A Non‐gaussian Family of State‐space Models With Exact Marginal Likelihood

Abstract

The Gaussian assumption generally employed in many state-space models is usually not satisfied for real time series. Thus, in this work, a broad family of non-Gaussian models is defined by integrating and expanding previous work in the literature. The expansion is obtained at two levels: at the observational level, it allows for many distributions not previously considered, and at the latent state level, it involves an expanded specification for the system evolution. The class retains analytical availability of the marginal likelihood function, uncommon outside Gaussianity. This expansion considerably increases the applicability of the models and solves many previously existing problems such as long-term prediction, missing values and irregular temporal spacing. Inference about the state components can be performed because of the introduction of a new and exact smoothing procedure, in addition to filtered distributions. Inference for the hyperparameters is presented from the classical and Bayesian perspectives. The results seem to indicate competitive results of the models when compared with other non-Gaussian state-space models available. The methodology is applied to Gaussian and non-Gaussian dynamic linear models with timevarying means and variances and provides a computationally simple solution to inference in these models. The methodology is illustrated in a number of examples. . The books by Durbin and Koopman (2001) and Fahrmeir and Tutz (2001) also present and discuss these models in addition to providing alternatives for analyzing non-Gaussian time series. The problem with this class of models is that the analytical form is easily lost, even using simple model components. Thus, the predictive distribution, which is essential for the inference process, can only be obtained approximately.Many researchers have worked in the last decades in auxiliary procedures in order to make inference for non-Gaussian SSM under the Bayesian approach. Gamerman (1998) proposed a GLM-based Markov chain Monte Carlo (MCMC) algorithm for inference, Frühwirth-Schnatter and Wagner (2006) used auxiliary mixture sampling and Pitt and Walker (2005) utilized auxiliary variables for constructing stationary time series. Andrieu and Doucet (2002) and Carvalho et al. (2010) are just a sample of a large community devoted to inference via sequential Monte Carlo or particle filtering in the SSM context. However, it is important to emphasize that these methods are approximate and the computing time can be large.The objective of this paper is to present a family of models that allows for exact, analytical computation of the marginal likelihood and thus the predictive distribution. This family is obtained by generalizations of results from Smith and Miller (1986). They considered an exponential observational model and an exact evolution equation that enables the analytical integration of the state and the attainment of the one-step-ahead predictive distributions. Harvey and Fernandes (1989) and Shephard (1994) showed that the same tractability...

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A simple model for learning in volatile environments

“…To build a generative model, we need to make some assumptions about the dynamics of this variable. Our approach here is essentially the same as that taken by Smith and Miller [25] and by Gamerman et al [26] (see also West et al [27]). Consider a problem in which the process variance dynamically changes.…”