186 Current Trends in Bayesian Methodology with Applications
An interval that will contain an obser ved value of y
i
with virtual certainty,
given that we know µ
i
and δ
i
, is µ
i
± δ
1/2
i
z
(1+p)/2
. L e t s
1i
and s
2i
be lower
and upper bounds respectively on the half-length of this interval. The values
s
1i
and s
2i
represent what we know about the scaling of y
i
and we choose
these so that s
1i
is not excessively small or s
2i
excessively large. These choices
imply that
s
−2
2i
z
2
(1+p)/2
≤ δ
−1
i
≤ s
−2
1i
z
2
(1+p)/2
. (9.7)
Therefore, σ
0
= max{(m
2i
−m
1i
)/2s
2i
: i = 1, . . . , p} is compatible with (9.6)
and (9.7), which specifies another hyperparameter. It is perhaps logical for
(m
2i
− m
1i
)/2s
2i
to be constant but this is not necessary.
Let G (α
0i
, β
0i
, ·) denote the gamma(α
0i
, β
0i
) cdf, ...