Biomedical Engineering Reference
In-Depth Information
2.10 Bayesian Derivation of the Minimum-Norm Method
2.10.1 Prior Probability Distribution and Cost Function
In this section, we derive theminimum-normmethod based onBayesian inference. As
in Eq. (
2.16
), we assume that the noise
ʵ
is independently and identically distributed
Gaussian, i.e.,
, ʲ
−
1
I
ʵ
∼
N(
ʵ
|
0
),
(2.54)
2
.
Thus, using Eq. (
2.14
), the conditional probability distribution of the sensor data for
agiven
x
,
p
ʲ
−
1
where the precision
ʲ
is used, which is the inverse of the noise variance,
=
˃
(
y
|
x
)
is
2
M
/
2
exp
2
−
2
p
(
y
|
x
)
=
y
−
Fx
.
(2.55)
ˀ
This conditional probability
p
in the arguments
in Sect.
2.5
. Since
x
is a random variable in the Bayesian arguments, we use the
conditional probability
p
(
y
|
x
)
is equal to the likelihood
p
(
y
)
.
Let us derive a cost function for estimating
x
. Taking a logarithm of the Bayes's
rule in Eq. (B.3) in the Appendix, we have
(
y
|
x
)
, instead of
p
(
y
)
log
p
(
x
|
y
)
=
log
p
(
y
|
x
)
+
log
p
(
x
)
+
C,
(2.56)
where
C
represents the constant terms. Neglecting
C
, the cost function
F(
x
)
in general
form is obtained as
2
F(
)
=−
(
|
)
=
ʲ
−
−
2log
p
(
x
).
(2.57)
x
2log
p
x
y
y
Fx
The first term on the right-hand side is a squared error term, which expresses how
well the solution
x
fits the sensor data
y
. The second term
is a constraint
imposed on the solution. The above equation indicates that the constraint term in the
cost function is given from the prior probability distribution in the Bayesian formu-
lation. The optimum estimate of
x
is obtained by minimizing the cost function
−
2log
p
(
x
)
F(
x
)
.
2.10.2 L
2
-Regularized Method
Let us assume the following Gaussian distribution for the prior probability distribu-
tion of
x
,
2
N
/
2
exp
2
−
2
p
(
x
)
=
x
.
(2.58)
ˀ
