Conservative Matrix-Level CRB Refinement

• Conservative Matrix-Level CRB Refinement is a technique that integrates geometric, algebraic, and optimization methods to provide tighter lower bounds on estimator variances.
• It refines the classical Cramér–Rao bound by leveraging higher-order derivatives and semidefinite programming to adjust for model curvature, singularities, and constraints.
• This approach offers practical insights for improving estimator design in complex, high-dimensional statistical models with nonlinear characteristics.

Conservative matrix-level refinement of the Cramér–Rao bound (CRB) refers to systematic techniques that yield provably tighter or more representative lower bounds for estimator variance/covariance matrices, especially in cases where standard CRB theory—relying on the inverse Fisher information matrix—may be optimistic due to model curvature, singularities, constraints, or inadmissible estimation subspaces. These refinements leverage geometric, algebraic, and optimization frameworks to supply correction terms or alternative formulations that more faithfully reflect achievable statistical accuracy.

1. Geometric Refinement of the Classical CRB

Matrix-level CRB refinements exploiting geometric properties begin by embedding the statistical model $\{f(\cdot;\theta)\}$ into the Hilbert space $L^2(\mu)$ via the square-root mapping $s_\theta = \sqrt{f(\cdot;\theta)}$. The tangent space at $\theta$ is generated by the first derivatives (jets) $\eta_1 = \partial_\theta s_\theta$, with higher-order jets $\eta_k = \partial_\theta^k s_\theta$ incorporating deeper model structure.

The classical scalar CRB is derived by orthogonal projection:

$$\text{Var}_\theta[T] = \|\text{Proj}_{\mathcal{T}_1} \tilde{Z}_0\|^2 + \|\mathcal{R}_1\|^2,\qquad \mathcal{T}_1 = \text{span}\{\eta_1\},$$

where $\tilde{Z}_0 = (T - \theta)s_\theta$ is the lifted error and $\mathcal{R}_1$ its orthogonal complement. The CRB corresponds to the variance attained by projecting the error onto $\mathcal{T}_1$. In nonlinear (curved) models, the residual $\mathcal{R}_1$ may be substantial, and is governed by the extrinsic model curvature—quantified by the second fundamental form:

$$II(\eta_1,\eta_1) = \eta_2 - \frac{\langle \eta_2, \eta_1\rangle}{\langle \eta_1, \eta_1\rangle}\eta_1.$$

As a result, the improved variance bound is given by

$$\text{Var}_\theta[T] \geq \frac{1}{\mathcal{J}(\theta)} + \frac{ \langle \tilde{Z}_0, II(\eta_1,\eta_1)\rangle^2 }{ \| II(\eta_1,\eta_1) \|^2 }.$$

This second term is strictly positive when the estimator error has a nonzero component in the normal (curvature) direction, thus quantifying the classical CRB’s underestimation in curved models (Krishnan, 22 Sep 2025).

2. Higher-Order and Multivariate Refinements

For models with higher-order structure or multivariate parameters, the jet space is expanded to include higher derivatives of $s_\theta$:

$$\mathcal{T}_m = \text{span}\{\eta_1, \ldots, \eta_m\}, \qquad \eta_k = \partial_\theta^k s_\theta, \quad k \geq 1.$$

Projection of the error onto $\mathcal{T}_m$ gives a hierarchy of Bhattacharyya-type bounds; curvature-aware corrections are then applied to the residuals in the normal space. The intricacy of these corrections is captured by the use of Faà di Bruno’s formula and complete exponential Bell polynomials:

$$\eta_k(\theta) = s_\theta \cdot B_k\left(\frac{1}{2}Y_1, \frac{1}{2}Y_2, \ldots, \frac{1}{2}Y_k\right),$$

where $Y_k$ are $k$-th derivatives of the log-density (raw scores).

For vector parameters $\theta\in\mathbb{R}^d$, the error covariance $\Sigma$ is classically lower bounded by $J^{-1}$, where $J$ is the Fisher information matrix. The refined bound incorporates the second fundamental form $\Pi(\partial_i, \partial_j)$ and, for any $v\in\mathbb{R}^d$,

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

with $Z_v$ the lifted error in direction $v$ and $\Pi_v$ an extrinsic curvature vector constructed from $\Pi$ with coordinate-corrected weights (Krishnan, 23 Sep 2025).

3. Matrix-Level Conservative Correction via SDP

In general, the directional corrections described above yield a family of quadratic inequalities that do not assemble into a single matrix correction in the absence of special structure. To resolve this, a semidefinite programming (SDP) approach is introduced: one seeks a symmetric matrix $\Delta\succeq 0$ such that

$\Sigma \succeq J^{-1} + \Delta$

and for all $v\in\mathbb{R}^d$,

$v^\top \Delta v \leq \mathcal{R}(v),$

where $\mathcal{R}(v)$ is the directional correction as above. By posing the difference

$$P_\Delta(v) := N(v)^2 - (v^\top \Delta v) D(v) \geq 0$$

(where $N(v)$ and $D(v)$ are numerators and denominators as per the explicit polynomial forms of the correction), a sufficient condition is obtained by requiring $P_\Delta(v)$ be a sum of squares. This is encoded via an SDP—an efficient computational certificate for the conservative correction (Krishnan, 23 Sep 2025).

4. Theoretical and Practical Implications

Curvature-aware corrections to the CRB are significant in practical statistical inference whenever the parametric model deviates from linearity or presents constraints that modify the local information geometry.

Key implications include:

• The refined CRB is always at least as large as the classical CRB, ensuring a conservative (i.e., pessimistic if not attained) but realistic lower bound on estimator variance.
• In models where the error is restricted to the tangent space (i.e., when estimators are "efficient" with respect to that geometry), the correction vanishes and the standard CRB is tight.
• The approach systematically quantifies the gap between attainable and classical bounds in models with parameter redundancy, singular Fisher information, or strong nonlinearity.
• The polynomial/SOS SDP technique for certifying conservative matrix corrections is broadly applicable and does not require closed-form solutions.

The approach also incorporates the use of higher-order derivatives and projective decompositions for multivariate models. Explicit examples, such as the “curved normal family” with parameter-dependent variance and curved Gaussian location models, illustrate that the curvature correction can be strictly positive and that the refined bound can be numerically and geometrically verified (Krishnan, 22 Sep 2025, Krishnan, 23 Sep 2025).

5. Concrete Example

In a bivariate Gaussian model with a nonlinear third coordinate, the extrinsic geometry approach yields a correction strictly improving on the classical CRB:

• For $X\sim\mathcal{N}(\mu(\theta),\sigma^2 I_3)$ with $$\mu(\theta) = (\theta_1,\theta_2, \alpha \theta_1^2)$$, tangent vectors at $\theta_1=0$ and the second fundamental form can be computed explicitly.
• For an unbiased estimator $T=(X_1, X_2 + \gamma (X_3-\mu_3))$, the only nonzero pairing in the correction term arises for $⟨\tilde{Z}^{(2)}, \Pi_{11}⟩ = \gamma\alpha$.
• The explicit corrected variance bound in direction $v$ becomes

$$v^\top (\Sigma - J^{-1}) v \geq \frac{ 16\sigma^4 v_2^2 v_1^4 (\gamma\alpha)^2 }{ 3(v_1^2 + v_2^2)^2 + 16\sigma^2\alpha^2 v_1^4 }$$

demonstrating tighter performance guarantees than the standard CRB (Krishnan, 23 Sep 2025).

6. Generalization and Future Directions

The extrinsic geometry-based refinement of the CRB opens further avenues:

• Systematic matrix-level corrections for non-exponential families, constrained estimation, or singular Fisher matrices.
• Computational methods (SDP/SOS) for verifying and implementing corrections in complex models.
• Impact assessment for estimator design and performance analysis, especially in high-dimensional or geometry-constrained statistical systems.

Such developments suggest a geometric perspective is essential for fully characterizing lower bounds in contemporary and high-dimensional statistical inference, offering both theoretical insight and practical design principles for estimator efficiency in nonlinear and multivariate settings.