Bayesian Analysis

Beyond Whittle: Nonparametric Correction of a Parametric Likelihood with a Focus on Bayesian Time Series Analysis

Claudia Kirch, Matthew C. Edwards, Alexander Meier, and Renate Meyer

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Nonparametric Bayesian inference has seen a rapid growth over the last decade but only few nonparametric Bayesian approaches to time series analysis have been developed. Most existing approaches use Whittle’s likelihood for Bayesian modelling of the spectral density as the main nonparametric characteristic of stationary time series. It is known that the loss of efficiency using Whittle’s likelihood can be substantial. On the other hand, parametric methods are more powerful than nonparametric methods if the observed time series is close to the considered model class but fail if the model is misspecified. Therefore, we suggest a nonparametric correction of a parametric likelihood that takes advantage of the efficiency of parametric models while mitigating sensitivities through a nonparametric amendment. We use a nonparametric Bernstein polynomial prior on the spectral density with weights induced by a Dirichlet process and prove posterior consistency for Gaussian stationary time series. Bayesian posterior computations are implemented via an MH-within-Gibbs sampler and the performance of the nonparametrically corrected likelihood for Gaussian time series is illustrated in a simulation study and in three astronomy applications, including estimating the spectral density of gravitational wave data from the Advanced Laser Interferometer Gravitational-wave Observatory (LIGO).

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Bayesian Anal., Volume 14, Number 4 (2019), 1037-1073.

First available in Project Euclid: 30 October 2018

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Bayesian nonparametrics frequency domain laser interferometric gravitational wave data spectral density stationary time series

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Kirch, Claudia; Edwards, Matthew C.; Meier, Alexander; Meyer, Renate. Beyond Whittle: Nonparametric Correction of a Parametric Likelihood with a Focus on Bayesian Time Series Analysis. Bayesian Anal. 14 (2019), no. 4, 1037--1073. doi:10.1214/18-BA1126.

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Supplemental materials

  • Supplementary material to ‘Beyond Whittle: Nonparametric Correction of a Parametric Likelihood with a Focus on Bayesian Time Series Analysis’. The electronic supplement contains the proofs for this paper, some comments on the Bayesian autoregressive sampler, the full conditional distribution of missing values as well as some additional simulation results.