Curvature-Aware Cramér-Rao Bound Refinements

• The paper introduces curvature corrections to the classical Cramér-Rao Bound by integrating extrinsic and intrinsic differential geometry for strictly tighter variance bounds.
• It leverages the second fundamental form and higher-order jet projections to account for curvature effects that remain hidden in first-order analyses.
• The refinements extend to vector, Bayesian, and quantum settings, unifying classical, higher-order, and Cartan-geometric frameworks for improved estimator efficiency.

Curvature-aware refinements of the Cramér-Rao Bound (CRB) represent a substantial extension of the classical information-theoretic lower bounds on estimator variance, incorporating higher-order geometric information from the statistical model manifold. These refinements leverage both extrinsic and intrinsic differential-geometric structures—such as the second fundamental form, jet bundles, and Cartan connections—to systematically account for curvature effects that are invisible to classical (first-order) information inequalities. This framework yields strictly tighter variance bounds for both scalar and vector parameter estimation, generalizes to higher-order (e.g., Bhattacharyya-type) inequalities, and admits invariant extensions in Bayesian statistics and quantum estimation.

1. Classical CRB and Statistical Embedding

Let $X$ be a random variable with density $f(x; \theta)$ (with respect to a fixed $\sigma$-finite measure $\mu$) and scalar parameter $\theta \in \Theta$. The classical CRB relates the variance of any unbiased estimator $\hat\theta$ of $\theta$ to the Fisher information $I(\theta)$: $$\operatorname{Var}_\theta[\hat\theta] \ge \frac{1}{I(\theta)},$$ where $$I(\theta) = \mathbb{E}_\theta [(\partial_\theta \log f(X; \theta))^2]$$ (Krishnan, 19 Nov 2025, Krishnan, 22 Sep 2025).

In the geometric formalism, the map $s_\theta(x) = \sqrt{f(x;\theta)}$ realizes the statistical model as a curve $\mathcal{M} = \{s_\theta : \theta \in \Theta\}$ in the Hilbert space $L^2(\mu)$, with the Fisher information identified as four times the squared norm of the tangent vector $\eta_1 = \partial_\theta s_\theta$: $I(\theta) = 4 \|\eta_1\|^2.$ This extrinsic perspective underpins all subsequent curvature-aware refinements (Krishnan, 22 Sep 2025).

2. Extrinsic Curvature and the Second Fundamental Form

To analyze how the statistical manifold $\mathcal{M}$ "curves" within $L^2(\mu)$, the second fundamental form becomes central. For the curve $s_\theta$:

• The tangent space at $\theta$ is $T_1 = \operatorname{span}\{\eta_1\}$.
• The second derivative $\eta_2 = \partial_\theta^2 s_\theta$ splits into tangent and normal components via orthogonal projection.
• The second fundamental form $\mathrm{II}(\eta_1, \eta_1)$ is the normal component: $$\mathrm{II}(\eta_1,\eta_1) = \eta_2 - \frac{\langle \eta_2, \eta_1 \rangle}{\|\eta_1\|^2} \eta_1,$$ with $\|\mathrm{II}(\eta_1, \eta_1)\|$ quantifying the extrinsic curvature of the model manifold at $\theta$ (Krishnan, 19 Nov 2025, Krishnan, 22 Sep 2025).

For unbiased estimators $T(X)$, the centered error $Z_0(X) = T(X) - \theta$ is "lifted" to $L^2$ as $\widetilde Z_0 = Z_0 s_\theta$. Classical first-order analyses project $\widetilde Z_0$ onto $T_1$, but the new insight is to further project onto the span of the curvature vector, thus bounding contributions of the estimator error that are invisible to first-order theory (Krishnan, 22 Sep 2025).

3. Curvature-Corrected and Higher-Order CRB Inequalities

The curvature correction emerges explicitly as an additional variance lower bound term. For a scalar parameter, the refined CRB is: $$\operatorname{Var}_\theta[\hat\theta] \ge \frac{1}{I(\theta)} + \frac{\|\mathrm{II}(\eta_1, \eta_1)\|^2}{\|\eta_1\|^4}$$ (Krishnan, 19 Nov 2025, Krishnan, 22 Sep 2025). The correction term is strictly positive unless the statistical curve is a straight line in $L^2(\mu)$, i.e., unless all normal curvature vanishes.

The extension to higher-order bounds builds on the Bhattacharyya–Pillai–Sinha framework, which projects the estimator error onto higher-order jets $$\mathcal{T}_m = \operatorname{span}\{\eta_1, \ldots, \eta_m\}$$. The jets are compactly represented via the Faà di Bruno formula and exponential Bell polynomials in terms of raw score functions: $$\eta_k = s_\theta\, B_k\Big(\tfrac12 Y_1, ..., \tfrac12 Y_k\Big),$$ where $B_k$ is the $k$th exponential Bell polynomial and $Y_k$ the $k$th derivative score (Krishnan, 22 Sep 2025).

The general second fundamental form $\Pi_m$ is given as: $$\Pi_m = \eta_{m+1} - \sum_{i=1}^{m} \Gamma_i\, \eta_i$$ where the coefficients $\Gamma_i$ arise from connection-induced projections. The curvature-refined bound,

$$\operatorname{Var}_\theta[T] \ge \|\mathrm{Proj}_{\mathcal T_m} \widetilde Z_0\|^2 + \|\mathrm{Proj}_{\operatorname{span}\{\Pi_m\}} \widetilde Z_0\|^2,$$

strictly improves upon classical higher-order bounds whenever the estimator error has a nontrivial component along $\Pi_m$ (Krishnan, 22 Sep 2025, Krishnan, 19 Nov 2025).

4. Intrinsic Cartan Geometric Reinterpretation

Cartan geometry provides an intrinsic, coordinate-free understanding of the curvature corrections by viewing parameter manifolds and their statistical embeddings through the lens of jet bundles and Ehresmann connections (Krishnan, 19 Nov 2025).

In this framework:

• The statistical jet bundle $J^m(E_{\mathrm{stat}})$ encodes functions and their derivatives up to order $m$, representing model jets as points $(\theta, s, s_1, ..., s_m)$.
• Canonical contact forms $\omega_k$ and the Cartan vector field $D_\theta$ endow this bundle with a natural 1-dimensional "Cartan distribution".
• The non-integrability (i.e., torsion) of the Cartan distribution captures the geometric obstruction underlying the curvature corrections; the failure of the jet prolongation $j^m s$ to remain horizontal reflects estimator inefficiency.

A central theorem establishes equivalence between the algebraic projection conditions (residual lies in the jet span) and geometric integrability (prolonged section is integral for the restricted Cartan vector field), connecting extrinsic variance bounds to intrinsic geometric flows (Krishnan, 19 Nov 2025).

5. Vector and Multivariate Extensions

For vector-valued parameters $\theta \in \mathbb{R}^d$, the curvature correction generalizes via the second fundamental form $\Pi_{ij}$ acting on the tangent subspace $$T_\theta = \mathrm{span}\{\eta_1, ..., \eta_d\} \subset L^2(\mu)$$. For any direction $v \in \mathbb{R}^d$, form the directional curvature vector

$$\Pi_v = \sum_{i,j} \widetilde v_i \widetilde v_j \Pi_{ij}$$

where $\widetilde v = G(\theta)^{-1} v$ and $G_{ij} = \langle \eta_i, \eta_j \rangle$.

The directional refinement is

$$v^\top [\Sigma(\theta) - J(\theta)^{-1}] v \ge \frac{\langle Z_v, \Pi_v \rangle^2}{\|\Pi_v\|^2}$$

with $Z_v = \sum_p v_p \widetilde Z^{(p)}$, for all unbiased estimators $T$, and matrix-level refinements can be certified via sums-of-squares semidefinite programs (SOS-SDP) (Krishnan, 23 Sep 2025).

6. Bayesian and Invariant Geometric Bounds

Differential-geometric approaches extend refinement philosophy to the Bayesian CRB via the Fisher metric $g_{ab} = F_{ab}(\theta)$ and the Levi-Civita connection $\nabla$. Bounds constructed are fully coordinate invariant and explicitly encode curvature effects: $$\mathrm{Cov}_\pi(\hat\theta)^{ab} \succeq [G^{ab}(\theta) + B^{ab}(\theta)]^{-1},$$ where the prior information tensor $B^{ab}$ and the Laplace–Beltrami operator $\Delta_g$ encode the model curvature and prior effects (Tsang, 2020).

The Laplace–Beltrami contribution ensures that the minimax Bayes risk is governed by the spectrum of a Schrödinger-type operator whose potential is Fisher information, with curvature of the parameter manifold entering via the Riemannian geometry. This manifests in both classical and quantum statistical models, as illustrated in optomechanical waveform estimation and subdiffraction incoherent imaging (Tsang, 2020).

7. Examples and Explicit Calculations

Closed-form curvature corrections can be obtained for exponential family and location-scale models. For instance, in an exponential family $f(x;\theta) = \exp\{\theta T(x) - \psi(\theta)\}$,

$$\operatorname{Var}_\theta[\hat\theta] \ge \frac{1}{\psi''(\theta)} + \frac{[\psi'''(\theta)]^2}{[\psi''(\theta)]^3}$$

where the numerator of the correction term directly involves the third central moment (Krishnan, 19 Nov 2025).

A concrete normal-family example shows that for $P_\theta \sim \mathcal{N}(\theta, 1 + \theta^2)$, the variance of the location estimator $T(X) = X$ satisfies

$$\operatorname{Var}_\theta[T] \ge \frac{(1+\theta^2)^2}{1+3\theta^2} + \frac{36 \theta^2 (\theta^8 + 4\theta^6 + 6\theta^4 + 4\theta^2 + 1)}{567\theta^8 + 792\theta^6 + 462\theta^4 + 144\theta^2 + 19}$$

so that whenever $\theta \neq 0$, the standard CRB is strictly strengthened by the curvature term (Krishnan, 22 Sep 2025).

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Curvature-aware Cramér-Rao refinements link estimator efficiency directly to the geometry of the embedded statistical model. Such refinements unify “first-order” variance bounds, higher-order Bhattacharyya–Pillai–Sinha bounds, and modern Cartan-geometric approaches, providing a comprehensive geometric picture in which inefficiency is measured by curvature and torsion in the appropriate jet bundle hierarchies (Krishnan, 19 Nov 2025, Krishnan, 22 Sep 2025, Krishnan, 23 Sep 2025, Tsang, 2020).