Bayesian modelling of ARFIMA processes by Markov chain Monte Carlo methods

Abstract

This article describes Bayesian inference for autoregressive fractionally integrated moving average (ARFIMA) models using Markov chain Monte Carlo methods. The posterior distribution of the model parameters, corresponding to the exact likelihood function is obtained through the partial linear regression coefficients of the ARFIMA process. A Metropolis-Rao-Blackwellizallization approach is used for implementing sampling-based Bayesian inference. Bayesian model selection is discussed and implemented.

Bayesian Inference for ARFIMA Models

“…The value of the p.d.f at d = 0 is 2.16, which coincides (as it must, because the support of d has measure 1) with the posterior odds ratio in favor of short memory. These findings are consistent with those of Koop et al () and Pai and Ravishanker (), which studied similar data.…”

“…It reflects the fact that as the long memory parameter d increases, and especially as it approaches the upper limit of stationarity d = 1/2, there is less information about the mean, which fails to exist at d = 1/2. This phenomenon is driven by the likelihood function and has been noted previously, for example, Pai and Ravishanker (). The posterior correlation between d and μ is −0.29: as d moves from 0.15 to 0.46, the conditional posterior mean of μ decreases substantially, from 0.033 to 0.012.…”

“…sample from the prior distribution, for evaluating the prior distribution at specified parameter values (the particles), and for evaluating the log likelihood function at specified parameter values. There are no ancillary computations analogous to the evaluation of derivatives in maximum likelihood (ML) (Sowell, ), creating and sampling from conditional posterior distribution in Gibbs sampling for Bayesian inference (Chib and Greenberg, ), or design of proposal distributions for importance sampling (Koop et al , ) or Metropolis chains (Pai and Ravishanker, ). Consequently there is often a great reduction in time devoted to tuning, tinkering, trial and error compared with alternative approaches. SABL approximates the marginal likelihood as a byproduct, together with a numerical standard error (NSE) of approximation just as it does for posterior moments.…”

Bayesian Inference for ARFIMA Models

“…The value of the p.d.f at d = 0 is 2.16, which coincides (as it must, because the support of d has measure 1) with the posterior odds ratio in favor of short memory. These findings are consistent with those of Koop et al () and Pai and Ravishanker (), which studied similar data.…”

“…It reflects the fact that as the long memory parameter d increases, and especially as it approaches the upper limit of stationarity d = 1/2, there is less information about the mean, which fails to exist at d = 1/2. This phenomenon is driven by the likelihood function and has been noted previously, for example, Pai and Ravishanker (). The posterior correlation between d and μ is −0.29: as d moves from 0.15 to 0.46, the conditional posterior mean of μ decreases substantially, from 0.033 to 0.012.…”

“…sample from the prior distribution, for evaluating the prior distribution at specified parameter values (the particles), and for evaluating the log likelihood function at specified parameter values. There are no ancillary computations analogous to the evaluation of derivatives in maximum likelihood (ML) (Sowell, ), creating and sampling from conditional posterior distribution in Gibbs sampling for Bayesian inference (Chib and Greenberg, ), or design of proposal distributions for importance sampling (Koop et al , ) or Metropolis chains (Pai and Ravishanker, ). Consequently there is often a great reduction in time devoted to tuning, tinkering, trial and error compared with alternative approaches. SABL approximates the marginal likelihood as a byproduct, together with a numerical standard error (NSE) of approximation just as it does for posterior moments.…”

Bayesian Wavelet-Based Methods for the Detection of Multiple Changes of the Long Memory Parameter

“…Our inferential procedure, however, could be generalized to include inference on other model parameters, such as the noise variance and the process mean, either by implementing a full Gibbs or by using a Rao-Blackwellization procedure. This would also make the investigation of prediction performances of the estimated models possible, for example, by approximating the predictive distribution via Monte Carlo integration using the output from the MCMC sampler; see, for example, Pai and Ravishanker [21], [22]. This is currently under investigation.…”

A review of predictive uncertainty estimation with machine learning

“…Autoregressive fractionally integrated moving average (ARFIMA) models: Bayesian modelling of ARFIMA models using non-informative priors has been done by Pai and Ravishanker (1996), while Durham et al (2019) propose solutions for large time series.…”

Bayesian Inference for ARFIMA Models

“…The value of the p.d.f at d = 0 is 2.16, which coincides (as it must, because the support of d has measure 1) with the posterior odds ratio in favor of short memory. These findings are consistent with those of Koop et al () and Pai and Ravishanker (), which studied similar data.…”

“…It reflects the fact that as the long memory parameter d increases, and especially as it approaches the upper limit of stationarity d = 1/2, there is less information about the mean, which fails to exist at d = 1/2. This phenomenon is driven by the likelihood function and has been noted previously, for example, Pai and Ravishanker (). The posterior correlation between d and μ is −0.29: as d moves from 0.15 to 0.46, the conditional posterior mean of μ decreases substantially, from 0.033 to 0.012.…”

“…sample from the prior distribution, for evaluating the prior distribution at specified parameter values (the particles), and for evaluating the log likelihood function at specified parameter values. There are no ancillary computations analogous to the evaluation of derivatives in maximum likelihood (ML) (Sowell, ), creating and sampling from conditional posterior distribution in Gibbs sampling for Bayesian inference (Chib and Greenberg, ), or design of proposal distributions for importance sampling (Koop et al , ) or Metropolis chains (Pai and Ravishanker, ). Consequently there is often a great reduction in time devoted to tuning, tinkering, trial and error compared with alternative approaches. SABL approximates the marginal likelihood as a byproduct, together with a numerical standard error (NSE) of approximation just as it does for posterior moments.…”

Bayesian Wavelet-Based Methods for the Detection of Multiple Changes of the Long Memory Parameter

“…Our inferential procedure, however, could be generalized to include inference on other model parameters, such as the noise variance and the process mean, either by implementing a full Gibbs or by using a Rao-Blackwellization procedure. This would also make the investigation of prediction performances of the estimated models possible, for example, by approximating the predictive distribution via Monte Carlo integration using the output from the MCMC sampler; see, for example, Pai and Ravishanker [21], [22]. This is currently under investigation.…”

A review of predictive uncertainty estimation with machine learning

“…Autoregressive fractionally integrated moving average (ARFIMA) models: Bayesian modelling of ARFIMA models using non-informative priors has been done by Pai and Ravishanker (1996), while Durham et al (2019) propose solutions for large time series.…”

Bayesian Inference for ARFIMA Models

“…The value of the p.d.f at d = 0 is 2.16, which coincides (as it must, because the support of d has measure 1) with the posterior odds ratio in favor of short memory. These findings are consistent with those of Koop et al () and Pai and Ravishanker (), which studied similar data.…”

“…It reflects the fact that as the long memory parameter d increases, and especially as it approaches the upper limit of stationarity d = 1/2, there is less information about the mean, which fails to exist at d = 1/2. This phenomenon is driven by the likelihood function and has been noted previously, for example, Pai and Ravishanker (). The posterior correlation between d and μ is −0.29: as d moves from 0.15 to 0.46, the conditional posterior mean of μ decreases substantially, from 0.033 to 0.012.…”

“…sample from the prior distribution, for evaluating the prior distribution at specified parameter values (the particles), and for evaluating the log likelihood function at specified parameter values. There are no ancillary computations analogous to the evaluation of derivatives in maximum likelihood (ML) (Sowell, ), creating and sampling from conditional posterior distribution in Gibbs sampling for Bayesian inference (Chib and Greenberg, ), or design of proposal distributions for importance sampling (Koop et al , ) or Metropolis chains (Pai and Ravishanker, ). Consequently there is often a great reduction in time devoted to tuning, tinkering, trial and error compared with alternative approaches. SABL approximates the marginal likelihood as a byproduct, together with a numerical standard error (NSE) of approximation just as it does for posterior moments.…”

Bayesian Wavelet-Based Methods for the Detection of Multiple Changes of the Long Memory Parameter

“…Our inferential procedure, however, could be generalized to include inference on other model parameters, such as the noise variance and the process mean, either by implementing a full Gibbs or by using a Rao-Blackwellization procedure. This would also make the investigation of prediction performances of the estimated models possible, for example, by approximating the predictive distribution via Monte Carlo integration using the output from the MCMC sampler; see, for example, Pai and Ravishanker [21], [22]. This is currently under investigation.…”

A review of predictive uncertainty estimation with machine learning

“…Autoregressive fractionally integrated moving average (ARFIMA) models: Bayesian modelling of ARFIMA models using non-informative priors has been done by Pai and Ravishanker (1996), while Durham et al (2019) propose solutions for large time series.…”