Bayesian Tracking of a Diffusing Target in Two and Three Dimensions
Abstract
We study Bayesian tracking of a diffusing target monitored by a noisy distributed sensor array. Building on an earlier mapping to KPZ growth with a moving defect (or an equivalent directed polymer pinning problem) we determine the phase structure, beyond the previously-studied one-dimensional case, for both Bayes-optimal and suboptimal inference. In d=2, theoretical analysis and numerical simulations both give a depinning transition between a successful tracking phase and a failure phase. Weak-coupling RG shows that Bayes-optimal tracking is always successful in d=2, but failure can arise from overconfident (suboptimal) inference. In d=3, tracking can succeed, or can fail in two distinct ways: the posterior probability distribution may delocalize (no detection), or may become sharply localized, but at the wrong position (a false detection). The two possibilities correspond to Edwards-Wilkinson or Kardar-Parisi-Zhang statistics for the log-posterior. The three phases meet at a Nishimori-like multicritical point on a Bayes-optimal line in a two-parameter phase diagram. (Model misspecification alone can drive depinning into either unpinned phase: underconfidence gives diffuse failure, while overconfidence gives localized-but-wrong failure.) We analyze the transitions between the various phases numerically and with renormalization group arguments. We show that some of these have unusual critical behavior, which the conventional ε expansion fails to describe. Recent rigorous results for directed polymers indicate an alternative scenario. Many of our results, including a scaling relation for exponents at pinning transitions and results for RG flows, are relevant to other phase transitions that involve surface growth or directed polymers in 2+1D or 3+1D.
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BibTeX
@misc{mcculloch2026bayesian,
title = {{Bayesian Tracking of a Diffusing Target in Two and Three Dimensions}},
author = {Ewan McCulloch and Adam Nahum},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.00144},
eprint = {2609.00144},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.00144},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Bayesian Tracking of a Diffusing Target in Two and Three Dimensions
AU - Ewan McCulloch
AU - Adam Nahum
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.00144
UR - https://arxiv.org/abs/2609.00144
AB - We study Bayesian tracking of a diffusing target monitored by a noisy distributed sensor array. Building on an earlier mapping to KPZ growth with a moving defect (or an equivalent directed polymer pinning problem) we determine the phase structure, beyond the previously-studied one-dimensional case, for both Bayes-optimal and suboptimal inference. In d=2, theoretical analysis and numerical simulations both give a depinning transition between a successful tracking phase and a failure phase. Weak-coupling RG shows that Bayes-optimal tracking is always successful in d=2, but failure can arise from overconfident (suboptimal) inference. In d=3, tracking can succeed, or can fail in two distinct ways: the posterior probability distribution may delocalize (no detection), or may become sharply localized, but at the wrong position (a false detection). The two possibilities correspond to Edwards-Wilkinson or Kardar-Parisi-Zhang statistics for the log-posterior. The three phases meet at a Nishimori-like multicritical point on a Bayes-optimal line in a two-parameter phase diagram. (Model misspecification alone can drive depinning into either unpinned phase: underconfidence gives diffuse failure, while overconfidence gives localized-but-wrong failure.) We analyze the transitions between the various phases numerically and with renormalization group arguments. We show that some of these have unusual critical behavior, which the conventional ε expansion fails to describe. Recent rigorous results for directed polymers indicate an alternative scenario. Many of our results, including a scaling relation for exponents at pinning transitions and results for RG flows, are relevant to other phase transitions that involve surface growth or directed polymers in 2+1D or 3+1D.
ER -