Geometrical interpretation of the population entropy maximum

• Published in: Stochastic models Vol. 40; no. 3; pp. 569 - 582
• Main Authors: Lazov, Igor, Lazov, Petar
• Format: Journal Article
• Language: English
• Published: Philadelphia Taylor & Francis 02-07-2024; Taylor & Francis Ltd
• Subjects: 60G05; Death; direct and inverse population; Entropy; inner product and covariance; Markov processes; maximum uncertainty point; Parameters; Poisson distribution; population information and entropy; Population size and utilization; Random variables
• ISSN: 1532-6349; 1532-4214
• DOI: 10.1080/15326349.2023.2297959

We consider a population of finite size M, where the current number N of entities,
N
∈
{
0
,
1
,
2
,
...
,
M
}
, determines its states. We geometrically analyze the amounts of information i
N
and i
M−N
, carried by the random variables N and (M−N), respectively, and population entropy
S
=
E
(
i
N
)
=
E
(
i
(
M
−
N
)
)
. The population is modeled by a family of Birth-Death Processes of size M+1, indexed by the population utilization parameter ρ (i.e., birth/death ratio), which determines its macrostates. Considering the quantities:
x
=
(
N
−
E
(
N
)
)
&
y
=
(
i
N
(
ρ
)
−
S
(
ρ
)
)
and
M
x
=
(
(
M
−
N
)
−
E
(
M
−
N
)
)
&
M
y
=
(
i
M
−
N
(
ρ
)
−
S
(
ρ
)
)
as vectors, it is shown that the angles:
φ
=
∠
(
x
;
y
)
and
ψ
=
∠
(
M
x
;
M
y
)
are supplementary ones, that is
φ
(
ρ
)
+
ψ
(
ρ
)
=
π
,
ρ
>
0
. Expressions for their inner products
<
x
,
y
>
and
<
M
x
,
M
y
>
, being the covariances of N&i
N
(ρ) and
(
M
−
N
)
&
i
M
−
N
(
ρ
)
, respectively, which sum equals zero, with respect to the parameter ρ are also obtained. Further, what is especially important, it is revealed that
φ
=
ψ
=
π
/
2
, that is both inner products equal zero, at the point ρ
M
max
, which is the value of parameter ρ at which the entropy has maximum value. Finally, for an information linear population only, it is shown that N obeys uniform distribution at the point ρ
M, max
, that is that
S
(
ρ
M
,
max
)
=
ln
⁡
(
M
+
1
)
. These results are further illustrated on populations described by truncated geometrical, binomial, and truncated Poisson distribution.