Statistical Methods in Medical Research 4ed

distributed. The methods developed for samples from a normal distribution
therefore provide a reliable approximation for samples from non-normal distributions,
unless the departure from normality is severe or the sample size is very
small. The same is true in Bayesian inference, and we shall concentrate here on
methods appropriate for samples from normal distributions.
Figure 4.1 describes the likelihood function for a single observation x from a
normal distribution with unit variance, N(m, 1). It is a function of m which takes
the shape of a normal curve with mean x and unit variance. This result can
immediately be extended to give the likelihood from a sample mean. Suppose
that x is the mean of a sample of size n from a normal distribution N(m, s2 ).
From §4.2, we know that x is distributed as N(m, s2 =n), and the likelihood
function is therefore a normal curve N(x, s2 =n).
Suppose now that m follows a normal prior distribution N(m0, s2 0 ). Then,
application of Bayes' theorem shows that the posterior distribution of m is
N x ‡ m0s2 =ns2 0
1 ‡ s2 =ns2 s
,
0
2 =n
1 ‡ s2 =ns2 : …6:1†
0
The mean of this distribution can be written in the form
x n
s2 ‡ m0 n 1
‡
s2 s
1
2 0
6.2 Bayesian inference for a mean 169
s 2 0
, …6:2†
which is a weighted mean of the observed mean x and the prior mean m0, the
weights being inversely proportional to the two variances of these quantities (the
sampling variance of x, s2 =n, and the prior variance of m, s2 0 ). Thus, the observed
data and the prior information contribute to the posterior mean in proportion to
their precision. The fact that the posterior estimate of m is shifted from the
sample mean x, in the direction of the prior mean m0, is an example of the
phenomenon known as shrinkage, to be discussed further in §6.4.
The variance of the posterior distribution (6.1) may be written in the form
…s2 =n†s2 0
…s2 =n†‡s2 , …6:3†
0
which is less than either of the two separate variances, s2 =n and s2 0 . In this sense,
precision has been gained by combining the information from the data and the
prior information.
These results illustrate various points made in §6.1. First, the family chosen for
the prior distributions, the normal, constitutes the conjugate family for the normal
likelihood. When the prior is chosen from a conjugate family, the posterior
distribution is always another member of the same family, but with parameters
altered by the incorporation of the likelihood. Although this is mathematically
very convenient, it does not follow that the prior should necessarily be chosen