INTRODUCTION TO NONINFORMATIVE PRIORS

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 p. This rule is a special form of a more general one derived by Jeffreys (1961) (see Section 5.4) we illustrate it here by two examples. 

The likelihood function for 9 in Examples 5.1 and 5.3 depends on the data only through the mean value y. that is. 

Since i 0 y,a) is data-translated in 0. the function p 0) here is simply 0. The noninformative prior is accordingly 

The resulting posterior density function is thus proportional to the likelihood. 

The standard deviation cr of a normally distributed variable y with known mean y has the likelihood function (see Eq. (5.1-11)) 

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