Geometrical interpretation of the population entropy maximum

Abstract

. 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.