JEFFREYS PRIOR FOR MULTIPARAMETER MODELS

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). 

When data are obtained, we assume that the log-likelihood function will have a local maximum (mode) at some point 6 = 6m- A second-order Taylor expansion of L(6) around that point will give 

The expectation of this expansion over the distribution of y predicted by the model is 

Here IniOu) is the value at 0m of Fisher s (1935) information matrix 

The matrix Xn (j ) then inherits positive definiteness from X (0), and the 0-analog of Eq. (5.5-2) gives the following log-likelihood contours, 

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