Discussion of the paper 'Bayesian Time Series' by Aguilar, Huerta, Prado and West

We congratulate the authors with this important work on capturing latent structures in time series. The paper and it's predecessors are prominent contributions in the exploration of the applicability of dynamic models. The dynamic approach to time series, longitudinal and spatio-temporal problems is in our opinion the natural one. It provides an appealing conceptualisation of the application in question and it may, due to the simplicity of the Kalman filter, have some computational advantages over e.g. "brute force" MCMC. Methods for obtaining estimates of hyper parameters and also methods for model diag-nostics lack attention, though. The adaptive nature of the Kalman filter ensures reasonable short-term forecasts, even under a slightly misspecified model. However, since the analysis essentially is interpreting which part of the random variation in the time series originates from the evolution noise and which part from the observational noise, the long-term predictions may be incorrect and could lead to erroneous conclusions under a misspecified model. In a situation in which a "learning data set" is available or in a prospective analysis, we suggest yet another application of the Kalman filter to separate the two variance components (the evolution noise and the observational noise) and to obtain estimates of these variances. By treating the variance estimation problem as a missing data problem, the latent process being the missing data, the EM-algorithm may be applied. Note that the "full data" likelihood factorises into three components corresponding to the observation equation, the system equation, and the initial information. Whether the variances Î Ø and Ï Ø are parametrized or left completely unspecified, the estimation of parameters based on the EM-algorithm utilises the conditional distribution of the latent process Ø given all data Ý ½ Ý Ò. The latter distribution is provided recursively by the Kalman smoother, and the E-step is hence straight forward. See e.g. Dethlefsen et al. (1997). Also, residuals for model diagnostics may be provided by use of the Kalman filter and smoother. We suggest the use of marginal residuals, ÓÓ××ÖÚ ÓÖ ÔÖ BLOCKINØØØ ÚÐÙÙ ÑÑÖÒÒÐ ÜÔÔ BLOCKINØØØØÓÒ ×ØØÒÒÒÖ ÖÖÓÖ for checking the structure of the observation regression (the vector Ø) and the evolution structure (the matrix Ø). By observed or predicted value we mean the observation Ý Ø , the posterior mean of Ø given current information (filtering), or the posterior mean of Ø given all information (smoothing). For checking distributional form and correlation structure over time (and …