Bayesian Inference in Partially Observed Critical Systems
Abstract
We ask how much of a large random system must be observed before a global average can be inferred with controlled uncertainty. For the random transposition shuffle, a conjugate Bayesian estimator of the average displacement, computed from m uniformly sampled cards, has relative uncertainty 1/√(2m), independent of the deck size n. The law of total variance exposes a second error, the floor Rn, which arises because the observed configuration is a single draw from equilibrium: no relative tolerance below κ√(Rn) can be certified, even by a census. For the shuffle we derive Rn=(2n2+7)/(5(n−1)2(n+1)) and show that the posterior remains calibrated. At the critical point of the two-dimensional Ising model the floor of the energy vanishes, whereas that of the magnetisation stays near $0.072$; its mean-field limit is √2−1. For correlated fields read by sensor networks, the usual 95% interval covers the ensemble mean in 7% of cases. Bounding the floor from one realisation, as in single-chain Monte Carlo diagnostics, requires an assumption on the decay of correlations.
Related
- PAC-Bayesian Bounds for Learning Partially Observed Stochastic Linear Time-Invariant State-Space Systems with Inputs and Sub-Gaussian Noise
- A Stochastic Maximum Principle for Partially Observed Jump-Diffusion Systems with State-Dependent Counting-Process Observations
- Defense Filters for Robust Learning of Partially Observed Dynamical Systems
- Modular Bayesian Inference for Partially Misspecified Mixtures with Uncertain Class Labels
- Causal Fingerprints Reveal Hidden Mechanisms in Partially-Observed Dynamical Systems
BibTeX
@misc{singhania2026bayesian,
title = {{Bayesian Inference in Partially Observed Critical Systems}},
author = {Mihika Singhania},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.35973},
eprint = {2610.35973},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.35973},
}
RIS
TY - UNPB
TI - Bayesian Inference in Partially Observed Critical Systems
AU - Mihika Singhania
PY - 2026
DA - 2026/10/28
PB - arXiv
DO - 10.48550/arxiv.2610.35973
UR - https://arxiv.org/abs/2610.35973
AB - We ask how much of a large random system must be observed before a global average can be inferred with controlled uncertainty. For the random transposition shuffle, a conjugate Bayesian estimator of the average displacement, computed from m uniformly sampled cards, has relative uncertainty 1/√(2m), independent of the deck size n. The law of total variance exposes a second error, the floor R_n, which arises because the observed configuration is a single draw from equilibrium: no relative tolerance below κ√(R_n) can be certified, even by a census. For the shuffle we derive R_n=(2n^2+7)/(5(n−1)^2(n+1)) and show that the posterior remains calibrated. At the critical point of the two-dimensional Ising model the floor of the energy vanishes, whereas that of the magnetisation stays near $0.072$; its mean-field limit is √2−1. For correlated fields read by sensor networks, the usual 95% interval covers the ensemble mean in 7% of cases. Bounding the floor from one realisation, as in single-chain Monte Carlo diagnostics, requires an assumption on the decay of correlations.
ER -