Bayesian Factor Analysis: A Versatile Framework for Denoising, Interference Suppression, and Source Localization - Electromagnetic Brain Imaging - page 93

F =

E ( A , u ) [

( , , ) − ( | ) − ( | ) ]

log p y u A log p u y log p A y

E ( A , u ) log p ) = ( y | u , A ) + log p ( u ) + log p ( A

+ H( ( | )) + H( ( | )).

p u y p A y (5.87)

Substituting

K

K
2

1
2

T

log p ( y | u , A ) = log | ʛ |−

1 (

y k − Au k ) ʛ ( y k − Au k ), (5.88)

k =

K

1
2

u k u k ,

( ) =−

log p u (5.89)

k = 1

M M

1
2

1
2

a j ʻ j ʱ log p ( A ) = log | ʻ j ʱ |− a j , (5.90)

j = 1 j = 1

and

K
2

H( p ( u | y )) =− log | ʓ | , (5.91)

M

1
2

H( p ( A | y )) =− log | ʻ j ʨ | , (5.92)

j = 1

into Eq. ( 5.87 ), we obtain

K K

K
2

1
2

1
2

T

u k u k F = log | ʛ |+ E −

1 (

y k − Au k ) ʛ ( y k − Au k ) −

( A , u )

k = k = 1

⊡ ⊤

M M M

1
2

1
2 E A

K
2

1
2

⊣

a j ʻ j ʱ

⊦ −

+ log | ʻ j ʱ |− a j log | ʓ |− log | ʻ j ʨ | .

j = 1 j = 1 j = 1

(5.93)

Note that the following relationships:

K K

k = 1 (

1
2

1
2

T

u k u k E − y k − Au k ) ʛ ( y k − Au k ) −

( A , u )

k = 1

K K

1
2

1
2

y k ʛ u k ʓ ¯ =− y k +

1 ¯

u k (5.94)

k = 1 k =