We investigate the physical consequences of imposing symmetry requirements to the Cramer–Rao inequality, and investigate in particular translation, inversion, and rotation and show the above relationremains invariant under these transformations, which adds additional flavor to the adjective shift invariant attached to the concept of Fisher measure. In particular, if the inequality is saturated, it remains so under any transformation represented by a square matrix, representative of a (physical) unitary operator, as is the case in the all important instance of the classical harmonic oscillator.

The Maximum Entropy Method, using physical statistics, chooses the most probable estimate consistent with limited measurements. Thermodynamic analogies and the degree of confidence are discussed.

Seeking the unknown dynamics obeyed by a particle gives rise to the de Broglie wave representation, without the need for physical assumptions specific to quantum mechanics. The only required physical assumption is conservation of momentum μ. The particle, of mass m, moves through free space from an unknown source-plane position a to an unknown coordinate x in an aperture plane of unknown probability density pX(x), and then to an output plane of observed position y=a+z. There is no prior knowledge of the probability laws p(a,M),p(a) or p(M), with M the particle momentum at the source. It is desired to (i)optimally estimate a, in the sense of a maximum likelihood (ML) estimate. The estimate is further optimized, by minimizing its error through (ii) maximizing the Fisher information about a that is received at y. Forming the ML estimate requires (iii) estimation of the likelihood law pZ(z), which (iv) must obey positivity. The relation pZ(z)=|u(z)|*2≥0 satisfies this. The same u(z) conveniently defines the Fisher channel capacity, a concept central to the principle of Extreme physical information (EPI). Its output u(z) achieves aims (i)–(iv). The output is parametrized by a free parameter K. For a choice K=0, the result is u(z)=δ(z), indicating classical motion. Or, for a finite, empirical choice K=h¯(Planck’s constant), u(z) obeys the familiar de Broglie representation as the Fourier transform of the particle’s probability amplitude function P(μ) on momentum μ. For a definite momentum μ, u(z) becomes a sinusoid of wavelength λ= h¯/μ, the de Broglie result.

The peak brightness of the solar spectrum is in the green when plotted in wavelength units. It peaks in the near-infrared when plotted in frequency units. Therefore the oft-quoted notion that evolution led to an optimized eye whose sensitivity peaks where there is most available sunlight is misleading and erroneous. The confusion arises when density distribution functions like the spectral radiance are compared with ordinary functions like the sensitivity of the eye. Spectral radiance functions, excepting very narrow ones, can change peak positions greatly when transformed from wavelength to frequency units, but sensitivity functions do not. Expressing the spectral radiance in terms of photons per second, rather than power, also causes a change in the shape and peak of the distribution, even keeping the choice of bandwidth units fixed. The confusion arising from comparing simple functions to distribution functions occurs in many parts of the scientific and engineering literature aside from vision, and some examples are given. The eye does not appear to be optimized for detection of the available sunlight, including the surprisingly large amount of infrared radiation in the environment. The color sensitivity of the eye is discussed in terms of the spectral properties and the photo and chemical stability of available biological materials. It is likely that we are viewing the world with a souvenir of the human evolutionary voyage

A black hole (BH) communicates information about its event area to an observer. This defines an information channel. We show that if the area is optimally encoded, the probability law on its square root follows approximate exponential decay, and the resulting entropy is precisely the Bekenstein–Hawking result including the important factor of 1/4. This result arises as well as the solution to: (a) maximum entropy subject to a fixed message length, or (b) minimum Fisher information subject to a normalization condition. A verifiable prediction is a randomly enlarged event area, implying a randomly enlarged irreducible mass value. These effects should be observable as an enhanced number of BH’s with large mass, resulting in an increased occurrence of gravitational lensing, and an enhanced ability to entrap relatively distant stellar objects.

A coarse-grained Wigner distribution p(W)(x,mu) obeying positivity derives out of information-theoretic considerations. Let p(x,mu) be the unknown joint probability density function (PDF) on position and momentum fluctuations x, mu for a particle in a pure state psi(x). Suppose that the phase part Psi(x,z) of its Fourier transform T-F[p(x,mu)]equivalent to parallel to G(x,z)parallel to exp[i Psi(x,z)] is constructed as a hologram. (Such a hologram is often used in heterodyne interferometry.) Consider a particle randomly illuminating this phase hologram. Let its two position coordinates be measured. Require that the measurements contain an extreme amount of Fisher information about true position, through variation of the phase function Psi(x,z). The extremum solution gives an output PDF p(x,mu) that is the convolution of the Wigner p(W)(x,mu) with an instrument function defining uncertainty in either position x or momentum mu. The convolution arises naturally out of the approach, and is one dimensional, in comparison with the ad hoc two-dimensional convolutions usually proposed for coarse graining purposes. The output obeys positivity, as required of a PDF, if the one-dimensional instrument function is sufficiently wide. The result holds for a large class of systems: those whose amplitudes psi(x) are the same at their boundaries [examples: states psi(x) with positive parity; with periodic boundary conditions; free particle trapped in a box].

We show, starting from first principles, that thermodynamics' first law can be microscopically obtained for Fisher's information measure without need of invoking the adiabatic theorem. Further, it is proved that enforcing the Fisher-first law requirements in a process in which the probability distribution is infinitesimally varied is equivalent to minimizing Fisher's information measure subject to appropriate constraints. (c) 2006 Elsevier B.V. All rights reserved.