Identifiability, Fisher Information, and Amortized Inference for Heterogeneous Diffusion from Discrete-Time Noisy Observations

Zhen Yuan Yeo
Proceedings of The 1st Symposium on Probabilistic Machine Learning, PMLR 327:290-313, 2026.

Abstract

Estimating diffusion coefficients from discrete-time noisy particle trajectories is a core problem in single-particle tracking , yet its statistical foundations remain incomplete. We establish exact identifiability structure for single-species and heterogeneous-population models, derive closed-form Fisher information bounds, and show how these results jointly define a feasibility phase diagram over the normalized diffusion scale $ \alpha = 2D\Delta t/\sigma^2$ and effective particle occupancy $ \beta = \rho\sigma^2$. A key finding is that one-step increment distributions alone cannot separate the diffusion coefficient from localization noise, but temporal autocorrelation structure resolves this ambiguity without additional calibration. For heterogeneous populations, identifiability holds under a variance separation condition, and Fisher information for rare components degrades quadratically with mixture weight, setting a fundamental limit on what unlabeled trajectory data can recover. We use these theoretical results to derive an amortized inference architecture: the time-averaging aggregation, factored posterior parameterization , and pretraining objective each follow directly from the theory.

Cite this Paper

BibTeX
@InProceedings{pmlr-v327-yeo26a, title = {Identifiability, Fisher Information, and Amortized Inference for Heterogeneous Diffusion from Discrete-Time Noisy Observations}, author = {Yeo, Zhen Yuan}, booktitle = {Proceedings of The 1st Symposium on Probabilistic Machine Learning}, pages = {290--313}, year = {2026}, editor = {Swaroop, Siddharth and Rügamer, David and Kristiadi, Agustinus}, volume = {327}, series = {Proceedings of Machine Learning Research}, month = {05 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v327/main/assets/yeo26a/yeo26a.pdf}, url = {https://proceedings.mlr.press/v327/yeo26a.html}, abstract = { Estimating diffusion coefficients from discrete-time noisy particle trajectories is a core problem in single-particle tracking , yet its statistical foundations remain incomplete. We establish exact identifiability structure for single-species and heterogeneous-population models, derive closed-form Fisher information bounds, and show how these results jointly define a feasibility phase diagram over the normalized diffusion scale $ \alpha = 2D\Delta t/\sigma^2$ and effective particle occupancy $ \beta = \rho\sigma^2$. A key finding is that one-step increment distributions alone cannot separate the diffusion coefficient from localization noise, but temporal autocorrelation structure resolves this ambiguity without additional calibration. For heterogeneous populations, identifiability holds under a variance separation condition, and Fisher information for rare components degrades quadratically with mixture weight, setting a fundamental limit on what unlabeled trajectory data can recover. We use these theoretical results to derive an amortized inference architecture: the time-averaging aggregation, factored posterior parameterization , and pretraining objective each follow directly from the theory. } }
Endnote
%0 Conference Paper %T Identifiability, Fisher Information, and Amortized Inference for Heterogeneous Diffusion from Discrete-Time Noisy Observations %A Zhen Yuan Yeo %B Proceedings of The 1st Symposium on Probabilistic Machine Learning %C Proceedings of Machine Learning Research %D 2026 %E Siddharth Swaroop %E David Rügamer %E Agustinus Kristiadi %F pmlr-v327-yeo26a %I PMLR %P 290--313 %U https://proceedings.mlr.press/v327/yeo26a.html %V 327 %X Estimating diffusion coefficients from discrete-time noisy particle trajectories is a core problem in single-particle tracking , yet its statistical foundations remain incomplete. We establish exact identifiability structure for single-species and heterogeneous-population models, derive closed-form Fisher information bounds, and show how these results jointly define a feasibility phase diagram over the normalized diffusion scale $ \alpha = 2D\Delta t/\sigma^2$ and effective particle occupancy $ \beta = \rho\sigma^2$. A key finding is that one-step increment distributions alone cannot separate the diffusion coefficient from localization noise, but temporal autocorrelation structure resolves this ambiguity without additional calibration. For heterogeneous populations, identifiability holds under a variance separation condition, and Fisher information for rare components degrades quadratically with mixture weight, setting a fundamental limit on what unlabeled trajectory data can recover. We use these theoretical results to derive an amortized inference architecture: the time-averaging aggregation, factored posterior parameterization , and pretraining objective each follow directly from the theory.
APA
Yeo, Z.Y.. (2026). Identifiability, Fisher Information, and Amortized Inference for Heterogeneous Diffusion from Discrete-Time Noisy Observations. Proceedings of The 1st Symposium on Probabilistic Machine Learning, in Proceedings of Machine Learning Research 327:290-313 Available from https://proceedings.mlr.press/v327/yeo26a.html.

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