Troyan V., Kiselev Y. Statistical Methods of Geophysical Data Processing
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Elements of applied functional analysis 273
The corresponding Euler’s equation has the form
ln p + αF + β1 = 0,
from which
ˆp = p
0
e
−αF
,
where p
0
= e
−β1
= const. Substituting F (ϕ) into ||ϕ||
2
H
, we obtain the finally form
of the weight function p(ϕ), which corresponds to the minimal arbitrariness in the
choice of this function
ˆp(ϕ) = p
0
e
−α||ϕ||
2
H
= p
0
e
−α(ϕ,Hϕ)
,
where α > 0.
The obtained function ˆp(ϕ) has all properties of the probability density, therefore
we have the possibility to make use of the probability measure properties and to
implement the statistical interpretation of all constructions including the weight
function p, for example, the mathematical expectation and variance:
Eϕ
∆
= (ϕ, p),
E[(ϕ − Eϕ)
2
]
∆
= ((ϕ − (ϕ, p))
2
, p).
The operation of the mathematical expectation E is linear in the explicit form:
E(λϕ
1
+ µϕ
2
) = λE(ϕ
1
) + µE(ϕ
2
),
therefore
ELϕ = LEϕ,
i.e. the operator of the mathematical expectation E computes with linear operators.
We shall introduce a correlation operator (correlator) by Eϕ = 0:
K
ϕϕ
: E(η, ϕ)(ξ, ϕ)
∗
= (η, K
ϕϕ
ξ) ∀ ξ, η.
It follows from the definition, that the correlation operator, which can be written
as K
ϕϕ
= Eϕϕ
∗
, is nonnegative:
(η, K
ϕϕ
η) = E(η, ϕ)(η, ϕ
∗
) = E|(η, ϕ)|
2
≥ 0.
It is possible to write the correlator for the linear transformed functions Aϕ and
(Bϕ)
∗
:
E(Aϕ)(Bϕ)
∗
= AK
ϕϕ
B
∗
,
as far as
E(η, Aϕ)(ξ, Bϕ)
∗
= E(A
∗
η, ϕ)(B
∗
ξ, ϕ)
∗
= (A
∗
η, K
ϕϕ
B
∗
ξ) = (η, (AK
ϕϕ
B
∗
) ξ).
In particular, if the correlator of the function ϕ is equal to K
ϕϕ
, then the correlator
of the linear transform Aϕ is equal to AK
ϕϕ
A
∗
:
Eψψ
∗
∆
= E(Aϕ)(Aϕ)
∗
= AK
ϕϕ
A
∗
.
274 STATISTICAL METHODS OF GEOPHYSICAL DATA PROCESSING
The correlation function k
ϕϕ
(x, x
0
), which is an integral kernel of the correlation
operator K
ϕϕ
, it is possible to represent as:
k
ϕϕ
(x, x
0
) = Eϕ(x)ϕ
∗
(x
0
),
since the function ϕ(x) in the points x and x
0
are determined by projection operators
(projectors) P
x
and P
x
0
:
P
x
(ϕ) =
Z
ϕ(y)δ(x − y)dy,
P
x
0
(ϕ) =
Z
ϕ(y)δ(x
0
− y)dy.
Then we write:
E(P
x
ϕ)(P
x
0
ϕ)
∗
= P
x
K
ϕϕ
P
x
0
= k
ϕϕ
(x, x
0
).
We introduce the Hilbert space with the scalar product:
(ϕ, ψ)
∆
= spEϕψ
∗
∆
= spK
ϕψ
.
Here the operator E is given by the measure dµ
ϕψ
= p(ϕ, ψ)dϕdψ; K
ϕψ
is called
the cross-correlator of the functions ϕ,ψ. We remember that the correlator K
ϕϕ
is
an auto-correlator. We shall check that the form (ϕ, ψ) is a scalar product in fact:
(1) (ϕ, ϕ) = spK
ϕϕ
> 0;
(2) (ϕ, ψ) = Eϕψ
∗
spK
ϕψ
=
R
k
ϕψ
(x, x) dx =
=
R
k
∗
ψϕ
(x, x) dx = spK
∗
ψϕ
= sp(Eψϕ
∗
)
∗
= (ψ, ϕ)
∗
;
(3) (ϕ, α
1
ψ
1
+ α
2
ψ
2
) = α
1
(ϕ, ψ
1
) + α
2
(ϕ, ψ
2
).
The orthogonal functions ϕ and ψ are called uncorrelated functions, if (ϕ, ψ) = 0.
We note, that it is possible to introduce the Hilbert space determining the scalar
product with the measure dµ
ϕ
= p(ϕ)dϕ. Then the scalar product for the elements
ϕ(x) and ϕ(x
0
) of the Hilbert space will be given by the correlation function at the
point (x, x
0
):
(ϕ|
x
, ϕ|
x
0
) = k
ϕϕ
(x, x
0
).
Based on the scalar product properties, it is possible to write the property of the
correlation function: we obtain from Schwarz’s inequality
|(ϕ|
x
, ϕ|
x
0
)| ≤ ||ϕ|
x
||
2
||ϕ|
x
0
||
2
,
i.e.
|k(x, x
0
)|
2
≤ σ
2
ϕ
(x)σ
2
ϕ
(x
0
), σ
2
ϕ
(x) = E|ϕ(x)|
2
.
Introducing a correlation coefficient r for a real function ϕ:
r =
k
ϕϕ
(x, x
0
)
σ
ϕ
(x)σ
ϕ
(x
0
)
,
we obtain the relation for the correlation coefficient: −1 ≤ r ≤ 1. We define
random fields ϕ as homogeneous, if the following condition k
ϕϕ
(x, x
0
) = k
ϕϕ
(x−x
0
)
Elements of applied functional analysis 275
is satisfied. This condition leads, in particular, to a constancy of the variance:
σ
ϕϕ
(x) = const.
As mentioned above, the maximum entropy function p(ϕ) is such that the func-
tion p ∼ e
−α(ϕ,Hϕ)
satisfies to the condition (ϕ, Hϕ) = C.
The function p(ϕ) it is possible to interpret as the Gaussian density function,
which for the ϕ ∈ R
n
has the form
p(ϕ) = (2π)
−n/2
|K
ϕϕ
|
−1/2
exp
−
1
2
(ϕ, K
−1
ϕϕ
ϕ)
.
We shall remember certain properties of the Gaussian integrals.
1. The Gaussian integral it is possible to represent in the following form:
Z
exp
−
1
2
(ϕ, Hϕ)
Dϕ
=
Z
exp
−
1
2
(Uϕ, UHU
∗
Uϕ)
Dϕ
DU ϕ
DU ϕ
= |Det U |
Z
exp
−
1
2
(ψ, λψ)
Dψ
=
Z
exp
−
1
2
X
α
λ
α
ψ
2
α
Y
α
dψ
α
. (9.19)
Here U is an unitary matrix, its rows are eigenvectors of the matrix H : Hϕ
α
=
λ
α
ϕ
α
, Hϕ
β
= λ
β
ϕ
β
(ϕ
α
ϕ
β
= δ
αβ
);
U
∗
U = I, DetU = 1, UHU
∗
= Λ;
Λ is a diagonal matrix; ψ = Uϕ.
We shall remember the single-valued integral (it will be necessary for the integral
(9.19) calculation):
I =
Z
exp
−
1
2
λx
2
dx.
We note that
I
2
=
Z
exp
−
λ
2
(x
2
+ y
2
)
dxdy =
∞
Z
0
exp
−
λ
2
r
2
2πrdr
=
2π
λ
0
Z
−∞
e
ρ
dρ =
2π
λ
, I =
r
2π
λ
.
From here
Z
exp
−
1
2
(ϕ, Hϕ)
dϕ = (2π)
n/2
(DetH)
−1/2
,
and the normalization factor A :
R
A exp
−
1
2
(ϕ, Hϕ)
dϕ = 1 is equal
A = (2π)
−n/2
(DetH)
1/2
.
276 STATISTICAL METHODS OF GEOPHYSICAL DATA PROCESSING
2. We shall find the relation between the operator H and the correlator of the
function ϕ:
Eϕϕ
∗
=
R
ϕϕ
∗
exp[−(ϕ, Hϕ)/2]dϕ
R
exp[−(ϕ, Hϕ)/2]dϕ
=
R
ψψ
∗
exp[−(ψ, Σψ)/2]Dψ
R
exp[−(ψ, Σψ)/2]dψ
.
We write the matrix element K
γδ
of the correlator K:
K
γδ
=
R
ψ
γ
ψ
δ
exp[−(1/2)
P
αβ
Σ
αβ
ψ
α
ψ
β
]
Q
α
dψ
α
(2π)
−n/2
(DetΣ)
−1/2
= −2(2π)
−n/2
(DetΣ)
1/2
∂
∂Σ
γδ
×
Z
exp
−
1
2
X
α,β
Σ
αβ
ψ
α
ψ
β
Y
α
dψ
α
.
Taking into account that
∂
∂Σ
γδ
Z
exp[−(ψ, Σψ)/2]Dψ
=
∂
∂Σ
γδ
[(2π)
n/2
(DetΣ)
−1/2
] = (2π)
n/2
−
1
2
(DetΣ)
−3/2
¯
Σ
γδ
,
where
¯
Σ
γδ
is an algebraic cofactor of the element Σ
γδ
of the matrix Σ, we obtain
K
γδ
=
¯
Σ
γδ
|DetΣ|
= [Σ
−1
]
γδ
,
i.e. K
ϕϕ
= H
−1
.
The obtained relation makes possible to write the Gaussian density using the
correlator K
ϕϕ
, in the form
p(ϕ) =
exp[−(ϕ, K
−1
ϕϕ
ϕ)/2]
R
exp[−(ϕ, K
−1
ϕϕ
ϕ)/2]dϕ
. (9.20)
The distribution is called a nondegenerate one, if K
ϕϕ
> 0.
We calculate the Shannon’s functional for the Gaussian distribution
−(p, ln p) = −
Z
A exp[−(ϕ, K
−1
ϕϕ
ϕ)/2]
ln A −
1
2
(ϕ, K
−1
ϕϕ
ϕ)
dϕ
= −ln A +
1
2
A
Z
(ϕ, K
−1
ϕ) exp[−(ϕ, K
−1
ϕ)/2]dϕ.
The second term, using change ψ = K
−1/2
ϕ, it is possible to represent in the form
A
Z
(ϕ, K
−1
ϕ) exp[−(ϕ, K
−1
ϕ)/2]dϕ
= A
Z
(ψ, ψ) exp[−(ψ, ψ)/2](DetK)
1/2
dψ
= (DetK)
−1/2
(DetK)
1/2
f(n) = f(n).
Elements of applied functional analysis 277
Here the function f (n) depends only on the space dimension. Finally, we write:
H = −(p, ln p) =
1
2
ln DetK + f(n). (9.21)
If the density function p(ϕ) is given in the form (9.20), then Eϕ = 0 due to the
p(ϕ)(p(ϕ) = p(−ϕ)) evenness. For the density function
p(ϕ) = A exp
−
1
2
(ϕ − ϕ
0
, K
−1
ϕϕ
(ϕ − ϕ
0
))
,
using change of the variables ψ = ϕ − ϕ
0
, we obtain Eϕ = ϕ
0
.
We remember the probability characteristics of the description of the system of
random functions ϕ and ψ. Let the joint density function ϕ and ψ is p(ϕ, ψ). The
projection of this density on the space Φ is called a marginal probability density :
p(ψ) =
Z
p(ϕ, ψ)dϕ.
Respectively the projection of p(ϕ, ψ) on the space ψ gives the marginal density
p(ϕ) =
Z
p(ϕ, ψ)dψ.
The normalized cross-sections of the density p(ϕ, ψ) determine the conditional den-
sity — the probability density ϕ by fixed ψ:
p(ϕ|ψ) =
p(ϕ, ψ)
p(ψ)
,
the probability density ψ by fixed ϕ:
p(ψ|ϕ) =
p(ϕ, ψ)
p(ϕ)
.
The random fields ϕ and ψ are called independent, if the following conditions
are fulfilled
p(ϕ|ψ) = p(ϕ), p(ψ|ϕ) = p(ψ).
As an example we consider the model u = Lϕ + ε, where ε and ϕ are described by
the Gaussian distribution ϕ ∈ N(0, K
ϕ
), i.e.
p(ϕ) ∼ A exp
−
1
2
(ϕ, K
−1
ϕ
ϕ)
.
This distribution is called a priori distribution with respect to the desired field ϕ
and ε ∈ N(0, K
ε
), i.e.
p(ε) ∼ A exp
−
1
2
(ε, K
−1
ε
ε)
.
We assume that ϕ and ε are statistically independent: p(ϕ, ε) = p(ϕ)p(ε). The
distributions p(ϕ, ε) and p(u) are Gaussian distributions, since
u = ||L
.
.
.I||
ϕ
ε
.
278 STATISTICAL METHODS OF GEOPHYSICAL DATA PROCESSING
The joint probability density p(ϕ, u) is determined by the correlator
K
u
ϕ
= E
u
ϕ
||u
∗
ϕ
∗
||
=
K
uu
K
uϕ
K
ϕu
K
ϕϕ
=
L
∗
K
ϕ
L + K
ε
LK
ϕ
K
ϕ
L
∗
K
ϕ
.
It is easy to check, that the inverse operator is
K
−1
u
ϕ
it is possible to represent in the form
K
−1
u
ϕ
=
K
−1
ε
−K
−1
ε
L
−L
∗
K
−1
ε
K
−1
δϕ
,
K
δϕ
= K
ϕ
− K
ϕ
L
∗
(LK
ϕ
L
∗
+ K
ε
)
−1
LK
ϕ
.
If the inverse operator K
−1
ϕ
exists, then the correlator K
δϕ
can be represented
in the form
K
δϕ
= (L
∗
K
−1
ε
L + K
−1
ϕ
)
−1
,
using the identity
(A + BDB
∗
)
−1
= A
−1
− A
−1
B(B
∗
A
−1
B + D
−1
)B
∗
A
−1
.
We write the conditional probability density ϕ at prescribed value of u = Lϕ+ε,
which is called a posteriori density:
p(ϕ|u) =
p(u|ϕ)
p(u)
∼ exp
−
1
2
(ϕ − ˆϕ, K
−1
δϕ
(ϕ − ˆϕ))
, (9.22)
where
ˆϕ =
Z
ϕp(ϕ|u)dϕ
∆
= E
u
ϕ = K
ϕ
L
∗
(LK
ϕ
L
∗
+ K
ε
)
−1
u; (9.23)
E
u
ϕ — conditional average of ϕ.
We represent the conditional probability density u at prescribed value of ϕ:
p(u|ϕ) =
p(u, ϕ)
p(ϕ)
∼ exp
−
1
2
(u − ˆu, K
−1
ε
(u − ˆu))
, (9.24)
where ˆu = E
ϕ
u = Lϕ.
It is easily to see that the conditional probability density p(u|ϕ) coincides with
the distribution p
ε
independently of the type of the joint distribution for the model
u = Lϕ + ε i.e.:
p(u|ϕ) = p
ε
(u − Lϕ).
We clarify, how it was obtained the expressions (9.22) and (9.24); for that we
write the quadratic form standing at the exponent of the joint distribution p(u, ϕ):
||u ϕ||K
−1
u
ϕ
= (u, K
−1
ε
u) − (u, K
−1
ε
Lϕ) − (ϕ, L
∗
K
−1
ε
u) + (ϕ, K
−1
δϕ
ϕ).
Elements of applied functional analysis 279
At prescribed value of u it is possible to extract the total square for the function ϕ:
(ϕ − K
δϕ
L
∗
K
−1
ε
u, K
−1
δϕ
(ϕ − K
δϕ
L
∗
K
−1
ε
u)),
and for the fixed value of ϕ the total square for u is written as
(u − K
ε
K
−1
ε
Lϕ, K
−1
ε
(u − K
ε
K
−1
ε
Lϕ)).
It should be noted that the operator K
δϕ
represents a difference of the two
positive operators: K
ϕ
> 0 and K
ϕ
L
∗
(LK
ϕ
L
∗
+ +K
ε
)
−1
LK
ϕ
> 0, and in this case
K
δϕ
> 0.
We consider statistical criteria for searching of the field ϕ estimate for the model
u = Lϕ+ε, using obtained expression for the conditional probability densities p(ϕ|u)
and p(u|ϕ).
1. Let it is known only a distribution of the random component (noise) ε, i.e.
it is given the density p(ε). This distribution can be obtained a priori by means
of the statistical analysis of a measurement error of the experiment. Then it is
reasonable as a solution to take a such function ˆϕ, that the difference u − L ˆϕ is
maximum “similar” to the noise ε, i.e. the function u −L ˆϕ = ˆε should be such that
the probability p
ˆε
would be maximal:
ˆϕ = arg sup p
ε
(u − Lϕ) = arg sup p(u|ϕ) = arg sup ln p
ε
(u − Lϕ). (9.25)
Such approach of the estimation is called the maximum likelihood method (Rao,
1972), in this case the function ln p(u|ϕ) is called the likelihood function. The
monotone nondecreasing function (logarithm) is chosen in connection with that
many empirical distributions are exponential, in particular this concerns to the
Gaussian distribution, which in accordance with the central limit theorem (Rao,
1972) is the most prevailing approximation of the distributions. The extremum
problem (9.25) solution can be obtained by one of the numerical methods. Here
we extract the Gaussian distribution. Since the likelihood function is a quadratic
functional in this case, and it is possible to write Euler’s equation in the explicit
form
ˆϕ = arg inf(u −Lϕ, K
−1
ε
(u − Lϕ)),
(L
∗
K
−1
ε
L)ϕ = L
∗
K
−1
ε
u . (9.26)
The analysis of the equation (9.26) shows that the maximum likelihood method is
identical to the method of least squares in the case of the Gaussian distribution and
has the same disadvantage: the solution instability since the operator L
∗
K
−1
ε
L is
compact operator. In the case of the Laplace distribution, the maximum likelihood
method leads to the extremum problem solution in the norm L
1
:
ˆϕ = arg inf ||u − Lϕ||
L
1
,
that is equivalent to the method of the least modulus:
ˆϕ = arg inf |u − Lϕ|.
280 STATISTICAL METHODS OF GEOPHYSICAL DATA PROCESSING
2. We can obtain the regularized solution ˜ϕ only in the case if we have a priori
information about the solution. Let along with distribution p(ε) it is given a priori
distribution p(ϕ). It seems advisable to choose as a solution that it provides the
validity of two conditions simultaneously: in the first place the function ˜ϕ should
be such that the difference u − L ˜ϕ would be as much as possible similar to the
noise ε (by analogy with the condition of the maximum likelihood method); in
the second place the function ˜ϕ should correspond as well as possible to a priori
representation of the distribution p(ϕ). Taking into account the suggestion about
the statistical independence ϕ and ε, the validity of these conditions is reduced to
the requirements:
˜ϕ = arg sup p
ε
(u − Lϕ)p(ϕ) = arg sup p(u|ϕ)p(ϕ). (9.27)
Accordingly to the Bayes formula
p(ϕ|u) =
p(ϕ) p(u|ϕ)
p(u)
,
it is possible to represent the estimate ˜ϕ (9.27) as
˜ϕ = arg sup p(ϕ|u). (9.28)
The obtained solution (9.28) is called an estimate of the maximum a posteriori
probability.
The Bayes strategy determines the solution ϕ in the following way:
˜ϕ = E
u
ϕ =
Z
ϕp(ϕ|u)dϕ, (9.29)
which is identical to the estimate ˜ϕ (9.28) in the case of the symmetric a posteriori
distribution p(ϕ|u).
In the case of the normal distributions p(ϕ), p(ε) it is possible to write the
explicit form of the regularized solution (Turchin et al., 1971; Fedotov, 1990):
˜ϕ = arg inf
h
||u − Lϕ||
2
K
−1
ε
+ ||ϕ||
2
K
−1
ϕ
i
= K
ϕ
L
∗
(LK
ϕ
L
∗
+ K
ε
)
−1
u. (9.30)
The analysis of the formula (9.30) shows, that by using a priori information
about the fields ϕ, the extremum problem contains the stabilization functional
||ϕ||
2
K
−1
ϕ
. In this case the correlator can be determined to within a multiplier and
then the extremum problem (9.30) has the form coinciding with the general regu-
larization scheme (9.16):
ˆϕ = arg inf[||u − Lϕ||
2
K
−1
ϕ
+ α||ϕ − ϕ
0
||
2
K
−1
ϕ
]. (9.31)
The solution ˜ϕ (see the formula (9.30)) is called the solution on the statistical regu-
larization method. How it is seen from the equality (9.23), the solution, obtained by
using the Bayes strategy, is interpreted as a conditional mean value. The statistical
approach gives a possibility together with the solution to obtain the distribution of
the solution errors:
E
u
((ϕ − ˜ϕ)(ϕ − ˜ϕ)
∗
) = K
δϕ
,
K
δϕ
= K
ϕ
− K
ϕ
L
∗
(LK
ϕ
L
∗
+ K
ε
)
−1
LK
ϕ
.
Elements of applied functional analysis 281
We consider a particular case of the correlators K
ϕ
and K
ε
representation:
K
ϕ
= σ
2
ϕ
I, K
ε
= σ
2
ε
I.
In this case the solution ˜ϕ (9.31) is found from the extremum problem
˜ϕ = arg inf[||u − Lϕ||
2
+ α
0
||ϕ||
2
],
where the coefficient α
0
= σ
2
ε
/σ
2
ϕ
is a regularization parameter, that has in this
case the obvious physical sense: in the first place α
0
−→ 0 in the case of σ
2
ϕ
→ ∞,
i.e. the convergence of the regularization parameter to zero is connected identically
with the removal of a priori restrictions; in the second place, the same degree of
the stability (regularization) of the extremum problem is obtained by the relation
σ
2
ϕ
= 0(σ
2
ε
). Taking into account these results it is justified physically a degree
of the convergence of the regularization parameter to zero in the methods of the
functional regularization: if
J(ϕ) = ||Lϕ − u||
2
L
2
+ α||ϕ||
2
H
and if ||Lϕ − u||
L
2
= δ, then α = O(δ
2
).
We turn to the maximum likelihood method again. At the point of the solution
ˆϕ, the first derivative of the functional becomes zero:
δ
δϕ
[ln p(u|ϕ)] = 0,
and a measure of the measurement sensitivity to variations of δϕ serves the second
derivative incoming to the quadratic form:
−
1
2
δϕ,
δ
2
δϕ
2
ln p(u|ϕ)
δϕ
.
Conducting an average on all set of the possible values u, we obtain the Fisher’s
information operator (Stratonovich, 1975):
F = −E
δ
2
δϕ
2
ln p(u|ϕ)
.
For the Gaussian distribution p and for our problem u = Lϕ + ε, we will have
the following expression for the Fisher’s operator:
F = E
δ
2
δϕ
2
(Lϕ − u, K
−1
ε
(Lϕ − u)) = L
∗
K
−1
ε
L.
We note that the Fisher’s operator determines the correspondence between distances
in the decision space ϕ and in the measurement space:
(δϕ, F δϕ) = (Lδϕ, K
−1
ε
Lδϕ) = (δu, K
−1
ε
δu),
i.e. it characterizes the discriminant metric in the decision space
||ϕ
1
− ϕ
2
||
2
F
= ||u
1
− u
2
||
2
K
−1
ε
.
282 STATISTICAL METHODS OF GEOPHYSICAL DATA PROCESSING
The behavior of the likelihood functional depends on the Fisher’s operator at
the extremum point. If the presentation of the operator F in the eigenbasis has the
form
F =
X
α
λ
α
ψ
α
ψ
∗
α
,
where (ψ
α
, ψ
β
) = δ
αβ
, λ = λ
1
≥ λ
2
≥ . . ., then directions in the functional space,
defined ψ
α
at big values of α, are such directions along that the likelihood functional
is closed to the stationary, i.e. the functions ϕ
1
and ϕ
2
are practically indistinguish-
able:
||ϕ
1
− ϕ
2
||
2
F
≈ 0,
if
α
0
X
α=1
|(ϕ
1
− ϕ
2
, ψ
α
)|
2
= ||ϕ
1
− ϕ
2
||
2
,
where the coefficient α
0
is the threshold value, at which the ratio λ
α0
/λ
max
is enough
small.
The proper basis of the Fisher’s operator is called the Korhunen–Loeve basis.
The normal distribution, which is written in this basis, decomposes on the products
of the probability densities (projects on the basis vectors):
p(ϕ) ∼ exp
−
1
2
(ϕ, F ϕ)
= exp
−
1
2
X
α
λ
α
|ϕ
α
|
2
=
Y
α
exp
−
1
2
λ
α
|ϕ
α
|
2
∼
Y
α
p(ϕ
α
),
where ϕ
α
= (ψ
α
, ϕ). Writing the Fisher’s operator decomposition in the form
F =
α
0
X
α=1
λ
α
ψ
α
ψ
∗
α
+
X
α=α
0
+1
λ
α
ψ
α
ψ
∗
α
,
the probability density p(ϕ) can be represented approximately as
p(ϕ) ≈ ˜p(ϕ) ∼
α
0
Y
α=1
exp
−
1
2
λ
α
|ϕ
α
|
2
,
i.e. the statistical assembly is described by the ort projection, corresponding to the
big eigenvalues λ
α
(α = 1 ÷α
0
) of the Fisher’s operator, completely enough in the
probability sense
We give additional interpretation of the statistical regularization and measure-
ment integration using the Fisher’s operator.
Let along with the model of indirect measurement of the field
u = Lϕ + ε, ε ∈ N (0, K
ε
),