Bayesian Factor Analysis: A Versatile Framework for Denoising, Interference Suppression, and Source Localization - Electromagnetic Brain Imaging - page 80

( | ʸ )

Based on the arguments in Sect. B.4 of the Appendix, p y is derived such that

K

p ( y | ʸ ) = p ( y k | ʸ ) where p ( y k | ʸ ) = N( y k | 0 , ʣ y ), (5.25)

k = 1

ʦ − 1 in Eq. (B.30), where ʣ y is the model data covariance. Setting A to H and I to

we derive ʣ y , such that

ʣ y = ʛ − 1 AA T + . (5.26)

According to Eq. (B.29), the log likelihood function is obtained as

K K

K
2

1
2

| ʣ − 1 y k ʣ − 1 log p ( y | ʸ ) = p ( y k | ʸ ) = log |− y k . (5.27)

y y

k = 1 k = 1

Using the matrix inversion formula in Eq. (C.91), we have

AA T − 1 ʣ − 1

y

ʛ − 1

= +

A A T I − 1

A T ʓ − 1 A T = ʛ − ʛ ʛ A + ʛ = ʛ − ʛ A ʛ , (5.28)

Using the equation above and Eq. ( 5.11 ), we get

y k =

y k ʣ − 1 y k ʓ − 1 A T y k ʛ u k ʓ ¯ y k = ʛ − ʛ A ʛ y k − ¯ u k .

y

Also, since the relationship

| ʣ − 1
y

ʓ − 1 A T ʓ − 1 A T || ʛ |=| ʓ − 1 |=| ʛ − ʛ A ʛ |=| I − ʛ A || ʛ | (5.29)

holds, 1 by substituting the above two equations into Eq. ( 5.27 ), we finally obtain the
expression for the likelihood:

K K K

K
2

log | ʛ |

1
2

1
2

y k ʛ u k ʓ ¯ log p ( y | ʸ ) = p ( y k | ʸ ) =

| ʓ | −

y k +

1 ¯

u k . (5.30)

k = 1 k = 1 k =

A more informative derivation of Eq. ( 5.30 ) uses the free energy expression in
Eq. (B.63) from the Appendix, which claims the following relationship holds:

1

ʓ − 1 A T ,wehave Defining C = I − ʛ A

A T C = A T

− A T ʛ A ʓ − 1 A T = A T − ( ʓ − I ) ʓ − 1 A T = ʓ − 1 A T .

|=| ʓ − 1 Taking the determinant of the equation above, we get | C | .