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Confidence intervals from the approximate posterior density 395

Confidence intervals from the approximate posterior density

The approximate posterior density π(θ,σ|y) for single-response data (8.116), describes
the joint uncertainty in both θ and σ. Of more interest is the marginal posterior density for
θ,
∞
'
π(θ|y) = π(θ,σ|y)dσ (8.117)
0
This is the posterior for θ, without regard to the exact value of σ. As we discard σ by
“integrating it out,” it is called a nuisance parameter.
For the approximate posterior (8.116), the marginal posterior density can be calculated
analytically,
∞ 1
' −N/2
T
T
π(θ|y) = π(θ,σ|y)dσ ∝ 1 + (θ − θ M ) [X X| θ M ](θ − θ M ) (8.118)
νs 2
0
Note again that for a linear model, this marginal posterior is exact.
Confidence interval for the mean of a population and
the t-distribution
We now form confidence intervals for the model parameters and the predicted responses
using this approximate marginal distribution. First, it is best to consider the simple model
[k]
(8.104), y [k] = θ + ε . After N measurements, the posterior density is
1 2 2
1
−(N+1)
p(θ, σ|y) ∝ σ exp − [νs + N(θ − ¯ y) ] (8.119)
2σ 2
where the sample mean and sample variance are
N 1 N

2
¯ y = N −1 y [k] s = y [k] − ¯ y 2 ν = N − 1 (8.120)
ν
k=1 k=1
The marginal posterior density for θ is
−(ν+1)/2
N 2
∞
'
p(θ|y) = p(θ, σ|y)dσ ∝ 1 + 2 (θ − ¯ y) (8.121)
0 νs
Defining the t-statistic,
¯ y − θ
t ≡ √ (8.122)
(s/ N)
whose distribution satisfies p(t|ν)dt = p(θ|y)dθ,wehave
−(ν+1)/2
t
2
p(t|ν) ∝ 1 + (8.123)
ν
This is the famous t-distribution of Student, a pseudonym for W. S. Gosset ( Gosset, 1908).
As the number of degrees of freedom v approaches infinity, the t-distribution reduces to a