242 Optimizing Optimization
with
∂ ∂ ∂ ∂ ∂ ∂
L
i
LL
iωωθμθθμω πθω Ω 12120010Consequently,
ωμθ
θΩ^11
2(,)i⎡⎣⎢
⎢⎤⎦⎥
⎥where
μμμ
μθ
θπ
ΩΩ
ΩΩ11
111
2 1i
iii⎡⎣⎢
⎢
⎢⎤⎦⎥
⎥
⎥⎡⎣⎢
⎢⎤⎦⎥
⎥⎡⎣⎢
⎢⎤⎦⎥
⎥That is,
ωμˆ ˆ (ˆ,)ˆπ
Ω^11
1iQ⎡⎣⎢
⎢⎤⎦⎥
⎥and therefore,
σωωπππ2111
11 ˆ ˆˆ ()ˆˆ ,Ω QpQ p p⎡⎣⎢
⎢⎤⎦⎥
⎥
⎡⎣⎢
⎢⎤⎦⎥
⎥
whereFrom our earlier results, pQ p W T NTpQ p
ˆ^1 |ˆ ( , )
1
μ∼ 1 ,^11 where
W 1 ( · ) is a Wishart of dimension 1. Here, Qi i
(,)μμˆˆΩ^1 (),,
and therefore,
pQ p ˆ^1 ∼χ()^2 TN 1 Ψ, where χm
2
is a chi-squared with m degree of freedom.
withΨ (^11)
T
pQ p