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Mathematical statistics and data analysis 3文档
LECTURE 12
Mid-term Test
Date: Thursday 18 March 2010
Location : LT22
Time : 4:15 pm to 5:45 pm
The mid-term test is closed book but calcu-
lators and up to 3 help sheets are allowed to
be brought in.
1
Case of unknown mean and un-
known variance
In this case, there are two parameters Θ
and Ξ. A Bayesian approach requires the
specification of a joint 2-dimensional prior
distribution.
For mathematical convenience, we assume
that Θ and Ξ are independent where
Θ ∼ N (θ0 , ξ−1 ),
prior
Ξ ∼ Γ(α, λ).
Here θ0 , ξprior , α and λ are all known con-
stants.
2
Then the posterior distribution of (Θ, Ξ)
given the data X = (X , . . . , X ) is
1 n
fΘ,Ξ |X (θ, ξ |x)
∝ fX |Θ,Ξ (x |θ, ξ)fΘ (θ)fΞ (ξ)
1 n/2 −(ξ/2) n (x −θ)2
= ξ e i=1 i
C
−(ξ /2)(θ−θ )2 α−1 −λξ
×e prior 0 ξ e ,
where C is a constant that does not depend
on either θ or ξ .
This posterior distribution is more compli-
cated than those of the previous examples.
In particular, the constant of proportional-
ity C can only be evaluated using numerical
integration where
∞ ∞ n/2 −(ξ/2) n (x −θ)2
C = ξ e i=1 i
−∞ 0
−(ξ /2)(θ−θ )2 α−1 −λξ
×e prior 0 ξ e dξdθ.
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