Information Technology Reference
In-Depth Information
factors. This is the continuous counterpart of the partition invariance prop-
erty of discrete Shannon entropy (see e.g. [62]).
Rényi's quadratic (differential) entropy of a partitioned PDF is expressed
as
ln
i
f
i
(
x
)
dx
;
a
i
H
R
2
(
f
)=
−
(C.4)
D
i
that is,
H
R
2
(
f
) is not decomposable as Shannon's counterpart. Nevertheless,
as the minimization of
H
R
2
(
f
) is equivalent to the maximization of
V
R
2
=
exp(
H
R
2
), we may use the decomposition of the information potential
V
R
2
which is readily expressed as
V
R
2
(
f
)=
i
−
a
i
V
R
2
(
f
i
)
.
(C.5)
The variance of
f
can also be decomposed as
V
[
f
]=
i
a
i
V
[
f
i
]+
i
μ
)
2
,
a
i
(
μ
i
−
(C.6)
where
μ
(
μ
i
) is the expected value of
f
(
f
i
).
