Computational aspects of Bayesian spectral density estimation

TL;DR

This paper introduces a computationally efficient Bayesian spectral density estimation method for Gaussian time-series models, combining approximate likelihoods with importance sampling and Sequential Monte Carlo to improve inference accuracy.

Contribution

It develops a novel fast approximation of the likelihood, a sampling scheme for the approximate posterior, and a Sequential Monte Carlo method to handle multi-modal distributions.

Findings

The variance of importance sampling weights diminishes with larger sample sizes.

The proposed method outperforms traditional approaches on simulated and real datasets.

Bayesian semi-parametric spectral density estimation yields more reasonable results than frequentist methods.

Abstract

Gaussian time-series models are often specified through their spectral density. Such models present several computational challenges, in particular because of the non-sparse nature of the covariance matrix. We derive a fast approximation of the likelihood for such models. We propose to sample from the approximate posterior (that is, the prior times the approximate likelihood), and then to recover the exact posterior through importance sampling. We show that the variance of the importance sampling weights vanishes as the sample size goes to infinity. We explain why the approximate posterior may typically multi-modal, and we derive a Sequential Monte Carlo sampler based on an annealing sequence in order to sample from that target distribution. Performance of the overall approach is evaluated on simulated and real datasets. In addition, for one real world dataset, we provide some numerical evidence that a Bayesian approach to semi-parametric estimation of spectral density may provide more reasonable results than its Frequentist counter-parts.