188 Exposure-Response Modeling: Methods and Practical Implementation
the po sterior distr ibutio ns have the simpler form
β|σ
2
∼ N((
ˆ
β + n
0
β
0
)/(n
0
+ 1), σ
2
S
−1
x
/(n
0
+ 1))
σ
2
∼ IG(a + (n − p)/2, b + (s
2
+ n
0
(
ˆ
β − β
0
)
T
S
x
(
ˆ
β − β
0
)/(n
0
+ 1))/2)
(7.11)
where p is the dimension of β. This prior has a simple relationship with EES,
as it means that the prior information on β is equivalent to n
0
“copies” of
the observed data. This is refle cted in the po sterior distribution, as its mean
is the weighted average of the LS and prior mean based on the observed and
“pseudo” s ample sizes and the inverse variances are pro po rtional to the total
sample size. One may specify a weak prior by specifying a prior equiva lent to,
say, 0.1 copy of the observed information.
For NLMM ...