Spectral subsampling MCMC for Lévy-driven continuous-time ARMA models with expensive likelihood contributions

2026-05-28

AI Summary

This study addresses the computational burden of likelihood evaluation in traditional MCMC methods for Lévy-driven continuous-time ARMA models with discrete observations, where aliasing in the frequency domain renders exact likelihood calculations prohibitively expensive. The authors propose a spectral subsampling MCMC approach that combines Whittle likelihood approximation with control variates to efficiently estimate the log-likelihood, substantially reducing computational cost. This work represents the first successful application of subsampling MCMC to Bayesian inference in the frequency domain for continuous-time processes affected by aliasing, overcoming the limitation of existing subsampling methods that are restricted to settings with cheap likelihood evaluations. The proposed method achieves significant acceleration while preserving inferential accuracy, demonstrating its efficiency and practicality in scenarios involving expensive likelihood computations.

Abstract

Subsampling-based Markov chain Monte Carlo (MCMC) algorithms aim to accelerate Bayesian inference by evaluating the likelihood using only a subset of the data at each iteration. However, in many standard tall-data applications, individual likelihood contributions are inexpensive to evaluate and the resulting reductions in actual computing time are often substantially smaller than the nominal reduction in data size due to computational overhead. We study a different computational regime arising in frequency-domain inference for continuous-time processes observed at equally spaced discrete time points. This gives rise to aliasing, whereby each contribution to the Whittle likelihood requires summation over shifted frequency components, unlike standard discrete-time spectral settings where spectral evaluations do not require such summation. We demonstrate that this structure makes subsampling MCMC, a subsampling-based MCMC approach that estimates the log-likelihood using data subsampling and efficient control variates, particularly effective for reducing computational cost. We illustrate the approach for Bayesian frequency-domain inference in discretely observed continuous-time autoregressive moving average models driven by finite second-moment Lévy processes.

Problem

• subsampling MCMC
• Lévy-driven CARMA
• Whittle likelihood
• frequency-domain inference
• aliasing

Innovation

• subsampling MCMC
• spectral inference
• Lévy-driven CARMA
• Whittle likelihood
• aliasing