Information Manifolds in Statistical Inference

• Information manifolds are smooth structures where each point represents a probability distribution, defined by the Fisher–Rao metric and dual affine connections.
• They employ geodesic flows and curvature tensors to quantify statistical distinguishability and dynamical complexity across diverse fields including quantum information and data geometry.
• The framework drives practical applications in optimal parameter estimation, entropic dynamics, and explainable AI by geometrically modeling data and statistical systems.

Information manifolds are smooth manifolds whose points correspond to parameterized probability distributions, equipped with geometrical structures—most notably the Fisher–Rao metric, dual affine connections, and associated curvature tensors—characterizing statistical distinguishability and governing entropic dynamics. This geometrization provides a unified analytic language for statistical inference, complexity, dynamical modeling, optimization, quantum information, and machine learning, rigorously connecting information-theoretic and physical properties of statistical systems.

1. Construction of Information Manifolds and the Fisher–Rao Metric

Given a parametric family of probability densities or mass functions,

$$M = \{ p(x\mid\theta) : \theta = (\theta^1,\ldots,\theta^n) \in \Theta \subset \mathbb{R}^n \},$$

the manifold structure arises by treating $\theta$ as local coordinates. The Fisher–Rao information metric,

$$g_{ij}(\theta) = \mathbb{E}_\theta \left[ \partial_i \log p(x\mid\theta) \, \partial_j \log p(x\mid\theta) \right] = \int_X p(x\mid\theta)\,\partial_i\log p(x\mid\theta)\,\partial_j\log p(x\mid\theta)\;dx,$$

uniquely characterizes infinitesimal statistical distinguishability. Equivalently, $g_{ij}$ is the Hessian of the relative entropy $$D_{\text{KL}}[p(\cdot\mid\theta)\Vert p(\cdot\mid\theta+\Delta\theta)]$$ at $\Delta\theta=0$ (Cafaro, 2013, Mishra et al., 2023, Nielsen, 2018). Cencov’s theorem shows the Fisher metric is distinguished by invariance under sufficient statistical mappings.

For exponential families $$p(x;\theta)=\exp[\theta\cdot F(x)-\phi(\theta)]\,\nu(dx)$$, the Fisher metric is expressed as $g_{ij}(\theta)=\partial_i\partial_j\phi(\theta)$. Dual coordinate systems (e-coordinates $\theta$, m-coordinates $\eta$) are canonically related via Legendre duality (Mishra et al., 2023).

2. Geodesics, Affine Connections, and Curvature

The geometric structure is enriched by dual affine (torsion-free) connections $(\nabla, \nabla^*)$, defined via a divergence function $D(\theta,\theta')$ by

$$\Gamma_{ij,k} = \frac12 T_{ijk}, \quad T_{ijk} = -\partial_i\partial_j\partial_{k'} D(\theta,\theta')\big|_{\theta'=\theta},$$

with dual connection indices related by index permutations. The Levi–Civita connection corresponds to the metric-compatible, torsion-free case, and its Christoffel symbols take the standard Riemannian form:

$$\Gamma^k_{ij} = \frac12 g^{k\ell}(\partial_i g_{\ell j} + \partial_j g_{\ell i} - \partial_\ell g_{ij}).$$

Statistical inference and entropic dynamics manifest as minimum-distance (geodesic) flows on $(M,g)$, with the geodesic equations

$$\frac{D\dot{\theta}^k}{Ds} = \ddot{\theta}^k + \Gamma^k_{ij}(\theta)\dot{\theta}^i\dot{\theta}^j = 0,$$

selecting the most probable macroscopic trajectory under maximum relative entropy principles (Cafaro, 2013, Mishra et al., 2023).

Curvature tensors ($R^k_{\,ij\ell}$) derived from $(M,g,\nabla)$ or its dual(s) control the separation of geodesics, model sensitivity, and complexity growth. In the Gaussian case, the scalar curvature is negative and constant only for the one-parameter family; in higher-dimensional models, curvature is generally variable (Li et al., 2014).

3. Complexity Measures on Information Manifolds

Complexity in information geometry is quantified by Riemannian lengths and statistical volumes traversed by dynamical geodesics:

$$C(\tau) = \int_0^\tau \sqrt{g_{ij}(\theta(s))\,\dot{\theta}^i(s)\,\dot{\theta}^j(s)}\,ds.$$

The Information Geometric Entropy (IGE)—the logarithm of the mean explored “statistical volume”—serves as a proxy for cumulative complexity (Cafaro, 2013, Gassner et al., 2019). For Gaussian models:

• Under the Fisher–Rao metric, $C(\tau)$ and IGE grow linearly in $\tau$ and geodesics converge exponentially.
• For alternative, e.g. $\alpha$-order entropy metrics, complexity growth is merely logarithmic and convergence polynomial, reflecting a trade-off between distinguishability and inference speed (Gassner et al., 2019).

Statistical embedding—constraining the manifold by priors, correlations, or uncertainty-type relations—lowers scalar curvature and softens complexity by slowing the rate of statistical volume growth (Cafaro, 2013).

4. Geometry of Quantum, Infinite-Dimensional, and Specialized Information Manifolds

Quantum information manifolds generalize the Fisher–Rao metric to quantum Fisher information, as characterized by the Bures metric on manifolds of quantum states; this structure is critical in quantum parameter estimation and quantum metrology (Marian et al., 2016). In the statistical manifolds of two-mode Gaussian states, the Bures metric is diagonal in natural parameters and scalar curvature is a function of input thermal photon numbers, determining the volume of quantum distinguishable regions.

Infinite-dimensional information manifolds, defined via balanced charts on measure spaces, retain core features of finite-dimensional counterparts: the $\alpha$-divergences are regular, the Fisher metric is defined (as a pseudo-Riemannian metric on the ambient Banach manifold, Riemannian on finite-dimensional submanifolds), and $\alpha$-covariant derivatives are well-posed up to the degree determined by integrability exponents (Newton, 2013).

In signal processing, Kähler information manifolds of linear filters in weighted Hardy spaces admit Hermitian metrics derived from squared weighted Hardy norms as Kähler potentials, providing closed-form metrics, connections, and curvature in terms of the poles and zeros of transfer functions. Classical Fisher–Rao and mutual-information metrics are recovered as special cases (Choi, 2021).

5. Information Manifolds in Machine Learning and Data Geometry

Information geometric structure extends beyond parameter space to data space in statistical learning. The data information matrix (DIM), defined for a fixed classifier as

$$D_{ij}(x) = \mathbb{E}_{y\mid x,\theta}\left[\partial_{x^i}\ln p(y\mid x,\theta)\,\partial_{x^j}\ln p(y\mid x,\theta)\right],$$

induces a local Riemannian metric on the data manifold. Level sets of constant DIM rank integrate to foliations, with leaves supporting nondegenerate metrics; on neural networks, this framework reveals low-dimensional data leaves corresponding to valid inputs, and curvature exposes class-reachability and robust directions (Tron et al., 18 Sep 2024, Grementieri et al., 2021).

Geodesic analysis in data space identifies minimal-information paths for valid data morphing, and the restricted metric offers explainable AI mechanisms by directly relating local curvature and metric structure to network decision boundaries and class confusion (Tron et al., 18 Sep 2024).

6. Holonomy, F-Manifolds, and Extensions

The global holonomy of the Fisher metric of statistical manifolds, for generic exponential families, is generically the full special orthogonal group $SO(n)$, reflecting irreducibility and absence of special structures except in rare Einstein or parallelizable cases (Li et al., 2014). This maximal holonomy implies that information manifolds of normal distributions or generic exponential families are as generic as possible in Riemannian geometry.

Classical information manifolds also admit F-manifold structure: a commutative, associative multiplication of tangent fields built from the Levi–Civita connection, paralleling developments in singularity theory and topological field theory (Combe et al., 2020). For exponential families, this multiplication is derived from the third derivatives of the partition function, and Frobenius manifold structure appears under additional flatness and potentiality conditions.

Extensions include the construction of information manifolds as spaces of entropic parameters (e.g., the $(c,d)$-manifold of Hanel–Thurner entropy), with the Fisher metric and curvature encoding degrees of non-extensivity and enabling classification of complex-system behaviors (Ghikas et al., 2018).

7. Applications, Impact, and Interdisciplinary Connections

Information manifolds unify concepts in statistical inference, complexity theory, and dynamical systems:

• The Fisher metric underlies the Cramér–Rao bound, natural gradient methods in learning, and optimal transport distances (Mishra et al., 2023).
• Geodesic analysis provides a geometric foundation for entropic dynamics, maximum entropy inference, and information-theoretic model selection (Cafaro, 2013, Gassner et al., 2019).
• In quantum systems, information geometry connects distinguishability, estimation theory, and entanglement measures (Marian et al., 2016).
• The pseudo-Riemannian (Lorentzian) extension of information geometry permits reinterpretation of gravitational dynamics, with Fisher geometry serving as the metric underpinning emergent Einstein equations, cosmological constants, and entropy–area relations (Alshal, 2023).
• In machine learning, the induced geometry on data spaces offers practical, explainable metrics for model interpretability, robustness, and data denoising (Tron et al., 18 Sep 2024, Grementieri et al., 2021).

Theoretical developments continue in multiple directions: generalization to infinite-dimensional contexts, the study of holonomy and global manifold structure, the role of F-manifold and Kähler geometry, and the connection of information geometry to physical theories via emergent spacetime notions and entropy-driven dynamics.