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Which of approaches - deterministic or probabilistic - is the more general?
The question sounds a bit scholastic, inasmuch as the two approaches have a
common philosophy. The point is that in the probabilistic formulation the inverse
problem remains unstable and still needs the regularization, which directly does
not follow from probabilistic formalism. Obviously, the principle definitions for
the probabilistic inverse problem should be derived from the general theory of
regularization.
Let us come back to operator equations (10.1). In a general form we can write
F
F
{
x
,
y
,
z
=
0
,
,
(
x
,
y
,
z
)
} =
,
(10
.
75)
where F is an operator of the forward problem, which depends parametrically on
x
and calculates the tensor or vector response function F from a given electrical
conductivity
,
y
,
z
), and F is this response function determined on the sets of
observation points
M
(
x
(
x
,
y
,
.
We will begin with the principle of regularization, which in the deterministic
terms is expressed by (10.64). These equations can be readily rewritten in proba-
bilistic terms. Concerning (10.75), we write
,
y
) and frequencies
with error
lim
P
→
{
−
˜
>
} =
¯
0
,
(10
.
76)
0
where
P
is the probability that the error of the solution (the deviation of approxi-
mate solutions ˜
from the exact model solution ¯
) is larger than an arbitrarily small
positive number
. Equation (10.76) expresses the stochastic principle of regular-
ization. An inversion operator constructed on the basis of this principle is called a
stochastically regularizing operator.
To exemplify the construction of the stochastically regularizing operator, we
consider the
method of maximum likelihood
(Goltsman, 1971; Kovtun, 1980;
Yanovskaja and Porokhova, 1983).
Following Goltsman (1971), we introduce the
likelihood function
as:
)
p
( F)
l
(
)
=
ln
p
(
,
(10
.
77)
where
p
(
) is the density of the a priori (unconditional) probability of the solution
and
p
( F) is the density of the a posteriori (conditional) probability of a response
function F for the given conductivity distribution
. It is reasonable to think that
if a solution
comes into being , the probability of this event is fairly great. We
can go a bit further and suppose that the most probable event characterized by
the maximum likelihood is the advent of a solution
that is close to the exact
model solution ¯
. This heuristic consideration says that the approximate solution to
the problem (10.75) can be found from the condition for the maximum likelihood
function
l
(˜
)
=
sup
∈
l
(
)
(10
.
78)
