Posterior mean and variance approximation for regression and time series problems

Abstract

This paper develops a methodology for approximating the posterior first two moments of the posterior distribution in Bayesian inference. Partially specified probability models, which are defined only by specifying means and variances, are constructed based upon second-order conditional independence, in order to facilitate posterior updating and prediction of required distributional quantities. Such models are formulated particularly for multivariate regression and time series analysis with unknown observational variancecovariance components. The similarities and differences of these models with the Bayes linear approach are established. Several subclasses of important models, including regression and time series models with errors following multivariate t, inverted multivariate t and Wishart distributions, are discussed in detail. Two numerical examples consisting of simulated data and of US investment and change in inventory data illustrate the proposed methodology.

Bayesian dynamic modelling and tracking control for flexible manoeuvring systems

“…Being inherently Bayesian, this approach has the advantage of providing the conditional distribution of parameters of the model as well as the predictive distribution of the response. This has similarities to the celebrated Kalman filter, but it extends it in order to incorporate learning from the data of hyperparameters, such as the variance of the innovations [4]. The experimental results provide a solid back-up to the proposed estimation method.…”

“…All these graphs illustrate the success of the parametric model, the validation of which is confirmed. More validation tests in a Bayesian setting including model comparison and model judgement can be found in Triantafyllopoulos and Harrison (2008). The power spectral density (PSD) profile of both the actual and predicted outputs are depicted in Fig.…”

Dynamic Non-parametric Monitoring of Air-Pollution

“…The components of the vector y t are likely to have the same distribution, albeit not a prespecified distribution such as the multivariate Gaussian distribution. Our study benefits by relaxing the distribution assumption and allowing a wider class of distributions, such as approximate Gaussian and Student t distributions; for a related discussion the reader is referred to Triantafyllopoulos and Harrison (2008).…”

Inference of Dynamic Generalized Linear Models: On‐Line Computation and Appraisal

“…One wonders why not the Bayes linear methodology be applied directly to the non‐linear model , as e.g. in Goldstein & Wooff (2007) and Triantafyllopoulos & Harrison (2008).…”