Basics of Bayesian Inference - Electromagnetic Brain Imaging - page 231

K

Hx k , ʛ − 1 , ʦ − 1

p ( y ) = N( y k | )N( x k | 0 ) d x k . (B.28)

k = 1

This p is called the marginal likelihood. This integral is computed in Sect. 4.3 .

Using Eq. ( 4.30 ) , we have

( y )

2 exp

y k

K

1
| ʣ y |

1
2 y k ʣ − 1

( ) ∝ − ,

p y (B.29)

y

1 /

k = 1

where ʣ y is expressed as

ʣ y = ʛ − 1 ʦ − 1 H T + H . (B.30)

This ʣ y is referred to as the model data covariance. Taking the normalization into

account, we get the marginal distribution p ( y ) , such that

K

p ( y ) = p ( y k ) where p ( y k ) = N( y k | 0 , ʣ y ). (B.31)

k = 1

Thus, ʣ y , the model data covariance, is the covariance matrix of the marginal dis-

tribution p ( y k ) .

B.5 Expectation Maximization (EM) Algorithm

B.5.1 Marginal Likelihood Maximization

The Bayesian estimate of the unknown x , ¯ x k , is computed using Eq. ( B.25 ). How-

¯

ever, when computing x k , ʦ and ʛ are needed. The precision matrix of the prior

ʦ ʛ

distribution are called the hyperparameters. Quite often, the

hyperparameters are unknown, and should also be estimated from the observation y .
The hyperparameters may be estimated by maximizing p

and that of the noise

( y | ʦ , ʛ ) , which is the like-

lihood with respect to is called the marginal likelihood or

the data evidence to discriminate it from the conventional likelihood p

ʦ and ʛ .This p ( y | ʦ , ʛ )

( y | x ) .

using 1 The marginal likelihood can be computed with p ( x | ʦ ) and p ( y | x , ʛ )

∞

p ( y | ʦ , ʛ ) = p ( y | x , ʛ ) p ( x | ʦ ) d x . (B.32)

−∞

However, computation of is generally quite trou-

blesome, as is demonstrated in detail in Chap. 4 . In the following, instead of directly

ʦ and ʛ that maximize p ( y | ʦ , ʛ )

1 The notation d x indicates d x 1 d x 2