<s>
Information	B-General_Concept
field	I-General_Concept
theory	I-General_Concept
(	O
IFT	O
)	O
is	O
a	O
Bayesian	O
statistical	O
field	O
theory	O
relating	O
to	O
signal	B-Algorithm
reconstruction	I-Algorithm
,	O
cosmography	O
,	O
and	O
other	O
related	O
areas	O
.	O
</s>
<s>
For	O
example	O
,	O
the	O
posterior	O
expectation	O
value	O
of	O
a	O
field	O
generated	O
by	O
a	O
known	O
Gaussian	B-General_Concept
process	I-General_Concept
and	O
measured	O
by	O
a	O
linear	O
device	O
with	O
known	O
Gaussian	O
noise	O
statistics	O
is	O
given	O
by	O
a	O
generalized	B-Algorithm
Wiener	I-Algorithm
filter	I-Algorithm
applied	O
to	O
the	O
measured	O
data	O
.	O
</s>
<s>
To	O
handle	O
this	O
,	O
an	O
information	O
theory	O
for	O
fields	O
is	O
needed	O
,	O
and	O
that	O
is	O
what	O
information	B-General_Concept
field	I-General_Concept
theory	I-General_Concept
is	O
.	O
</s>
<s>
The	O
simplest	O
prior	O
for	O
a	O
field	O
is	O
that	O
of	O
a	O
zero	O
mean	O
Gaussian	O
probability	O
distributionThe	O
determinant	O
in	O
the	O
denominator	O
might	O
be	O
ill-defined	O
in	O
the	O
continuum	B-Algorithm
limit	I-Algorithm
,	O
however	O
,	O
all	O
what	O
is	O
necessary	O
for	O
IFT	O
to	O
be	O
consistent	O
is	O
that	O
this	O
determinant	O
can	O
be	O
estimated	O
for	O
any	O
finite	O
resolution	O
field	O
representation	O
with	O
and	O
that	O
this	O
permits	O
the	O
calculation	O
of	O
convergent	O
expectation	O
values	O
.	O
</s>
<s>
The	O
posterior	O
meanis	O
also	O
known	O
as	O
the	O
generalized	B-Algorithm
Wiener	I-Algorithm
filter	I-Algorithm
solution	O
and	O
the	O
uncertainty	O
covariance	O
as	O
the	O
Wiener	O
variance	O
.	O
</s>
<s>
The	O
classical	O
field	O
minimizes	O
the	O
information	O
Hamiltonian	O
,	O
and	O
therefore	O
maximizes	O
the	O
posterior:The	O
classical	O
field	O
is	O
therefore	O
the	O
maximum	B-General_Concept
a	I-General_Concept
posteriori	I-General_Concept
estimator	I-General_Concept
of	O
the	O
field	O
inference	O
problem	O
.	O
</s>
<s>
Minimizing	O
this	O
with	O
respect	O
to	O
,	O
in	O
order	O
to	O
get	O
its	O
maximum	B-General_Concept
a	I-General_Concept
posteriori	I-General_Concept
power	O
spectrum	O
estimator	O
,	O
yieldswhere	O
the	O
Wiener	O
filter	O
mean	O
and	O
the	O
spectral	O
band	O
projector	O
were	O
introduced	O
.	O
</s>
<s>
The	O
maximum	B-General_Concept
a	I-General_Concept
posteriori	I-General_Concept
estimator	I-General_Concept
for	O
the	O
power	O
spectrum	O
is	O
therefore	O
It	O
has	O
to	O
be	O
calculated	O
iteratively	O
,	O
as	O
and	O
depend	O
both	O
on	O
themselves	O
.	O
</s>
<s>
In	O
an	O
empirical	B-General_Concept
Bayes	I-General_Concept
approach	O
,	O
the	O
estimated	O
would	O
be	O
taken	O
as	O
given	O
.	O
</s>
<s>
As	O
a	O
consequence	O
,	O
the	O
posterior	O
mean	O
estimate	O
for	O
the	O
signal	O
field	O
is	O
the	O
corresponding	O
and	O
its	O
uncertainty	O
the	O
corresponding	O
in	O
the	O
empirical	B-General_Concept
Bayes	I-General_Concept
approximation	O
.	O
</s>
<s>
The	O
generalization	O
of	O
the	O
power	O
spectrum	O
estimation	O
formula	O
as	O
exhibits	O
a	O
perception	O
thresholds	O
for	O
,	O
meaning	O
that	O
the	O
data	O
variance	O
in	O
a	O
Fourier	O
band	O
has	O
to	O
exceed	O
the	O
expected	O
noise	O
level	O
by	O
a	O
certain	O
threshold	O
before	O
the	O
signal	B-Algorithm
reconstruction	I-Algorithm
becomes	O
non-zero	O
for	O
this	O
band	O
.	O
</s>
<s>
Whenever	O
the	O
data	O
variance	O
exceeds	O
this	O
threshold	O
slightly	O
,	O
the	O
signal	B-Algorithm
reconstruction	I-Algorithm
jumps	O
to	O
a	O
finite	O
excitation	O
level	O
,	O
similar	O
to	O
a	O
first	O
order	O
phase	O
transition	O
in	O
thermodynamic	O
systems	O
.	O
</s>
<s>
The	O
generalized	B-Algorithm
Wiener	I-Algorithm
filter	I-Algorithm
,	O
that	O
emerges	O
in	O
free	O
IFT	O
,	O
is	O
in	O
broad	O
usage	O
in	O
signal	O
processing	O
.	O
</s>
<s>
Minimizing	O
the	O
Gibbs	O
free	O
energy	O
provides	O
approximatively	O
the	O
posterior	O
mean	O
field	O
whereas	O
minimizing	O
the	O
information	O
Hamiltonian	O
provides	O
the	O
maximum	B-General_Concept
a	I-General_Concept
posteriori	I-General_Concept
field	O
.	O
</s>
<s>
The	O
term	O
information	B-General_Concept
field	I-General_Concept
theory	I-General_Concept
was	O
coined	O
by	O
Torsten	O
Enßlin	O
.	O
</s>
