Fisher information measure

More recently, Fisher information has been studied as an intrinsic accuracy measure for concrete atomic models and densities [43, 44] and also for quantum mechanics central potentials [45]. Also, the concept of phase space Fisher information, where position and momentum variables are included, was analyzed for hydrogenlike atoms and the isotropic harmonic oscillator [46]. The net Fisher information measure is found to correlate well with the inverse of the ionization potential and dipole polarizability [44]. 

Interesting new results in the IT studies of the elementary reaction mechanisms have recently been obtained in the Granada group [92-94]. Both the global (Shannon) and local (Fisher) information measures have been used in these investigations. The course of two representative reactions has been examined of the radical abstraction of hydrogen (nvo-step mechanism),... 

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. 

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... 

In this work, we will also analyze, apart from C(LMC) and C(FS), the Cramer-Rao complexity C(CR), also as the product of a local and a global measure, keeping the first one as the Fisher information I, and replacing the Shannon entropy exponential by the variance V, giving rise to... 

Analyzing the main information-theoretic properties of many-electron systems has been a field widely studied by means of different procedures and quantities, in particular, for atomic and molecular systems in both position and momentum spaces. It is worthy to remark the pioneering works of Gadre et al. [62,63] where the Shannon entropy plays a fundamental role, as well as the more recent ones concerning electronic structural complexity [27, 64], the connection between information measures (e.g., disequilibrium, Fisher information) and experimentally accessible quantities such as the ionization potentials or the static dipole polarizabilities [44], interpretation of chemical phenomena from momentum Shannon entropy [65, 66], applications of the LMC complexity [36, 37] and the quantum similarity measure [47] to the study of neutral atoms, and their extension to the FS and CR complexities [52, 60] as well as to ionized systems [39, 54, 59,67]. 

Consider the Fisher information for locality [1,2], called intrinsic accuracy, which historically predates the Shannon entropy by about 25 years being proposed at about the same time that the final form of quantum mechanics was shaped. This local measure of the information content of the continuous (normalized) classical probability density p r) reads... 

The Fisher information, reminiscent of von Weizsacker s [70] inhomogeneity correction to electronic kinetic energy in the Thomas-Fermi theory, charactoizes the compactness of the probability density. For example, the Fisher information in normal distribution measures the inverse of its variance, called invariance, while the complementary Shannon entropy is proportional to the logarithm of variance, thus monotonically increasing with the spread of Gaussian distribution. Therefore, Shannon entropy and intrinsic accuracy describe complementary facets of the probability density the former reflects the distribution s ( spread ( disorder , a measure of uncertainty), while the latter measures its narrowness ( order ). 

Then, this second derivative can be interpreted as a measure of how quickly the probability distribution function (pdf) changes when we change the parameter in the proximity of the MLE estimation. The expectation of the square of this value gives us a more robust version of this measure if Fisher information is large, then the distribution quickly changes when we move from the MLE value of the parameter (i.e., the distribution with parameter 6 is different from the distributions with parameters close to 0) on the other hand, if Fisher information is small, this means that the distribution corresponding to 0 is Very similar to distributions with values of the parameter not so close to. In the former case, we should be able to robustly estimate 0 based on the data, whereas in the latter case the robustness of the estimation is worse. 

Information theory has been shown to provide a novel and attractive perspective on the entropic origins of the chemical bond. It also offers a complementary outlook on the transformation of the electronic information content in the elementary chemical reactions. In this short overview, we have first introduced the key IT concepts and techniques to be used in such a complementary analysis of electron distributions in molecular systems. They have been subsequently applied to explore the bonding pattern in typical molecules in terms of the information distribution, the bond localization/multiphcity, and its ionic/covalent composition. The use of the information densities as local probes of electronic distributions in molecules has been advocated and the importance of the nonadditive entropy/information measures in extracting subtle changes due to the bond formation has been stressed. The use of the CG density, of the nonadditive Fisher information (electronic kinetic energy) in the AO resolution, as an efficient localization probe of the direct chemical bonds has been validated. 

Fisher 1998 demonstrates that the more that is learned about the biomarker (half-life, time course in blood or urine, and development of PBPK model) and the exposed population (age, body weight, pharmacoge-netic traits, behavioral factors that affect exposure, and time between exposure and sample measurement), the more refined dose estimates can become. Without such information, a highly transient metabolite like TCE is not a reliable marker of exposure, unless exposure is nearly continuous and uniform. That may not be the case in the general population, so TCE in blood may not be a good biomarker for assessment of general-population exposure, although PBPK models are available to extrapolate from biomarker concentration to external dose in both animals and humans (Clewell et al. 2000). 

The characterization of the total mass thus obtained, performed on representative samples taken during the vial filling procedure, gave additional information on the properties of the material. The average moisture content, measured by Karl-Fisher titration, was lower than 3%. The particle size distribution of the whole mass, measured using the Sympatec analyzer with Helos measuring device, was to peak at 80-90 pm with an upper limit of 435 pm. 

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