Bayesian Factor Analysis: A Versatile Framework for Denoising, Interference Suppression, and Source Localization - Electromagnetic Brain Imaging - page 94

and

M M M M

j

log ʻ

| ʱ |

M log | ʱ |
| ʨ |

log | ʻ j ʱ |− log | ʻ j ʨ |=

| ʨ | =

(5.95)

M
j

ʻ

j = 1 j = 1 j = 1

hold. The proof of Eq. ( 5.94 ) is presented in Sect. 5.7.2 . Also, we can show

⊡ ⊤ ⊡ ⊤

M M L

⊣ ⊦ = ⊣ ⊦

a j ʻ j ʱ

1 ʻ j ʱ A j ,

E A a j E A

j = 1 j = 1 =

M L M L

1
ʻ j [ ʨ − 1

1 ʻ j ʱ A j , +

=

1 ʻ j ʱ

] ,

j = 1 = j = 1 =

M L L

1 ʻ j ʱ A j , + 1 ʱ [ ʨ − 1

= M ] ,

j = 1 = =

M tr ʱʨ − 1

j = 1 ¯

M

a j ʻ j ʱ ¯ = a j + . (5.96)

By substituting the above three equations into Eq. ( 5.93 ), we obtain the expression
for the free energy:

K K

K
2

log | ʛ |

1
2

1
2

y k ʛ u k ʓ ¯

F =

| ʓ | −

y k +

1 ¯

u k

k = 1 k =

M

tr ʱʨ − 1

M
2

log | ʱ |

1
2

M
2

a j ʻ j ʱ ¯ +

| ʨ | −

1 ¯

a j − . (5.97)

j =

Note here that

tr tr ʛ ¯ A

j = 1 ¯

M

ʱ ¯ A T ʱ ¯ A T a j ʻ j ʱ ¯

¯ A

a j = ʛ = . (5.98)

Thus, we have

M

M tr ʱʨ − 1 tr

¯ A T

ʨ − 1 ʛ ¯ A a j ʻ j ʱ ¯

1 ¯

a j + = ʱ + M

j =

tr ʱ − 1 = ʱ M = ML . (5.99)