one derived by Jeffreys (1961) (see Section 5.4) we illustrate it here by two examples. 

Jeffreys (1961) gave a formal method for constructing noninformative priors... 

Two classes of parameters are needed in models of observations location parameters 6i to describe expected response values and scale parameters dg to describe distributions of errors. Jeffreys treated 6i and dg separately in deriving his noninformative prior this was reasonable since the two types of parameters are unrelated a priori. Our development here will parallel that given by Box and Tiao (1973, 1992), which provides a fuller discussion. The key result of this section is Eq. (5.5-8). 

In this chapter, Bayesian and likelihood-based approaches have been described for parameter estimation from multiresponse data with unknown covariance matrix S. The Bayesian approaches permit objective estimates of 6 and E by use of the noninformative prior of Jeffreys (1961). Explicit estimation of unknown covariance elements is optional for problems of Types 1 and 2 but mandatory for Types 3 and 4. 

While there exists a technique, due to Jeffreys, for generating noninformative priors directly from the likelihood function, here we simply propose a transformation T](a) = Ina, and then show that its likelihood is data-translated. For a discussion of automated procedures for constructing noninformative priors, consult a general text on Bayesian statistics such as Box Tiao (1973). 

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