JEFFREYS PRIOR FOR ONE-PARAMETER MODELS

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

Let 9m denote the mode (initially unknown) of the log-likelihood function L 9 y) for the single parameter 9. A second-order Taylor expansion of L around 9m, for a given model and pattern of experiments, gives 

Here A is a constant for a given model and pattern of n experiments, and din(OM) is the value at 9m of the function 

To obtain a noninformative prior, we require that the interval 4 — 4 m C have the same expected posterior probability content, wherever its center d M may turn out to be when data are analyzed. In other words, the integral f p (p) exp[iF - (0 - 5m) /2] d(j from (j M - C to j M + C should be independent of 4 m, so that p (j ) should be a constant. Equation (5.4-6) then gives 

Since 2 (0) is independent of 9 here, the Jeffreys prior p 9) is uniform. This result agrees with Eq. (5.3-2). 

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