fisher information analysis

This work proposes a novel experimental design framework for dynamic systems that addresses two key limitations of existing approaches: the neglect of process noise and the reliance on unknown true parameters for computing the Fisher information matrix (FIM). By integrating Bayesian averaging with an adaptive updating mechanism, the method jointly accounts for both process and measurement noise through Kalman filtering. The FIM is computed via Bayesian averaging over the parameter prior and is continuously updated in real time as new data become available, thereby optimizing subsequent experimental inputs. This approach achieves, for the first time, robust and real-time experimental design in linear dynamic systems with process noise without requiring knowledge of the true system parameters, significantly enhancing both the information efficiency and robustness of system identification.

This study investigates how artificial neural networks (ANNs) internally encode and transmit Fisher information during parameter estimation. We propose an information-aware training framework based on layer-wise monitoring of Fisher information flow. Through theoretical modeling and empirical analysis, we characterize the decay of parameter correlation information along both forward and backward propagation paths. Leveraging this insight, we devise a validation-free, model-agnostic early-stopping criterion: optimal estimation coincides with the peak of Fisher information transmission, while overfitting induces substantial information loss. This criterion provides an interpretable, quantifiable metric for training dynamics. Evaluated on experimental imaging physics data, our method enables real-time monitoring of training progression, significantly improving parameter estimation accuracy and model interpretability. The approach establishes a novel paradigm for AI-driven precision measurement systems.

This work addresses the computational intractability of the Fisher–Rao distance on statistical manifolds. Methodologically, it establishes the first systematic framework for computing tight, analytically tractable upper and lower bounds by integrating differential and information geometry—leveraging curvature constraints and parameterization invariance—and designing a low-complexity approximation algorithm via Taylor expansion and asymptotic analysis. Experimentally, the proposed bounds achieve over 40% improvement in tightness compared to state-of-the-art methods, while substantially reducing computational overhead. The contribution is twofold: (i) it advances the geometric understanding of the Fisher–Rao metric by revealing its curvature-dependent structural properties; and (ii) it delivers an efficient, robust metric tool applicable to statistical inference, model comparison, and generative learning—bridging theoretical insight with practical scalability.

This paper addresses the limitation in information geometry that the Fisher metric is defined only on the tangent bundle, hindering its direct connection to statistical quantities such as variance. It introduces, for the first time, an intrinsic definition of the Fisher cometric—the dual of the Fisher metric on the cotangent bundle—without relying on the original tangent-bundle Fisher metric. By establishing a natural correspondence between cotangent vectors and random variables, the variance/covariance structure is directly embedded into the cotangent space. Methodologically, the work integrates information geometry, statistical manifold theory, and invariance analysis under Markov morphisms. Key contributions include: (1) a cotangent-space analogue of the Čencov characterization theorem, simultaneously characterizing variance and covariance; (2) rendering the Cramér–Rao inequality a trivial consequence of the cometric structure; and (3) proving the invariance of the Fisher cometric under sufficient statistics and data processing, thereby providing a more fundamental geometric foundation for statistical inference.

This work addresses large-scale spatiotemporal systems with unknown or missing sensor models by proposing an inverse sensing architecture that synthesizes measurement likelihoods under prescribed accuracy constraints. The method minimizes information injection into the dynamic prior while ensuring the synthesized likelihood satisfies a specified error bound. Its core innovation lies in a unified maximum-entropy posterior framework for likelihood synthesis, which leverages relative entropy minimization and Radon–Nikodym derivatives to accommodate diverse discrepancy measures—including Wasserstein distance, maximum mean discrepancy (MMD), and f-divergences—and establishes a direct mapping between accuracy budgets and physical sensor configurations. Combining particle filtering with convex optimization, experiments validate the effectiveness of accuracy-constrained synthesis across four discrepancy measures, reveal how the choice of measure influences both the quantity and spatial distribution of injected information, and demonstrate successful distillation of nonparametric likelihoods into parametric forms.

This study clarifies a common misconception that Gaussian distributions always yield the largest Cramér-Rao bound (CRB). By leveraging Fisher information matrix theory and CRB analysis, together with carefully constructed counterexamples, it demonstrates for the first time that the Gaussian distribution maximizes the CRB only under restrictive conditions—specifically, when the mean and covariance parameters are decoupled, the parameters of interest reside solely in the mean vector, and no additive interference is present. Under more general settings, non-Gaussian distributions can produce strictly larger CRBs. This work challenges conventional wisdom by precisely delineating the narrow regime in which the Gaussian assumption guarantees maximal CRB, thereby providing rigorous theoretical guidance for distributional modeling in parametric estimation problems.

Classical point estimation theory suffers from fundamental limitations, including parameter dependence, undefined estimators at boundary points, the absence of unbiased estimators in certain settings, and a lack of rigorous theoretical grounding for maximum likelihood estimation. This work reframes estimators as functions on the parameter space and, adopting Fisher’s perspective of continuous null hypotheses within the Bahadur framework, introduces a generalized estimation mapping that coherently resolves these issues. By leveraging Hilbert space geometry and score function analysis, the approach elevates maximum likelihood to an exact result concerning the score function, naturally yielding the Cramér–Rao bound and sufficiency. It also provides valid estimates at boundary samples, demonstrates the nonexistence of uniformly minimum-variance unbiased estimators, and derives the Fisher information bound in a unified, streamlined manner—subsuming classical results as special cases.

This work investigates the impact of input distribution shift and parameter quantization on the spectral properties of the empirical Fisher Information Matrix (FIM), with a focus on the behavior of its largest eigenvalue. Leveraging Weyl’s inequality, assumptions from differential geometry, matrix perturbation theory, and statistical modeling, the authors derive two-sided theoretical bounds for the largest eigenvalue under structured perturbations and propose a computationally tractable approximation method with formal guarantees. Extensive experiments across 12 models and 1,080 training trajectories confirm that quantization substantially inflates the largest eigenvalue—reaching up to 244 times the full-precision baseline under 4-bit quantization—aligning closely with theoretical predictions and revealing the pronounced disruptive effect of low-bit quantization on the FIM’s spectral structure.

This study addresses the efficient and exact computation of the score function and observed Fisher information matrix in a Gaussian hidden Markov model (HMM) with Gaussian observation noise, where the latent state follows a Gaussian random walk. Leveraging Oakes’ identity in conjunction with the forward–backward algorithm, the authors derive, for the first time, a closed-form analytical expression for the observed Fisher information matrix and achieve linear-time exact computation of both the score function and the information matrix. This approach substantially enhances the efficiency and statistical accuracy of parameter estimation and enables rapid construction of confidence intervals. Experimental results across multiple simulated scenarios demonstrate that the proposed method reliably and efficiently performs parameter estimation, confirming its theoretical advantages and practical utility.