Bayesian Factor Analysis: A Versatile Framework for Denoising, Interference Suppression, and Source Localization - Electromagnetic Brain Imaging - page 114

5.7.2 Proof of Eq. ( 5.94 )

We provide a proof of Eq. ( 5.94 ), which is repeated here for convenience.

K K

1
2

1
2

T

u k u k E −

1 (

y k − Au k ) ʛ ( y k − Au k ) −

( A , u )

k = k = 1

K K

1
2

1
2

y k ʛ u k ʓ ¯

=−

y k +

1 ¯

u k .

j = 1 j =

The left-hand side of the equation above is changed to:

K K

1
2

1
2

T

u k u k E −

1 (

y k − Au k ) ʛ ( y k − Au k ) −

( A , u )

k = k = 1

K

1
2

y k ʛ =− y k

k = 1

− u k u k

K

1
2

u k A T y k ʛ u k A T

− E ( A , u ) ʛ y k − Au k + ʛ Au k + .

k = 1

(5.174)

On the right-hand side of Eq. ( 5.174 ), the following relationship:

u k

u k u k E u u k

E A A T

A I u k A T E ʛ Au k + = ʛ +

( A , u )

E u u k I u k

¯ A T

ʛ ¯ A ʨ − 1

= + M +

E u u k ʓ u k E u tr u k u k ʓ

= =

tr E u u k u k tr

u k + ʓ − 1 = ʓ = ( ¯ u k ¯ ) ʓ

u k ʓ ¯ = ¯ u k (5.175)

holds, where constant terms are omitted. In the equation above, we use Eq. ( 5.56 ).
Also, the relationship,

y k ʛ

y k

¯ A T

u k A T y k ʛ ¯ A u k E Au k + ʛ = u k + ¯ ¯ ʛ y k

( A , u )

¯ A T

y k ʛ ¯ A ʓ − 1 u k ʓʓ − 1 = ʓ ¯ u k + ¯ ʛ y k

u k ʓ ¯ = 2 ¯ u k (5.176)

holds. Substituting Eqs. ( 5.175 ) and ( 5.176 )into( 5.174 ), we get Eq. ( 5.94 ).