Spectral divide-and-conquer MCMC for long stationary time series
Abstract
The temporal dependence inherent in time series models poses a fundamental challenge for distributed Bayesian inference, as it precludes embarrassingly parallel algorithms based on naive independence assumptions in the time domain. We propose a frequency-domain framework for scalable Bayesian inference in stationary time series that exploits the asymptotic independence underlying the Whittle likelihood. To exploit parallel computing resources, we develop a distributed fast Fourier transform and integrate it with embarrassingly parallel Markov chain Monte Carlo (MCMC) algorithms within a modern cluster-computing framework. This enables the analysis of time series that exceed the memory capacity of a single computational node or for which computation time is a bottleneck. The proposed methodology is compatible with a broad class of existing divide-and-conquer algorithms for independent data by applying them to frequency-domain rather than time-domain partitions. We establish that the error of the spectral divide-and-conquer MCMC posterior approximation relative to the exact time-domain posterior converges to zero in probability in a shrinking neighbourhood of the full-data Whittle posterior mode. The corresponding convergence rate is also derived. Across several experiments, we demonstrate that our approach provides accurate approximations to the full-data Whittle posterior. The proposed method is shown to outperform the current state-of-the-art time-domain divide-and-conquer methodology, particularly for highly persistent processes. The methodology is further illustrated by fitting a semi-long range model to a long meteorological time series.
Related
- Scalable Bayesian Inference for Time Series via Divide and Conquer
- Two moments suffice for Bessel approximation: conditioned differences, contingency tables, and probabilistic divide-and-conquer
- Divide-and-Conquer Neural Bayes Estimation for Spatial and Spatiotemporal Point Process Models
- Scaling Spatially Varying Distributed Lag Nonlinear Models for Nationwide Small-Area Data: A Divide-and-Conquer Approach
- On Detecting Multiple Simultaneous Change-points in High Dimensional Non-Stationary Time Series
BibTeX
@misc{wang2026spectral,
title = {{Spectral divide-and-conquer MCMC for long stationary time series}},
author = {Zixuan Wang and Matias Quiroz and Feng Li and Mattias Villani and Robert Kohn},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.23985},
eprint = {2609.23985},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.23985},
}
RIS
TY - UNPB
TI - Spectral divide-and-conquer MCMC for long stationary time series
AU - Zixuan Wang
AU - Matias Quiroz
AU - Feng Li
AU - Mattias Villani
AU - Robert Kohn
PY - 2026
DA - 2026/09/21
PB - arXiv
DO - 10.48550/arxiv.2609.23985
UR - https://arxiv.org/abs/2609.23985
AB - The temporal dependence inherent in time series models poses a fundamental challenge for distributed Bayesian inference, as it precludes embarrassingly parallel algorithms based on naive independence assumptions in the time domain. We propose a frequency-domain framework for scalable Bayesian inference in stationary time series that exploits the asymptotic independence underlying the Whittle likelihood. To exploit parallel computing resources, we develop a distributed fast Fourier transform and integrate it with embarrassingly parallel Markov chain Monte Carlo (MCMC) algorithms within a modern cluster-computing framework. This enables the analysis of time series that exceed the memory capacity of a single computational node or for which computation time is a bottleneck. The proposed methodology is compatible with a broad class of existing divide-and-conquer algorithms for independent data by applying them to frequency-domain rather than time-domain partitions. We establish that the error of the spectral divide-and-conquer MCMC posterior approximation relative to the exact time-domain posterior converges to zero in probability in a shrinking neighbourhood of the full-data Whittle posterior mode. The corresponding convergence rate is also derived. Across several experiments, we demonstrate that our approach provides accurate approximations to the full-data Whittle posterior. The proposed method is shown to outperform the current state-of-the-art time-domain divide-and-conquer methodology, particularly for highly persistent processes. The methodology is further illustrated by fitting a semi-long range model to a long meteorological time series.
ER -