Fisher information

The Fisher information Ip in a single measurement of a physical system is given by... 

Frieden s theory is that any physical measurement induces a transformation of Fisher information J I connecting the phenomenon being measured to intrinsic data. What we call physics - i.e. our objective description of phenomenologically observed behavior - thus derives from the Extreme Physical Information (EPI) principle, which is a variational principle. EPI asserts that, if we define K = I — J as the net physical information, K is an extremum. If one accepts this EPI principle as the foundation, the status of a Lagrangian is immediately elevated from that of a largely ad-hoc construction that yields a desired differential equation to a measure of physical information density that has a definite prior significance. 

An analytical description of the photon-economy and additive noise could be carried out by the estimation of the Fisher-information matrix of the used estimators [34],... 

Furthermore, under symplectic transformations, it is relatively easy to show, using the Hessian formula for calculating the Fisher information matrix, that the measurement covariance matrix transforms as... 

Now we show that there is a surprising relation between Fisher s fundamental theorem of natural selection and other theory developed by Fisher, the likelihood theory in statistics and Fisher information [21], As far as we know, the present chapter is the first publication in the literature pointing out the connections between these two problems formulated and studied by Fisher. 

By interpreting the state vector u in y (x t) and the position vector x as a parameter, In y (x, f) can be interpreted as a likelihood function and / (x, t) as a vector of statistical scores as expected from likelihood theory, the average score is zero, Xu( > 0 = Pq. (4)]. Moreover, the covariance matrix of the relative rates of evolution plays the role of a Fisher information metric ... 

The Fisher information metric (23) or (25) is the ideal tool for solving inverse problems for reaction transport systems with incomplete knowledge of the parameters. Since the reaction-transport systems are described by local, partial differential equations, considering small space variations, the differential... 

The generalized Fisher theorems derived in this section are statements about the space variation of the vectors of the relative and absolute space-specific rates of growth. These vectors have a simple natural (biological, chemical, physical) interpretation They express the capacity of a species of type u to fill out space in genetic language, they are space-specific fitness functions. In addition, the covariance matrix of the vector of the relative space-specific rates of growth, gap, [Eq. (25)] is a Riemannian metric tensor that enters the expression of a Fisher information metric [Eqs. (24) and (26)]. These results may serve as a basis for solving inverse problems for reaction transport systems. 

B. Roy Frieden, Science from Fisher Information, 2nd ed., Cambridge University Press, Cambridge, 2004. 

Using the notation of experimental design, F represents the extended design matrix, where the elements of its k x I row-vectors, f, are known functions of x. The matrix (FT) is the Fisher information matrix and its inverse, (FT)-1, is the dispersion matrix of the regression coefficients. 

R.F. Nalewajski, P. de Silva, J. Mrozek, Kinetic-energy/Fisher-information indicators of chemical bonds, in A. Wang, T.A. Wesoxowski, (Eds.), Kinetic Energy Functional, World Scientific, Singapore, in press. 

Provided the estimator is unbiased, this expression indicates that the diagonal elements of the inverse information matrix, the Cramer-Rao bound (CRB), give the highest attainable variance of the unknown parameters. For zero-mean noise and a correct model r, no difference is expected between the values of the estimates and the true values. These minimal variances depend on the pdf of the signal, because the Fisher information matrix I is given by... 

Retout, S. Duffull, S.B. Mentre, E. Development and implementation of the population fisher information matrix... 

An analysis based upon the Fisher information matrix showed that the process model is highly sensitive to the Henry coefficients, mass transfer resistances and reaction rate. These parameters are, therefore, re-estimated online at every cycle. Figure 9.8 compares the concentration profiles collected in the recycling line with the simulated ones. The parameters were initialized with the values given in Table 9.1. 

For studies that appear to be exchangeable, the weighting is provided by the reciprocal of the estimation variance (which is the precision or Fisher information). 

S. Retout, F. Mentre, and R. Bruno, Fisher information matrix for non-linear mixed-effects models evaluation and application for optimal design of enoxaparin population pharmacokinetics. StatMed 21 2623-2639 (2002). 

PFIM-4 Population Fisher information matrix D-optimai design (4 sampies/subject) 0.5, 4, 50.5, and 60... 

S. Retout and E. Mentre, Eurther developments of the Fisher information matrix in nonlinear mixed effects models with evaluation in population pharmacokinetics. J Biop harm Stat 13 209-227 (2003). 

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