In multi-target tracking (MTT), the model parameters of a multi-object dynamical system must be learned from noisy observations whose origin among the targets is unknown, and where targets may be missed (detection failures) and spurious false alarms occur. Houssineau, Singh and Jasra (2019) established identifiability, consistency and asymptotic normality of the maximum likelihood estimator of the fixed-dimensional MTT model parameters, and showed through the Fisher information matrix that unknown data association and detection failures cause a strict loss of information — but their likelihood marginalizes over all possible observation-to-track associations. In their conclusion they explicitly left open the association-aware counterpart: 'the study of identifiability of specific observation-to-track associations, instead of marginalising over all possibilities', noting that 'the parameters to be learned increase in dimensionality with time, so that it is not a special case of the results presented here'. This record states that open problem: characterize when specific observation-to-track associations are identifiable in the association-aware formulation, in which the time-varying association variables are treated as unknowns estimated jointly with the model parameters, delineating the regimes in which association uncertainty is resolvable from those in which it is not.

Background

Multi-target tracking (MTT) is the problem of estimating the time-varying states of an unknown and possibly changing number of moving objects (targets) from a sequence of noisy sensor measurements, some of which originate from no target at all (false alarms, usually modeled as a Poisson clutter process), and some of which correspond to a target that was detected with some probability of detection pDp_D. The combinatorial difficulty at the heart of MTT is data association: deciding, at each time step, which observation (if any) originates from which target. A standard statistical formulation, used by Houssineau, Singh and Jasra, treats the multi-object system as a collection of partially observed Markov processes with dynamics and observation laws parameterized by a fixed-dimensional parameter θ\theta (shared, e.g., across targets), with detection failures, observation noise, and false alarms; the likelihood of the observed data is then obtained by marginalizing — summing — over the finite set of all admissible observation-to-track association sequences. Identifiability means that distinct parameter values induce distinct distributions of the observed data; consistency means that the maximum likelihood estimator (MLE) converges to the true parameter as the amount of data grows; the Fisher information matrix characterizes the asymptotic variance of a consistent MLE via the Cramér–Rao lower bound and asymptotic normality. Within this framework, Houssineau, Singh and Jasra proved that identifiability transfers from the single-target to the multi-target model, that the MLE of θ\theta is consistent and asymptotically normal under verifiable conditions on the single-target dynamics and observations, and (their Theorem 4) that data-association uncertainty and detection failures induce a strictly positive loss of Fisher information, quantified in special cases such as a single static target with false alarms and fully unknown association. This marginalized treatment is statistically clean but does not estimate the association itself, which is often the object of practical interest: the association pattern is what identifies object trajectories and identities (as in multiple-hypothesis tracking practice), and Bayesian computational methods for joint tracking and parameter learning already exist (e.g., Jiang, Singh and Yıldırım, 2015) without such guarantees. The closest prior result on the association side is due to Storlie, Hannig and Lee (2011), who proved almost-sure consistency of a maximum likelihood data-association estimate — but in a different model class (Gaussian-process tracks with splitting and merging, and a growing number of observations over a fixed time interval) with no shared dynamical parameter θ\theta, so their technique does not transfer. The question explicitly left open by the 2019 paper is the identifiability of specific observation-to-track associations in the association-aware formulation, in which the time-varying assignment variables are treated as unknowns estimated jointly with θ\theta rather than marginalized over. This is delicate because the number of association variables grows with the number of observations, so the total parameter dimension grows with the sample size, which breaks the fixed-dimensional asymptotics on which the 2019 results (and the classical hidden-Markov-model MLE theory they build on) rest; the problem is essentially one of asymptotic statistics with increasing-dimensional parameters over a combinatorially growing discrete space.

Problem Statement

In the multi-object dynamical system framework of Houssineau, Singh and Jasra (2019) — an unknown, time-varying number of targets, each a partially observed Markov process, with dynamics and observation laws depending on a fixed-dimensional parameter θ\theta, in the presence of detection failures, observation noise, and Poisson false alarms — consider the association-aware formulation in which the time-varying observation-to-track association variables are treated as unknowns and estimated jointly with the model parameters, rather than marginalized over as in that paper. Give conditions under which specific observation-to-track associations are identifiable in this formulation (separately from, or jointly with, θ\theta), and characterize the regimes — e.g., in terms of target separation relative to the observation noise, detection probability, and false-alarm rate — in which association uncertainty is resolvable or unresolvable. As the source paper notes, the association-aware approach involves additional challenges because the association variables to be learned increase in dimensionality with the observations, so this question is not a special case of that paper's fixed-dimensional results. Consistency (and asymptotic normality) of joint estimation in this growing-dimensional regime, and an association-aware Fisher-information analysis, are related directions suggested by the source's results for the marginalized model, but the open question explicitly stated by the source — and the target here — is the identifiability of specific observation-to-track associations.

References

  1. Jérémie Houssineau, Sumeetpal S. Singh, Ajay Jasra, "Identification of MultiObject Dynamical Systems: Consistency and Fisher Information", SIAM Journal on Control and Optimization 57(4):2603–2627 (2019). DOI: 10.1137/17m113873x, https://doi.org/10.1137/17m113873x (arXiv:1707.04371, https://arxiv.org/abs/1707.04371)
  2. Curtis B. Storlie, Jan Hannig, Thomas C.M. Lee, "Statistical consistency of the data association problem in multiple target tracking", Electronic Journal of Statistics 5:1227–1275 (2011). DOI: 10.1214/11-ejs639, https://doi.org/10.1214/11-ejs639
  3. Jérémie Houssineau, Jiajie Zeng, Ajay Jasra, "Uncertainty modelling and computational aspects of data association", Statistics and Computing 31(5), article 59 (2021). DOI: 10.1007/s11222-021-10039-1, https://doi.org/10.1007/s11222-021-10039-1 (arXiv:2009.02517, https://arxiv.org/abs/2009.02517)
  4. Jérémie Houssineau, Ajay Jasra, Sumeetpal S. Singh, "On the loss of Fisher information in some multi-object tracking observation models" (2018). arXiv:1803.09496, https://arxiv.org/abs/1803.09496
  5. Jérémie Houssineau, Neil K. Chada, Emmanuel Delande, "Elements of asymptotic theory with outer probability measures" (2019). arXiv:1908.04331, https://arxiv.org/abs/1908.04331
  6. A. Marie d'Avigneau, Sumeetpal S. Singh, Raimund J. Ober, "Limits of Accuracy for Parameter Estimation and Localization in Single-Molecule Microscopy via Sequential Monte Carlo Methods", SIAM Journal on Imaging Sciences 15(1):139–171 (2022). DOI: 10.1137/21m1422823, https://doi.org/10.1137/21m1422823
  7. Daniel E. Clark, "Kullback's inequality and Cramer Rao bounds for point process models", TechRxiv preprint (2022). DOI: 10.36227/techrxiv.19085531, https://doi.org/10.36227/techrxiv.19085531
  8. M. Phil Lowney, Yaakov Bar-Shalom, Tod E. Luginbuhl, Peter Willett, "Target Motion Analysis in Clutter With ML-PDA and ML-PMHT Using Partial Prior Information", IEEE Transactions on Aerospace and Electronic Systems 62:1092–1106 (2026). DOI: 10.1109/TAES.2025.3628313, https://doi.org/10.1109/TAES.2025.3628313
  9. Lan Jiang, Sumeetpal S. Singh, Sinan Yildirim, "Bayesian Tracking and Parameter Learning for Non-Linear Multiple Target Tracking Models", IEEE Transactions on Signal Processing 63(21):5733–5745 (2015). DOI: 10.1109/TSP.2015.2454474, https://doi.org/10.1109/TSP.2015.2454474