Relative Fisher Information Research Articles

Why are information-theoretic descriptors powerful predictors of atomic and molecular polarizabilities.

We rationalize the excellent performance of information-theoretic descriptors for predicting atomic and molecular polarizabilities. It seems that descriptors which capture information about the change in valence-shell structure, especially the relative Fisher information measures, are particularly useful. Using this, we can rationalize why the G3 form of the relative Fisher information, which measures the deviation of effective nuclear charge between an atom-in-a-molecule and the reference pro-atom, is especially effective as a predictor of molecular polarizability. There are no methods used in this paper, which relies on mathematical derivation and analysis.

Probing correlations in two-mode bosonic fields via Gaussian noise channels

Gaussian noise channels arise naturally in many physical situations and play an important role in information transmission involving continuous-variable quantum systems. In this work we employ Gaussian noise channels to probe and characterize correlations in two-mode bosonic fields. Based on the fact that local channels often cause more decoherence for global states than for local states due to correlations, we introduce a notion of conditional coherence relative to a local channel, which is defined as the difference between the global and local decoherence caused by the channel. We discuss its connections with quantum discord and relative quantum Fisher information and propose a class of correlation quantifiers for two-mode bosonic states in terms of conditional coherence relative to local Gaussian noise channels. We prove that any nonproduct state exhibits such correlations and show that these correlation quantifiers are measures of total correlations, which may include both classical and quantum parts. We further illustrate these correlation quantifiers through several typical two-mode bosonic states and make a comparative study between these correlation quantifiers and other known ones such as entanglement and discord.

Some inequalities on Riemannian manifolds linking Entropy, Fisher information, Stein discrepancy and Wasserstein distance

For a complete connected Riemannian manifold M let V∈C2(M) be such that μ(dx)=e−V(x)vol(dx) is a probability measure on M. Taking μ as reference measure, we derive inequalities for probability measures on M linking relative entropy, Fisher information, Stein discrepancy and Wasserstein distance. These inequalities strengthen in particular the famous log-Sobolev and transportation-cost inequality and extend the so-called Entropy/Stein-discrepancy/Information (HSI) inequality established by Ledoux, Nourdin and Peccati (2015) for the standard Gaussian measure on Euclidean space to the setting of Riemannian manifolds.

A trajectorial approach to relative entropy dissipation of McKean–Vlasov diffusions: Gradient flows and HWBI inequalities

We formulate a trajectorial version of the relative entropy dissipation identity for McKean–Vlasov diffusions, extending recent results which apply to non-interacting diffusions. Our stochastic analysis approach is based on time-reversal of diffusions and Lions’ differential calculus over Wasserstein space. It allows us to compute explicitly the rate of relative entropy dissipation along every trajectory of the underlying diffusion via the semimartingale decomposition of the corresponding relative entropy process. As a first application, we obtain a new interpretation of the gradient flow structure for the granular media equation, generalizing a formulation developed recently for the linear Fokker–Planck equation. Secondly, we show how the trajectorial approach leads to a new derivation of the HWBI inequality, which relates relative entropy (H), Wasserstein distance (W), barycenter (B) and Fisher information (I).

Cumulative past Fisher information measure and its extensions

In this work, we define the cumulative past Fisher (CPF) information and the relative cumulative past Fisher (RCRF) information measures for parameter as well as for the distribution function of the underlying random variables. We show that these cumulative past Fisher information measures can be expressed in terms of the reversed hazard rate function. We also define three extensions of the CPF information measure. Further, we study these cumulative information measures and their Bayes versions for some well-known models used in reliability, economics and survival analysis. The associated results reveal some interesting connections between the proposed Fisher type information measures with some well-known information divergences and reliability measures.

Quantitative approximate independence for continuous mean field Gibbs measures

Many Gibbs measures with mean field interactions are known to be chaotic, in the sense that any collection of k particles in the n-particle system are asymptotically independent, as n→∞ with k fixed or perhaps k=o(n). This paper quantifies this notion for a class of continuous Gibbs measures on Euclidean space with pairwise interactions, with main examples being systems governed by convex interactions and uniformly convex confinement potentials. The distance between the marginal law of k particles and its limiting product measure is shown to be O((k∕n)c∧2), with c proportional to the squared temperature. In the high temperature case, this improves upon prior results based on subadditivity of entropy, which yield O(k∕n) at best. The bound O((k∕n)2) cannot be improved, as a Gaussian example demonstrates. The results are non-asymptotic, and distance is quantified via relative Fisher information, relative entropy, or the squared quadratic Wasserstein metric. The method relies on an a priori functional inequality for the limiting measure, used to derive an estimate for the k-particle distance in terms of the (k+1)-particle distance.

Skewed Jensen—Fisher Divergence and Its Bounds

A non-uniform (skewed) mixture of probability density functions occurs in various disciplines. One needs a measure of similarity to the respective constituents and its bounds. We introduce a skewed Jensen–Fisher divergence based on relative Fisher information, and provide some bounds in terms of the skewed Jensen–Shannon divergence and of the variational distance. The defined measure coincides with the definition from the skewed Jensen–Shannon divergence via the de Bruijn identity. Our results follow from applying the logarithmic Sobolev inequality and Poincaré inequality.

Cumulative Residual and Relative Cumulative Residual Fisher Information and Their Properties

In this work, we propose cumulative residual Fisher information and relative cumulative residual Fisher information measures and establish some of their properties. We first show that these cumulative Fisher measures can be expressed based on the hazard function. We then define extended versions of cumulative residual entropy and cumulative residual Fisher information measures based on the Jensen inequality. We also discuss connections between these information measures based on a new version of de Bruijn's identity for survival functions.

Variational principle for stochastic mechanics based on information measures

Stochastic mechanics is regarded as a physical theory to explain quantum mechanics with classical terms such that some of the quantum mechanics paradoxes can be avoided. Here, we propose a new variational principle to uncover more insights on stochastic mechanics. According to this principle, information measures, such as relative entropy and Fisher information, are imposed as constraints on top of the least action principle. This principle not only recovers Nelson’s theory and, consequently, the Schrödinger equation but also clears an unresolved issue in stochastic mechanics on why multiple Lagrangians can be used in the variational method and yield the same theory. The concept of forward and backward paths provides an intuitive physical picture for stochastic mechanics. Each path configuration is considered as a degree of freedom and has its own law of dynamics. Thus, the variation principle proposed here can be a new tool to derive more advanced stochastic theory by including additional degrees of freedom in the theory. The structure of Lagrangian developed here shows that some terms in the Lagrangian are originated from information constraints. This suggests that a Lagrangian may need to include both physical and informational terms in order to have a complete description of the dynamics of a physical system.

The Deficit in the Gaussian Log-Sobolev Inequality and Inverse Santaló Inequalities

Abstract We establish dual equivalent forms involving relative entropy, Fisher information, and optimal transport costs of inverse Santaló inequalities. We show in particular that the Mahler conjecture is equivalent to some dimensional lower bound on the deficit in the Gaussian logarithmic Sobolev inequality. We also derive from existing results on inverse Santaló inequalities some sharp lower bounds on the deficit in the Gaussian logarithmic Sobolev inequality. Our proofs rely on duality relations between convex functionals (introduced in [ 16] and [ 62]) related to the notion of moment measure.

Fisher information and density functional theory

AbstractRelationship of Fisher information and density functional theory is reviewed. Links between the Fisher and Shannon information, the local wave‐vector and the relative information are displayed. Euler equations for the Fisher and relative Fisher information are presented. A combined information theoretical and thermodynamic view of density functional theory is analyzed. The extremum of the Shannon and Fisher information results a constant temperature and for Coulomb systems a simple relation between the total energy and phase‐space Fisher information. Relations for phase‐spase fidelity, fidelity susceptibility and relative information are also presented.

Quantifications and Applications of Relative Fisher Information in Density Functional Theory.

Though density functional theory is widely accepted as one of the most successful developments in theoretical chemistry in the past few decades, the knowledge of how to apply this new electronic structure theory, to help us better understand chemical processes and transformations, is still an unaccomplished task. The information-theoretic approach is emerging as a viable option for that purpose in the recent literature, providing new insights about steric effect, cooperativity, electrophilicity, nucleophilicity, stereoselectivity, homochirality, etc. In this work, based on the result from a recent paper by one of us [ J. Chem. Phys, 2019, 151, 141103], we present two quantifications of the relative Fisher information and discuss their physiochemical properties and possible applications. To that end, their analytical properties have been elucidated. They have also been applied to six categories of systems to illustrate their applicability. A better descriptor to quantify the single bond rotation barrier has been obtained. The relative Fisher information can also simultaneously determine electrophilicity and nucleophilicity, and effectively describe helical structures with different homochiral and heterochiral propensities. As integral parts of the information-theoretic approach, these newly introduced quantities will provide us with more analytical tools toward the long-term goal of crafting a chemical reactivity theory in the density-based language.

About the Use of Entropy Production for the Landau\u2013Fermi\u2013Dirac Equation

In this paper, we present new estimates for the entropy dissipation of the Landau–Fermi–Dirac equation (with hard or moderately soft potentials) in terms of a weighted relative Fisher information adapted to this equation. Such estimates are used for studying the large time behaviour of the equation, as well as for providing new a priori estimates (in the soft potential case). An important feature of such estimates is that they are uniform with respect to the quantum parameter. Consequently, the same estimations are recovered for the classical limit, that is the Landau equation.

Fisher Information of Free-Electron Landau States.

An electron in a constant magnetic field has energy levels, known as the Landau levels. One can obtain the corresponding radial wavefunction of free-electron Landau states in cylindrical polar coordinates. However, this system has not been explored so far in terms of an information-theoretical viewpoint. Here, we focus on Fisher information associated with these Landau states specified by the two quantum numbers. Fisher information provides a useful measure of the electronic structure in quantum systems, such as hydrogen-like atoms and under some potentials. By numerically evaluating the generalized Laguerre polynomials in the radial densities, we report that Fisher information increases linearly with the principal quantum number that specifies energy levels, but decreases monotonically with the azimuthal quantum number m. We also present relative Fisher information of the Landau states against the reference density with , which is proportional to the principal quantum number. We compare it with the case when the lowest Landau level state is set as the reference.

Constraints on stochastic heat probability prescribed by exchange fluctuation theorems

The amount of heat transfer from one body to another during a finite time contact is a stochastic variable. The observed probability of heat can be prescribed by the exchange fluctuation theorem and its extensions; however, it should be stipulated in probability theory. We point out some constraints on the heat probability. We also note that relative Fisher information between forward and reverse heat probabilities is quantified by the difference in temperature of two systems before the contact.

Relating Relative Entropy, Optimal Transport and Fisher Information: A Quantum HWI Inequality

Quantum Markov semigroups characterize the time evolution of an important class of open quantum systems. Studying convergence properties of such a semigroup and determining concentration properties of its invariant state have been the focus of much research. Quantum versions of functional inequalities (like the modified logarithmic Sobolev and Poincaré inequalities) and the so-called transportation cost inequalities have proved to be essential for this purpose. Classical functional and transportation cost inequalities are seen to arise from a single geometric inequality, called the Ricci lower bound, via an inequality which interpolates between them. The latter is called the HWI inequality, where the letters I, W and H are, respectively, acronyms for the Fisher information (arising in the modified logarithmic Sobolev inequality), the so-called Wasserstein distance (arising in the transportation cost inequality) and the relative entropy (or Boltzmann H function) arising in both. Hence, classically, the above inequalities and the implications between them form a remarkable picture which relates elements from diverse mathematical fields, such as Riemannian geometry, information theory, optimal transport theory, Markov processes, concentration of measure and convexity theory. Here, we consider a quantum version of the Ricci lower bound introduced by Carlen and Maas and prove that it implies a quantum HWI inequality from which the quantum functional and transportation cost inequalities follow. Our results hence establish that the unifying picture of the classical setting carries over to the quantum one.

An inequality connecting entropy distance, Fisher Information and large deviations

In this paper we introduce a new generalisation of the relative Fisher Information for Markov jump processes on a finite or countable state space, and prove an inequality which connects this object with the relative entropy and a large deviation rate functional. In addition to possessing various favourable properties, we show that this generalised Fisher Information converges to the classical Fisher Information in an appropriate limit. We then use this generalised Fisher Information and the aforementioned inequality to qualitatively study coarse-graining problems for jump processes on discrete spaces.

On the time-dependent Fisher information of a density function

Fisher information is a very important and fundamental criterion in statistical inference especially in optimal and large sample studies in estimation theory. It also plays a key role in physics, thermodynamic, information theory and other applications. In the literature there have been defined two forms of Fisher information: one for the parameters of a distribution function and one for the density function of a distribution. In this paper, we consider a nonnegative continuous random (lifetime) variable $X$ and define a time-dependent Fisher information for density function of the residual random variable associated to $X$. We also propose a time-dependent version of Fisher information distance (relative Fisher information) between the densities of two nonnegative random variables. Several properties of the proposed measures and their relations to other statistical measures are investigated. To illustrate the results various examples are also provided.

Relative Fisher information in some central potentials

Relative Fisher information (IR) was pursued for 1D quantum harmonic oscillator (QHO), 3D isotropic QHO, hydrogen atom and pseudoharmonic potential (PHP) in both r and p spaces. In 1D case, the n=0 state is chosen as reference, whereas for a central potential, the respective circular (corresponding to lowest radial quantum number nr) state of a given l quantum number, is selected. Starting from their exact wave functions, expressions of IR in both r and p spaces are obtained in closed analytical forms in all these systems. For the 1D QHO, IR in r,p spaces increases linearly with n. For 3D QHO and PHP, it varies with single power of nr in both spaces. But, in H atom they depend on both principal (n) and azimuthal (l) quantum numbers. However, at a fixed l, IR initially advances with rise of n and then falls off; also for a given n, it always decreases with l.

Relative Fisher information for Morse potential and isotropic quantum oscillators

The relative Fisher information associated with quantum-mechanical states for two prototypical models is presented; the radial density functions that characterize the Morse oscillator and the d-dimensional isotropic quantum oscillators are analyzed in terms of this relative information by specifying the ground state as the reference. We find that the information increases as the eigenvalue (i.e., the degree of the associated Laguerre polynomials) increases for the Morse density function, but for the isotropic quantum harmonic models, it decreases with the dimension and increases almost linearly with the increasing quantum numbers. We present also the explicit expression for the information associated with the angular density as a function of the orbital quantum number and the magnetic quantum number for the three-dimensional isotropic quantum oscillator.