Fisher-Tracking: Estimation & Inverse Tracking

• Fisher-Tracking Strategy is an estimation-driven method that uses the Fisher Information Matrix to optimize tracking, parameter inversion, and sensor placement.
• It reverses conventional estimation flows by exploiting tracking outputs to deduce observer characteristics through nonlinear least-squares optimization and geometry-driven initialization.
• The approach finds applications in sonar tracking, adaptive experimental design, quantum metrology, and financial modeling, demonstrating near-optimal performance under theoretical bounds.

A Fisher-Tracking Strategy is an estimation-driven methodology that leverages Fisher information to optimize, interpret, or invert tracking and parameter estimation workflows. Originally developed in the context of inverse problems in passive sonar target motion analysis, the Fisher-tracking approach reverses the conventional direction of information flow: rather than using Fisher information solely to bound the accuracy of a tracking system, it can also be exploited to deduce properties of an observer or measurement platform from estimation outputs. More broadly, Fisher-tracking strategies appear in sensor placement, adaptive testing, sensor network control, quantum metrology, and information-theoretic performance benchmarking, unified by the role of Fisher information as a central quantitative metric for estimation precision and observability.

1. Fundamental Principles of Fisher-Tracking

The core of a Fisher-Tracking Strategy is the computation and use of the Fisher Information Matrix (FIM), which quantifies the sensitivity of the likelihood (or probability) of observed data with respect to underlying parameters of interest, such as target position, platform trajectory, or system state. The FIM,

$$J(\theta) = \mathbb{E}\left[ \left( \frac{\partial}{\partial\theta} \log p(y;\theta) \right) \left( \frac{\partial}{\partial\theta} \log p(y;\theta) \right)^\top \right]$$

sets the ultimate accuracy limit via the Cramér–Rao lower bound (CRLB) for unbiased estimators, as $\text{Cov}(\hat{\theta}) \succeq J(\theta)^{-1}$.

In tracking applications, Fisher-tracking uses this structure not only as a performance metric but as a signature that encodes the interplay between system geometry, measurement model, and even adversarial observability. For instance, in the bearings-only tracking problem, the FIM depends intricately on both the target trajectory and the (possibly unknown) observer platform trajectory, allowing for inversion if the FIM and target trajectory estimates are intercepted (Ciuonzo et al., 2013).

Key mathematical properties:

• Additivity: Cumulative Fisher information across independent measurements is additive.
• Transformation behavior: Sufficient statistics preserve Fisher information; nonsufficient or stochastic transformations reduce it.
• Data-processing inequality: Stochastic mappings or many-to-one reductions do not increase the Fisher information.

2. Inverse Tracking and Observability via FIM

A prototypical Fisher-Tracking Strategy is analyzed in depth for passive bearings-only sonar using maximum likelihood probabilistic data association (ML-PDA) (Ciuonzo et al., 2013). Here, the FIM is defined for the collection of angle measurements as a function of both the target state $x_t$ and observer (platform) state $x_p$:

$$J(x_t, x_p, \alpha_{\theta}) = \alpha_{\theta} \sum_{i=1}^{n} [\nabla_{x_t} \theta_i]^{}[\nabla_{x_t} \theta_i]^\top,$$

with $\alpha_{\theta}$ encapsulating measurement noise variance and information reduction due to clutter/missed detections.

The Fisher-tracking strategy inverts this mapping: An eavesdropper who intercepts estimated target state and FIM (from the ML-PDA output) seeks to solve for the platform trajectory $x_p$ by minimizing

$$\mathcal{F}(x_p^s, \alpha_{\theta}) = \|J^{\text{obs}} - J(x_p^s, \alpha_{\theta})\|_F^2,$$

where $x_p^s$ is a reduced-dimension parameterization of the platform's path. This is cast as a nonlinear least-squares optimization, boosted by a “geometry-driven” initialization that leverages FIM structure. Fundamental results about observability include:

• Under two-leg, constant-speed motion and certain geometric requirements (specifically, change in platform heading not orthogonal to the target’s velocity), the observer's trajectory is locally observable from the FIM and ML-PDA output.
• If the two platform legs have arbitrary speeds, there exists an affine subspace of indistinguishable solutions, resulting in unobservability for the inverse problem unless additional constraints are imposed.

This inversion demonstrates that Fisher information can act both as a bound on estimation precision and as a signature that allows “tracking the tracker.”

3. Applications in Sensing and Sensor Placement

Beyond inverse tracking, Fisher-tracking strategies are widely used to direct resource allocation and sensor placement for optimal tracking performance. In trajectory and sensor path optimization (Kirchner et al., 2023), the FIM accumulated along the sensor path,

$$\text{CFIM}(t, x, u(\cdot)) = Q_0 + \int_0^t Q(\gamma(s; x, u(\cdot)))\, ds,$$

serves as the performance objective, with its log-determinant minimized (D-optimal design) to reduce the volume of the estimation error ellipse. The metric guides both path planning and active sensor placement by quantifying the expected marginal gain in information for different actions, trajectories, or sensor configurations:

• Hamilton–Jacobi partial differential equations are used to solve for the optimal path, circumventing computational intractability by a hybrid grid/ODE approach that exploits the additive nature of information gain (Kirchner et al., 2023).
• In controlled sensing for discrete-state Markov systems, Fisher-tracking selects sensing actions that, at each time step, maximize a generalized Fisher information measure tailored to the discrete-observation model (Zois et al., 2014).

4. Adaptive Query Selection and Experimental Design

Fisher-tracking is directly operationalized in adaptive experimental design and active learning. In sequential binary-response adaptive testing (Kim et al., 9 Oct 2025), the Fisher-tracking strategy (FIT-Q) adaptively queries the subject by selecting, at each time $t$,

$$X_t = \operatorname{proj}_\mathcal{X}(\hat{\theta}_{t-1} - z_*),$$

where $z_*$ maximizes local Fisher information for the probability model $f(\theta_* - x)$, and $\hat{\theta}_{t-1}$ is the current method-of-moments estimator. This strategy is shown to be optimal in both fixed-confidence and fixed-budget regimes, achieving sample complexity and error probability that saturate the information-theoretic lower bounds dictated by the Fisher information at the optimal query. Mathematical structure:

• Test statistics for stopping and confidence certification are constructed using cumulant-generating function bounds and are computable in closed-form at just two points $\hat{\theta}_t \pm \epsilon$, ensuring computational tractability (Kim et al., 9 Oct 2025).

5. Information Loss and Observability in Multi-Object Tracking

Fisher-tracking principles are central to understanding limitations imposed by sensor imperfections, stochastic data association, and information bottlenecks in multi-object tracking. A rigorous framework shows:

• Random permutation of associations, independent thinning (random loss of objects), and superposition with clutter independently reduce Fisher information, as captured by

$$I'(\theta) \leq I(\theta) \qquad \text{(permuted or thinned)}$$

where $I'(\theta)$ is the Fisher information after randomization and $I(\theta)$ before (Houssineau et al., 2018).

• The additivity and decomposability of Fisher information enable precise accounting of how much information is lost at each processing or sensing stage, aiding in the design and analysis of robust tracking algorithms.

6. Extensions: Quantum Tracking and Market Applications

In quantum sensing, the Fisher-tracking strategy is embodied in monitoring the quantum Fisher information (QFI) during interaction with structured environments. The evolution of QFI in non-Markovian spin–boson systems reveals “information backflow” from the environment, nontrivial phase estimation behaviors, and sudden changes associated with entanglement decay (Hao et al., 2013). In quantum metrology, protocol adjustment based on real-time QFI constitutes a Fisher-tracking control mechanism for achieving and maintaining sub-shot-noise-limited estimation.

In financial markets, the principle of minimum Fisher information is invoked for strategy design: the market dynamics are modeled by variational minimization of Fisher information under a fixed risk, with quantized optimal strategies corresponding to harmonic oscillator eigenstates and their quantum superpositions (Makowski et al., 2022).

7. Simulation Studies and Real-World Performance

Simulation studies across domains (sonar tracking, sensor networks, molecular imaging) confirm that Fisher-tracking strategies, when implemented with robust estimators and proper initialization, achieve near-optimal performance as predicted by Cramér–Rao bounds:

• ML-PDA inversion with FIM matching converges to true platform trajectories even under low SNR, given suitable observability and initialization (Ciuonzo et al., 2013).
• Sensor placement strategies based on maximizing cumulative FIM minimize estimation error volumes in nonlinear, nonconvex environments (Kirchner et al., 2023).
• Adaptive testing algorithms using Fisher tracking minimize the number of queries required for a prescribed confidence margin, outperforming standard fixed-design procedures (Kim et al., 9 Oct 2025).

These results demonstrate that Fisher-tracking strategies are not only theoretically grounded but also practically effective across disparate tracking and estimation problems.

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In summary, the Fisher-Tracking Strategy is characterized by the exploitation of Fisher information—as both a constraint and a design criterion—for inversion, observation, control, and resource allocation in tracking and estimation problems. Its mathematical underpinnings span from optimal experiment design to information-theoretic observability and adaptive learning, highlighting the central role of Fisher information in both classical and emerging tracking paradigms.