Relative Fisher Measures in Information Geometry

• Relative Fisher measures are defined as comparisons of Fisher information relative to reference densities, metrics, or statistical models, unifying diverse constructions.
• They include forms such as the Fisher–Rao distance, score-based discrepancies, and marginal-weighted metrics, each providing intrinsic sensitivity and geometric insights.
• Applications span geometric inference, dynamical convergence analysis, complexity evaluation, thermodynamics, and quantum estimation, offering practical criteria for statistical distinguishability.

Searching arXiv for recent and foundational papers on Relative Fisher Measures. Relative Fisher measures are a family of constructions that quantify discrepancy, complexity, or local sensitivity relative to a reference law, reference geometry, or reference model. Across the literature, the term covers several distinct but related objects: the Fisher–Rao distance on spaces of probability measures induced by the Fisher information metric; relative Fisher information defined through score differences; marginal-weighted Fisher metrics on conditional probability polytopes; scaling-invariant biparametric relative Fisher functionals tied to Rényi and Kullback–Leibler divergences; and Fisher-geometric set-complexity measures such as Fisher width. What unifies these notions is that Fisher information is not used absolutely, but relative to another density, another metric background, or another statistical structure, so that distinguishability is measured intrinsically rather than by ambient Euclidean coordinates (Itoh et al., 2017).

1. Fisher–Rao geometry and the metric notion of relative Fisher measure

On a connected compact smooth manifold $M$ equipped with a fixed probability reference measure $\lambda$, the space

$$\mathcal{P}(M)=\left\{\mu=p\,\lambda \mid p\in C^0_+(M),\ \int_M p\,d\lambda=1\right\}$$

carries the Fisher information metric

$$G_\mu(\sigma,\tau)=\int_M \frac{\sigma(x)\tau(x)}{p(x)}\,d\lambda(x).$$

A tangent vector at $\mu=p\,\lambda$ is a signed measure $\tau=q\,\lambda$ with zero total mass and continuous Radon–Nikodym derivative $q/p$ (Itoh et al., 2017).

The central geometric device is the square-root embedding

$\Phi(\mu)=\sqrt{p},$

which maps $\mathcal{P}(M)$ into the unit sphere of $L^2(M,\lambda)$. The pullback of the $\lambda$0 inner product by $\lambda$1 is $\lambda$2, so Fisher–Rao geometry is isometric, up to a constant factor, to spherical geometry in $\lambda$3. If $\lambda$4, the spherical angle $\lambda$5 between $\lambda$6 and $\lambda$7 is determined by

$\lambda$8

The corresponding relative Fisher measure is the Fisher–Rao distance

$\lambda$9

Its inner term is the Bhattacharyya coefficient. Because

$$\mathcal{P}(M)=\left\{\mu=p\,\lambda \mid p\in C^0_+(M),\ \int_M p\,d\lambda=1\right\}$$0

the Fisher–Rao distance is a monotone transform of the Hellinger distance: $$\mathcal{P}(M)=\left\{\mu=p\,\lambda \mid p\in C^0_+(M),\ \int_M p\,d\lambda=1\right\}$$1 This distance is symmetric, nonnegative, vanishes exactly when the densities coincide almost everywhere, satisfies the triangle inequality, and has diameter $$\mathcal{P}(M)=\left\{\mu=p\,\lambda \mid p\in C^0_+(M),\ \int_M p\,d\lambda=1\right\}$$2 on $$\mathcal{P}(M)=\left\{\mu=p\,\lambda \mid p\in C^0_+(M),\ \int_M p\,d\lambda=1\right\}$$3 (Itoh et al., 2017).

The same spherical model gives explicit minimal geodesics. If

$$\mathcal{P}(M)=\left\{\mu=p\,\lambda \mid p\in C^0_+(M),\ \int_M p\,d\lambda=1\right\}$$4

then the geodesic from $$\mathcal{P}(M)=\left\{\mu=p\,\lambda \mid p\in C^0_+(M),\ \int_M p\,d\lambda=1\right\}$$5 to $$\mathcal{P}(M)=\left\{\mu=p\,\lambda \mid p\in C^0_+(M),\ \int_M p\,d\lambda=1\right\}$$6 is

$$\mathcal{P}(M)=\left\{\mu=p\,\lambda \mid p\in C^0_+(M),\ \int_M p\,d\lambda=1\right\}$$7

Any two distinct measures are joined by a unique Fisher geodesic segment, and all such geodesics are globally minimizing. The normalized geometric mean

$$\mathcal{P}(M)=\left\{\mu=p\,\lambda \mid p\in C^0_+(M),\ \int_M p\,d\lambda=1\right\}$$8

governs this geometry: it is symmetric, appears explicitly in the geodesic formulas, and the tangent lines at the endpoints intersect at $$\mathcal{P}(M)=\left\{\mu=p\,\lambda \mid p\in C^0_+(M),\ \int_M p\,d\lambda=1\right\}$$9 (Itoh et al., 2017).

A common misconception is to identify the Fisher–Rao distance with a divergence such as KL. In this framework it is a bona fide Riemannian distance with exact geodesic interpretation and minimizing properties, whereas KL is not a metric. The manifold-level invariance is also explicit: the Fisher metric and the distance are invariant under push-forward by homeomorphisms or diffeomorphisms of $$G_\mu(\sigma,\tau)=\int_M \frac{\sigma(x)\tau(x)}{p(x)}\,d\lambda(x).$$0 (Itoh et al., 2017).

2. Relative Fisher information as a score-based discrepancy

A second major meaning of relative Fisher measure is relative Fisher information, defined for $$G_\mu(\sigma,\tau)=\int_M \frac{\sigma(x)\tau(x)}{p(x)}\,d\lambda(x).$$1 on $$G_\mu(\sigma,\tau)=\int_M \frac{\sigma(x)\tau(x)}{p(x)}\,d\lambda(x).$$2 by

$$G_\mu(\sigma,\tau)=\int_M \frac{\sigma(x)\tau(x)}{p(x)}\,d\lambda(x).$$3

In this formulation, discrepancy is measured through score-function differences rather than through geodesic distance. Weighted variants $$G_\mu(\sigma,\tau)=\int_M \frac{\sigma(x)\tau(x)}{p(x)}\,d\lambda(x).$$4 and second-order Fisher information $$G_\mu(\sigma,\tau)=\int_M \frac{\sigma(x)\tau(x)}{p(x)}\,d\lambda(x).$$5 also appear naturally in evolution identities (Wibisono, 8 Feb 2025).

For a target $$G_\mu(\sigma,\tau)=\int_M \frac{\sigma(x)\tau(x)}{p(x)}\,d\lambda(x).$$6 that is $$G_\mu(\sigma,\tau)=\int_M \frac{\sigma(x)\tau(x)}{p(x)}\,d\lambda(x).$$7-strongly log-concave and $$G_\mu(\sigma,\tau)=\int_M \frac{\sigma(x)\tau(x)}{p(x)}\,d\lambda(x).$$8-log-smooth, relative Fisher information is linked to KL through the de Bruijn identity along Langevin dynamics: $$G_\mu(\sigma,\tau)=\int_M \frac{\sigma(x)\tau(x)}{p(x)}\,d\lambda(x).$$9 Thus $\mu=p\,\lambda$0 is the squared Wasserstein gradient norm of KL, and under $\mu=p\,\lambda$1-LSI one has $\mu=p\,\lambda$2. A corresponding time-derivative identity along general Fokker–Planck channels shows that if $\mu=p\,\lambda$3 and $\mu=p\,\lambda$4 evolve under the same drift-diffusion equation, then $\mu=p\,\lambda$5 is the sum of a negative Hilbert–Schmidt Hessian term and a curvature-weighted first-order term (Wibisono, 8 Feb 2025).

This score-based notion also appears in thermodynamic form. For canonical equilibrium phase-space densities $\mu=p\,\lambda$6 and $\mu=p\,\lambda$7 corresponding to forward and backward processes, the relative Fisher information used there is

$\mu=p\,\lambda$8

With

$\mu=p\,\lambda$9

one gets

$\tau=q\,\lambda$0

hence

$\tau=q\,\lambda$1

This identifies relative Fisher information with the phase-space mean-square gradient of dissipated work. Under a logarithmic Sobolev inequality,

$\tau=q\,\lambda$2

so a local gradient-level irreversibility measure is bounded below by a global entropy production term (Yamano, 2013).

A third score-based strand develops a variational theory of relative Fisher information with respect to a Gibbs reference $\tau=q\,\lambda$3. In one dimension,

$\tau=q\,\lambda$4

satisfies

$\tau=q\,\lambda$5

When $\tau=q\,\lambda$6, extremization under normalization and moment constraints yields a Schrödinger-like equation

$\tau=q\,\lambda$7

with

$\tau=q\,\lambda$8

This framework supports reciprocity relations, a generalized Euler theorem, and a Legendre-transform structure analogous to thermodynamics; the same relations were later derived from the Hellmann–Feynman theorem, which ties the multipliers $\tau=q\,\lambda$9 directly to expectation constraints and supports inference of PDFs and energy eigenvalues [(Venkatesan et al., 2013); (Venkatesan et al., 2014)].

3. Relative Fisher measures in conditional probability polytopes

In the geometry of stochastic matrices, relative Fisher measure acquires yet another precise meaning. For the polytope

$q/p$0

each row is a point in a simplex, so $q/p$1. A tangent vector is an $q/p$2 matrix with row sums equal to zero (Montufar et al., 2014).

Three Fisher-type metrics arise. The unscaled product metric is

$q/p$3

coming from the exponential-family embedding of the product polytope into a simplex. A scaled product metric

$q/p$4

is singled out by invariance under homogeneous conditional embeddings. No metric is invariant under the full class of non-homogeneous conditional embeddings (Montufar et al., 2014).

The relative Fisher measure in this setting appears when a marginal $q/p$5 on the row variable is fixed and the conditional polytope is embedded into the simplex of joint distributions by

$q/p$6

Pulling back the Fisher metric from the joint simplex gives

$q/p$7

Rows are thus weighted by their reference marginal masses $q/p$8. This is the precise sense in which the paper interprets a marginal-weighted product metric as a relative Fisher measure: it measures sensitivity of the conditional distributions $q/p$9 relative to the occurrence probabilities of the rows (Montufar et al., 2014).

The covariance law under conditional embeddings is equally important. If $\Phi(\mu)=\sqrt{p},$0 is a conditional embedding and the row marginal transforms as $\Phi(\mu)=\sqrt{p},$1, then

$\Phi(\mu)=\sqrt{p},$2

A uniqueness theorem states that among continuous families of metrics satisfying this covariance for all conditional embeddings, the marginal-weighted product Fisher metric is unique up to an overall constant (Montufar et al., 2014).

This construction clarifies a potential ambiguity in the phrase “relative Fisher.” Here the relativity is neither to a target density in score space nor to geodesic distance in the probability manifold, but to an externally specified row marginal. The reference distribution determines which conditional rows contribute more heavily to the local information geometry (Montufar et al., 2014).

4. Scale-invariant biparametric relative Fisher measures and sharp inequalities

A more recent development introduces a biparametric family of relative Fisher measures for one-dimensional densities $\Phi(\mu)=\sqrt{p},$3 and $\Phi(\mu)=\sqrt{p},$4 with common support $\Phi(\mu)=\sqrt{p},$5, based on the relative differential-escort transformation. For $\Phi(\mu)=\sqrt{p},$6, this transform is

$\Phi(\mu)=\sqrt{p},$7

Its transformed support length is

$\Phi(\mu)=\sqrt{p},$8

where $\Phi(\mu)=\sqrt{p},$9 is the exponential of a Rényi divergence (Iagar et al., 23 Jul 2025).

For $\mathcal{P}(M)$0 and $\mathcal{P}(M)$1, the relative Fisher divergence is defined by

$\mathcal{P}(M)$2

and the normalized relative Fisher measure is

$\mathcal{P}(M)$3

In contrast with earlier relative Fisher functionals such as

$\mathcal{P}(M)$4

this family is invariant under simultaneous affine transformations $\mathcal{P}(M)$5 of both densities (Iagar et al., 23 Jul 2025).

The escort transform linearizes the relation between this relative Fisher measure and Rényi/KL quantities. Two identities are central: $\mathcal{P}(M)$6 and

$\mathcal{P}(M)$7

These permit transfer of sharp single-density Stam and moment-entropy inequalities into the relative setting (Iagar et al., 23 Jul 2025).

The resulting inequalities are sharp. Under the paper’s parameter constraints, one obtains a moment-entropy-type bound

$\mathcal{P}(M)$8

and a Stam-like inequality

$\mathcal{P}(M)$9

In the Shannon limit $L^2(M,\lambda)$0, these reduce to KL-based inequalities such as

$L^2(M,\lambda)$1

The minimizers are expressed through inverse relative differential-escort transforms of stretched Gaussians, and generalized trigonometric or hyperbolic functions enter the explicit formulas for fixed-target adapted inequalities (Iagar et al., 23 Jul 2025).

A closely related 2026 extension places non-relative, relative, and cross informational functionals into a unified inequality theory. There the same scaling-invariant relative Fisher measure

$L^2(M,\lambda)$2

appears in sharp product inequalities involving Rényi entropy power, Rényi cross entropy, generalized cross-Fisher functionals, and moment-like deviations. The minimizers of the Stam-like inequality are in certain cases pairs of Gaussian or stretched Gaussian densities, whereas the moment-only inequality is minimized by generalized Beta distributions (Iagar et al., 9 Jul 2026).

5. Relative Fisher measures in dynamics, asymptotics, and discrete settings

Relative Fisher information is also used as a dynamical control functional. For the Proximal Sampler targeting a strongly log-concave, log-smooth distribution $L^2(M,\lambda)$3, one iteration is the composition of a forward Gaussian channel

$L^2(M,\lambda)$4

and a reverse Gaussian Bayes channel

$L^2(M,\lambda)$5

A strong data processing inequality along the forward channel gives

$L^2(M,\lambda)$6

while the reverse channel is non-expansive: $L^2(M,\lambda)$7 Consequently,

$L^2(M,\lambda)$8

With $L^2(M,\lambda)$9, this yields high-accuracy complexity

$\lambda$00

under the paper’s initialization and rejection-sampling assumptions. Here relative Fisher information gives a stronger guarantee than KL and explains the discrete-time convergence of the Proximal Sampler in a form matching continuous-time Langevin decay (Wibisono, 8 Feb 2025).

In the low-temperature analysis of reversible diffusions, the relevant functional is Fisher information relative to a Gibbs reference

$\lambda$01

For $\lambda$02,

$\lambda$03

As $\lambda$04, this functional admits a full $\lambda$05-development reflecting metastability. The first limit is

$\lambda$06

which vanishes exactly on measures supported on critical points. The next scale,

$\lambda$07

detects mass on non-minimum critical points through the local oscillator offsets $\lambda$08. Subsequent exponentially small scales

$\lambda$09

capture tunneling between wells with Eyring–Kramers prefactors $\lambda$10. This makes relative Fisher information a multiscale descriptor of metastable concentration, first on critical points, then on local minima, and finally across exponentially rare inter-well transitions (Gesù et al., 2016).

For discrete orthogonal-polynomial ensembles, relative Fisher information is defined on Rakhmanov distributions

$\lambda$11

through the forward difference operator: $\lambda$12 Exact formulas are available for the classical discrete families: $\lambda$13 for Charlier,

$\lambda$14

for Meixner, and corresponding hypergeometric formulas for Kravchuk and Hahn. In every case the functional is nonnegative, vanishes at degree $\lambda$15, and quantifies the oscillatory roughness of the polynomially weighted discrete law relative to its baseline weight $\lambda$16 (Dehesa et al., 2013).

6. Relative Fisher measures as complexity, inference criteria, and distinguishability limits

A geometric-complexity interpretation is provided by Fisher width. For a set $\lambda$17 and a Fisher information tensor $\lambda$18, the Fisher width at $\lambda$19 is

$\lambda$20

Equivalently, in coordinate-free form on a statistical manifold $\lambda$21,

$\lambda$22

for a standard Gaussian tangent vector $\lambda$23. This quantity is invariant under smooth reparameterizations, satisfies monotonicity, positive homogeneity, convex-hull invariance, and subadditivity, and obeys the spectral sandwich

$\lambda$24

It thus measures set complexity relative to local statistical distinguishability rather than Euclidean size. For Fisher-Lipschitz hypothesis classes it controls the generalization gap through

$\lambda$25

up to the universal constant stated in the paper (Ky, 16 Jun 2026).

In classical likelihood inference, “relative Fisher measures” can also refer to the comparison between observed and expected Fisher information matrices as competing interval-construction devices. With

$\lambda$26

and their per-observation versions $\lambda$27 and $\lambda$28, the paper compares approximate componentwise confidence levels induced by $\lambda$29 and $\lambda$30. Under regularity conditions, the main asymptotic result is

$\lambda$31

with strict inequality under an additional variability condition. The expected Fisher information is therefore asymptotically no worse, and typically better, than the observed Fisher information for unconditional Wald-type interval accuracy under the paper’s MSE criterion (Jiang, 2021).

In quantum information, relative Fisher measures arise through the right logarithmic derivative quantum Fisher information and its relation to geometric Rényi relative entropy. For a differentiable density family $\lambda$32,

$\lambda$33

when the support condition holds, and for channels the corresponding quantity admits an explicit Choi-operator formula. A chain rule holds: $\lambda$34 which implies amortization collapse for channel estimation. Combined with a meta-converse, this shows that if the channel RLD Fisher information is finite, Heisenberg scaling is unattainable for general sequential estimation protocols. The same paper establishes chain rules and amortization collapse for geometric Rényi channel divergences, yielding improved Chernoff and Hoeffding bounds for channel discrimination (Katariya et al., 2020).

These disparate constructions show that “relative Fisher measures” is not a single standardized term. It names a family of Fisher-based objects whose relativity may be to a reference density, a target distribution, a row marginal, a local Riemannian metric, a baseline inference criterion, or an alternative quantum channel. The common principle is that Fisher information acquires meaning through comparison: to another law, to another parametrization, or to another geometry.