Statistical Methods in Medical Research 4ed

to the likelihood, is a multivariate normal distribution, with mean b 0 and
dispersion matrix D. The posterior distribution of b is multivariate normal,
with dispersion M 1 and mean
where
16.3 The Bayesian linear model 539
M 1 …s 2 X T y ‡ D 1 b 0†, …16:4†
M ˆ s 2 X T X ‡ D 1 : …16:5†
Notice that these formulae are multivariate analogues of the univariate version,
where the posterior mean was derived as a sum weighted by precisions. The term
s 2 X T X in (16.5), being the inverse of (11.51), can be thought of as the multivariate
precision of the parameter, and the second term is the prior precision.
The analogue of the sample mean is b, defined by (11.49), and from this it follows
that X T y ˆ…X T X†b. Substituting this in (16.4) clearly demonstrates that the
posterior mean has the form of a weighted sum, analogous to the univariate
case. The situation where no prior information is available could be modelled by
using an improper prior in which, formally, D 1 ˆ 0. Consequently, from (16.4)
and (16.5), the posterior mean and variance reduce to the quantities (11.49) and
(11.51), which are familiar from frequentist analysis. Of course, as b is now
considered to be a random variable, there would be subtle differences in the
way the uncertainty in the estimate of the regression coefficients would be
expressed.
This result, and for that matter the simpler univariate version adduced in
Chapter 6, is not, in itself, very useful because it supposes s 2 is known and has
not been included as a random quantity in the Bayesian analysis. How to include
this parameter will now be discussed but, as will be seen, realistic approaches
lead to complications.
Conjugate and related analyses of normal samples with unknown
variance
The analysis in §6.2 is sufficiently realistic and elegant for it to be natural to
attempt to extend the use of conjugacy arguments to accommodate unknown
variance. When conjugate informative priors are sought for this problem some
difficulties arise, and these will be described below. However, some insight into
the form of distribution that will be required can be gained from the case where
an improper uninformative prior is used. From §16.2, the uninformative prior
for …m, s 2 † is p…m, s 2 †ˆ1=s 2 and from this it follows that the posterior is
p…m, s 2 jy† /s n 2 exp
1
2s2 ‰…n 1†s 2 ‡ n…y m† 2 Š , …16:6†
where y is the mean of a sample of size n which has variance s2 .