Bayesian Factor Analysis: A Versatile Framework for Denoising, Interference Suppression, and Source Localization - Electromagnetic Brain Imaging - page 115

5.7.3 Proof of Eq. ( 5.103 )

Next, the proof of Eq. ( 5.103 ) is presented, which is
E A AR uu A T = ¯ AR uu ¯ A T + ʛ − 1 tr ( R uu ʨ ) .

The proof is quite similar to the one provided in Sect. 5.7.1 .First,wehave

AR uu A T [ ] i , j = A i , k [ R uu ] k , A j , = [ R uu ] k , A i , k A j , .

k , k ,

Using Eq. ( 5.168 ),we get

E A ] i , j

1
ʻ j [ ʨ − 1

AR uu A T R uu ] k , A i , k

A j , +

[ = [ [ R uu ] k , ʴ i , j ] k , .

k , k ,

(5.177)

The first term on the right-hand side of the equation above is rewritten as

A j , =[ ¯ AR uu ¯ A T

R uu ] k , A i , k [ ] i , j .

k ,

Considering

R uu ] k , [ ʨ − 1 R uu ʨ − 1 [ ] k , = tr ( ), (5.178)

k ,

the second term in Eq. ( 5.177 ) is rewritten as

1
ʻ j [ ʨ − 1

] k , =[ ʛ − 1 R uu ʨ − 1 [ R uu ] k , ʴ i , j ] i , j tr ( ). (5.179)

k ,

Consequently, we get the following relationship:

]= ¯ AR uu ¯ A T

AR uu A T + ʛ − 1 tr R uu ʨ − 1 E A [ ( ). (5.180)

5.7.4 Proof of Eq. ( 5.166 )

Next, the proof of Eq. ( 5.166 ) is presented. To derive this equation, we should consider
log p ( y | z , A ) and log p ( A ) , because ʛ is contained in these two terms. We have