Bayesian inference in nonparametric dynamic state-space models

Abstract

We introduce state-space models where the functionals of the observational and evolutionary equations are unknown, and treated as random functions evolving with time. Thus, our model is nonparametric and generalizes the traditional parametric state-space models. This random function approach also frees us from the restrictive assumption that the functional forms, although time-dependent, are of fixed forms. The traditional approach of assuming known, parametric functional forms is questionable, particularly in state-space models, since the validation of the assumptions require data on both the observed time series and the latent states; however, data on the latter are not available in state-space models.

We specify Gaussian processes as priors of the random functions and exploit the “look-up table approach” of Bhattacharya (2007) to efficiently handle the dynamic structure of the model. We consider both univariate and multivariate situations, using the Markov chain Monte Carlo (MCMC) approach for studying the posterior distributions of interest. We illustrate our methods with simulated data sets, in both univariate and multivariate situations. Moreover, using our Gaussian process approach we analyze a real data set, which has also been analyzed by Shumway & Stoffer (1982) and Carlin, Polson & Stoffer (1992) using the linearity assumption. Interestingly, our analyses indicate that towards the end of the time series, the linearity assumption is perhaps questionable.