Minimum Fisher Information Research Articles

Derivation of the equations of nonrelativistic quantum mechanics using the principle of minimum Fisher information

The many-particle time-dependent Schr\odinger equation is derived using the principle of minimum Fisher information. This application of information theory leads to a physically well motivated derivation of the Schr\odinger equation, which distinguishes between subjective and objective elements of the theory.

Incompressible turbulence and minimum Fisher information: Correlations between velocity and pressure

The principle of minimum Fisher information (MFI) and the theory of random Gaussian fields are used to work out the joint distribution function of the density and velocity in homogeneous, isotropic, stationary, nearly incompressible turbulence, in the case where the velocity and pressure are correlated. The appropriate Fisher variables seem to be the mass flux, the density, and the generalized heat function (enthalpy) or pressure head. It is shown that simple constraints on the minimization may be chosen to give a good fit to the pressure distribution function found in recent direct numerical simulations and experiments, where the PDF is exponential for negative p and roughly exp[−(p/p0)3/2]p−1/2 for positive p. In this case, the fit is an improvement on a past MFI calculation, in which the correlations between p and u were not accounted for. In addition, the form of the conditional average 〈p|u〉 as found from direct numerical simulations is taken into consideration. The theory of random Gaussian velocity fields predicts 〈p|u〉=〈p|0〉−βu2, where u2≡u⋅u and β⩽1/8 is a constant. In conjunction with this theory, MFI predicts a specific dependence of the conditional average 〈ux2|p〉 on p, where ux is a typical velocity component. The conditional PDF P(ux|p) is slightly non-Gaussian, but P(ux) is Gaussian. The relation 2〈u2δp〉2=〈u2〉[〈u2(δp)2〉−〈u2〉〈(δp)2〉] is predicted.

On the trimmed mean and minimax‐variance L‐estimation in Kolmogorov neighbourhoods

AbstractWe consider the properties of the trimmed mean, as regards minimax‐variance L‐estimation of a location parameter in a Kolmogorov neighbourhood K() of the normal distribution:We first review some results on the search for an L‐minimax estimator in this neighbourhood, i.e. a linear combination of order statistics whose maximum variance in Kt() is a minimum in the class of L‐estimators. The natural candidate – the L‐estimate which is efficient for that member of Kt,() with minimum Fisher information – is known not to be a saddlepoint solution to the minimax problem. We show here that it is not a solution at all. We do this by showing that a smaller maximum variance is attained by an appropriately trimmed mean. We argue that this trimmed mean, as well as being computationally simple – much simpler than the efficient L‐estimate referred to above, and simpler than the minimax M‐ and R‐estimators – is at least “nearly” minimax.

Minimum Fisher information of moment-constrained distributions with application to robust blind identification

In this paper, the minimization of the Fisher information measure under constraints on higher-order moments of the distribution is treated. It is well known that in the case of minimizing the moment-constrained Shannon information, i.e. maximizing the entropy, the solution, if exists, it is always of generalized exponential-type matching the given moments. Applying the same procedure to the Fisher information yields a set of densities which satisfy the Riccati differential equation. Although in the general case the solution can be found by numerical means only, it is shown that a closed form approximate solution can be easily obtained if we constrain the solutions to generalized exponential distributions. The analysis of this class is provided and limitations are identified. If applied to distributions with infinite support, the approximation holds for moment sequences of sub-Gaussian distributions only. In order to cover the class of super-Gaussian distributions, Gaussian mixtures are introduced as an alternative distribution class for which moment matching can be easily done. The developed theory is applied to the problem of robust accuracy improvement of a moment-based blind identification algorithm. The information about the moments is used to improve the criterion function of the inverse filtering algorithm. The new blind identification procedure delivering more accurate parameter estimates is verified on simulated examples.

Electron density profile reconstruction from multichannel microwave interferometer data at W7-AS

A multichannel microwave interferometer has been built for the W7-AS fusion experiment in order to study temporal phenomena in plasma density. For this reason, electron density profile reconstruction methods based on regularization functionals were studied. Simulations showed, that the minimum Fisher-information approach (adapted from tomography) gives better results than the maximum entropy approach. This is due to the fact, that in the minimum Fisher-information method the reconstructed distribution is inherently assumed to be continuous, whereas the maximum entropy method the distribution is assumed to be noncontinuous. The profiles reconstructed from multichannel interferometer data are in good agreement with the profiles measured by other diagnostic systems.

Modeling nearly incompressible turbulence with minimum Fisher information

The principle of minimum Fisher information (MFI) is used to work out the joint distribution function of the density and velocity in homogeneous, isotropic, stationary, nearly incompressible turbulence. We assume that the Mach number is very low. It is shown that simple constraints on the minimization may be chosen to give a good fit to the pressure distribution function found in recent direct numerical simulations and experiments, where the PDF is exponential for negative p and roughly exp[−(p/p0)3/2]p−1/2 for positive p. The appropriate constraints in the MFI problem are on the skewness of the PDF both by itself and weighted by the internal energy density of the fluid. MFI then predicts that in the limit of very low Mach number, the pressure and the fluid velocity u are independent of each other, as random variables; i.e., P(p,u)→P(p)P(u). Also, P(u) is a Gaussian.

ASYMPTOTIC MINIMAX PROPERTIES OF L-ESTIMATORS OF SCALE

Summary This paper asks whether or not the efficient L-estimator of scale corresponding to the least informative distribution in e-contamination and Kol-mogorov neighbourhoods of certain distributions possesses the saddlepoint property. This is of interest since the saddlepoint property implies the mini-max property, namely, that the supremum of the relative asymptotic variance of an L-estimator is minimized by the efficient estimator corresponding to that member of the distributional class with minimum Fisher information for scale. Our findings are negative in all cases investigated.

Fisher information as the basis for Maxwell's equations

Maxwell's equations of classical electrodynamics may be derived on the following statistical basis. Consider a gedanken experiment whereby the mean space-time coordinate for photons in an electromagnetic field is to be determined by observation of one photon's space-time coordinate. An efficient (i.e. optimum) estimate obeys a condition of minimum Fisher information, or minimum precision, according to the second law of thermodynamics. The Fisher information I is a simple functional of the probability law governing space-time coordinates of the “particles” of the field. This probability law is modeled as the source-free Poynting energy flow density, i.e., the ordinary local intensity in the optical sense, or, the square of the four-vector potential. When the Fisher information is extremized subject to an additive constraint term in the total interaction energy, Maxwell's equations result.

Fisher information and the complex nature of the Schr�dinger wave equation

We show that the minimum Fisher information (MFI) approach to estimating the probability law p(x) on particle position x, over the class of all two-component laws p(x), yields the complex Schrodinger wave equation. Complexity, in particular, traces from an “efficiency scenario” (demanded by MFI) where the two components of p(x) are so separated that their informations add.

Derivation of the vectorial wave equation from a variational point of view

The electromagnetic field in a linear, inhomogeneous, isotropic, and nonmagnetic dielectric is considered. It is shown that the vectorial wave equation for the electric field can apparently be derived on the basis of a principle of minimum divergence of the field. It is argued that this principle is preferable to the recently suggested method of obtaining the scalar wave equation by a principle of minimum Fisher information.

Asymptotic minimax properties of M-estimators of scale

We ask whether or not the saddlepoint property holds, for robust M-estimation of scale, in gross-errors and Kolmogorov neighbourhoods of certain distributions. This is of interest since the saddlepoint property implies the minimax property — that the supremum of the asymptotic variance of an M-estimator is minimized by the maximum likelihood estimator for that member of the distributional class with minimum Fisher information. Our findings are exclusively negative — the saddlepoint property fails in all cases investigated.

Fisher information as the basis for the Schrödinger wave equation

It is shown that the assumption that nature acts to make optimum estimates of position maximally in error leads to the Schrödinger (energy) wave equation (SWE). In this way, the SWE follows from a simple statement of uncertainty. The approach grows out of probability estimation theory, in particular the principle of minimum Fisher information (or maximum Cramer–Rao bound). The minimized Fisher information turns out to be proportional to the mean kinetic energy of the particle. This approach is an attractive supplement to conventional ways of introducing quantum mechanics to students, since it avoids the use of imperfect physical models (such as vibrating strings) or the immediate need for complex momentum operators, and follows from a plausible assumption as to how nature works.

Fisher information as the basis for diffraction optics

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.

Stochastic approaches to inversion problems in optics

Image restoration, image smoothing, and probability law estimation are important inverse problems in optics. Within the past 10 or so years, much progress has been made toward their resolution. These problems are often ill-posed mathematically. However, such concepts as maximum entropy, maximum likelihood, binay decision discrimination, median window filtering, MAP estimation, and minimum Fisher information have proven invaluable in regularising, or reducing, the ill-posed nature of the problems. In particular, those concepts which measure uncertainty or disorder have played a central role. Given noise-prone data, concepts that describe maximal disorder (maximum entropy, minimum Fisher information, minimal bina y discrimination) exert a smoothing influence on the solutions that drastically reduces noise propagation into the output. Given insufficient but noise-free data, as in the probability estimation problem, the principle of minimum Fisher information, in particular, creates smooth estimates which, of...

Applications to Optics and Wave Mechanics of the Criterion of Maximum Cramer-Rao Bound

Abstract 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 Schrodinger 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.