An unknown probability law p(x) can be efficiently estimated using a smoothness enforcing principle I=∫dxp'(x)2/p(x) = minimum, called minimum Fisher information (MFI). An unknown diffraction pattern is an unknown probability law (on photon position x). Consider a best estimate of the centroid position in the pattern from observation of a single-photon position. It is known that the mean-square error e2 in such an estimate must exceed the Cramer-Rao bound 1II,I above. Therefore, if we demand that e2 be a maximum, we have constructed a physical rationale for the use of MFI for estimating p(x). To bring in the known refractive-index profile n(x) of the diffracting medium, demand as a constraint that the mean-square spatial phase (2π/λ)2∫dxp2(x)p(x)=ϕ2 be generally nonzero. Constructing an overall principle I — ϕ 2 = minimum, a variational problem in p(x) results. The Euler-Lagrange solution is the Helmholtz wave equation on a real amplitude function q(x), where p(x) = q(x) 2 . Thus a diffraction pattern is seen to be nature’s way of maximally smearing out spatially the probability law on position for each photon forming the pattern.
An unknown probability law p(x) can be efficiently estimated using a smoothness enforcing principle I=∫dxp'(x)2/p(x) = minimum, called minimum Fisher information (MFI). An unknown diffraction pattern is an unknown probability law (on photon position x). Consider a best estimate of the centroid position in the pattern from observation of a single-photon position. It is known that the mean-square error e2 in such an estimate must exceed the Cramer-Rao bound 1II,I above. Therefore, if we demand that e2 be a maximum, we have constructed a physical rationale for the use of MFI for estimating p(x). To bring in the known refractive-index profile n(x) of the diffracting medium, demand as a constraint that the mean-square spatial phase (2π/λ)2∫dxp2(x)p(x)=ϕ2 be generally nonzero. Constructing an overall principle I — ϕ 2 = minimum, a variational problem in p(x) results. The Euler-Lagrange solution is the Helmholtz wave equation on a real amplitude function q(x), where p(x) = q(x) 2 . Thus a diffraction pattern is seen to be nature’s way of maximally smearing out spatially the probability law on position for each photon forming the pattern.
It is shown that the Helmholtz wave equation follows from a new uncertainty principle: Given, as data, the position of a photon in an unknown diffraction pattern, the estimated position of the centroid of the pattern will suffer minimum precision. This implies a maximally spread out diffraction pattern, obeying a principle of minimum Fisher information. The minimum is constrained by knowledge of the refractive-index function n(x, y, z) of the medium through a requirement that the mean-square spatial phase gradient across the medium should be generally nonzero. Operationally the principle works directly with intensities and not complex amplitudes. As a practical matter the numerical use of the intensity-based principle might permit a widening of the known scope of solutions to diffraction problems.
Optical signals may be regarded as probability laws (on photon events). The problem of optical signal recovery may therefore be viewed as a problem in estimating a probability law p(x). Maximum entropy has, in the past, been used for this purpose. Here, we discuss instead the use of minimum Fisher information I = ∫ d x [ p ′ ( x ) ] 2 / p ( x ) . Consider a physical system consisting of a random aggregate of particles or pho tons. Consider the experiment of measuring one particle's coordinate y, and from this, best estimating the mean coordinate θ of the ensemble, where y = θ + x. As time passes, the particles randomly become more spread out, so that the error in the estimate should increase. But the error goes inversely as I. Therefore, I should statistically decrease with time. With no constraint on the system, as t → ∞, I → 0, defining a maximally random law p(x) = constant. However, in the presence of a physical constraint, I should approach a finite value, obeying I = minimum. When the constraint is linear in the mean kinetic energy of the system, the solution p(x) obeys the stationary differential equation for the system. In this way, the Schrodinger (energy) wave equation, Helmholtz wave equation, diffusion equation, and Maxwell-Boltzmann law may be derived from one classical principle of disorder.
A wide class of physical problems requires the estimation of probability laws. Diffraction patterns and quantum mechanical probability laws on position are examples. Minimum Fisher information is one approach to estimating such laws; maximum entropy is another. In this paper, we show that the minimum Fisher information approach may be derived from a prior principle of maximum Cramer-Rao bound (MCRB). The MCRB-Fisher approach is then applied to some fundamental physical problems, including diffraction theory and quantum mechanics. It is found to give the correct physical solutions, that is the Helmholtz and Schrödinger wave equations respectively. By comparison, the maximum entropy approach gives incorrect solutions to these problems, except in the special (thermodynamic) case of a harmonic oscillator potential.
The principle of minimum Fisher information (MFI) provides a basis for deriving most of the laws of physics.1 The Fisher I is defined as (mean-squares error)-1 in the efficient determination of the classical position θ of a particle. For a one-component pdf p(x) = q2(x), the Fisher information is a one-term functional I = 4 ʃdx[q’(x)]2. Thus, I decreases as q(x) becomes smoother, or as disorder increases, consistent with the second law of thermodynamics. In the above, p(x) was specifically assumed to have a single component q2(x). Then MFI gives a real-only wave equation. Here we consider multicomponent cases p(x) = q12(x) + … + qN2(x), where the qn(x) are independent degrees of freedom. We seek the Fisher I for the case N = 2 of nonrelativistic quantum mechanics or of scalar optics. A benchmark statistical situation arises when the N components of qn(x) do not overlap. Then their informations add, and the resulting mean-squares error value is efficient, i.e. an absolute minimum.
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.
Consider an isolated statistical system specified by a coordinate x and its probability density p(x). A functional of p(x) called ``Fisher information'' can be used to measure the degree of disorder of the system due to the spread in p(x). Fisher information may be minimized, subject to a physical constraint, to attain a temporal equilibrium solution p(x). When the constraint is linear in the mean kinetic energy of the system, the equilibrium solution p(x) often obeys the correct differential equation for the system. In this way, the Schr\"odinger (energy) wave equation, Klein-Gordon equation, Helmholtz wave equation, diffusion equation, Boltzmann law, and Maxwell-Boltzmann law may be derived from one classical principle of disorder. The convergence rate for Fisher information is about that for alternative use of maximum entropy (in problems where both have the same equilibrium solution). This suggests that Fisher information defines an arrow of time. The arrow points in the direction of decreasing accuracy for the determination of the mean, or ideal, value of a parameter.